Communications In Mathematical Physics - Volume 275

Commun. Math. Phys. 275, 1–36 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0293-4

Communications in

Mathematical Physics

Exponential Times in the One-Dimensional Gross–Pitaevskii Equation with Multiple Well Potential
Dario Bambusi1 , Andrea Sacchetti2 1 Dipartimento di Matematica, Universitá degli studi di Milano, Via Saldini 50, Milano 20133, Italy.

E-mail: [email protected]

2 Dipartimento di Matematica Pura ed Applicata, Universitá degli studi di Modena e Reggio Emilia,

Via Campi 213/B, Modena 41100, Italy. E-mail: [email protected] Received: 1 August 2006 / Accepted: 15 February 2007 Published online: 20 July 2007 – © Springer-Verlag 2007

Abstract: We consider the Gross-Pitaevskii equation in 1 space dimension with a N -well trapping potential. We prove, in the semiclassical limit, that the finite dimensional eigenspace associated to the lowest N eigenvalues of the linear operator is slightly deformed by the nonlinear term into an almost invariant manifold M. Precisely, one has that solutions starting on M, or close to it, will remain close to M for times exponentially long with the inverse of the size of the nonlinearity. As heuristically expected the effective equation on M is a perturbation of a discrete nonlinear Schrödinger equation. We deduce that when the size of the nonlinearity is large enough then tunneling among the wells essentially disappears: that is for almost all solutions starting close to M their restriction to each of the wells has norm approximatively constant over the considered time scale. In the particular case of a double well potential we give a more precise result showing persistence or destruction of the beating motions over exponentially long times. The proof is based on canonical perturbation theory; surprisingly enough, due to the Gauge invariance of the system, no non-resonance condition is required.

1. Introduction In this paper we study the dynamics of low energy states of the one-dimensional GrossPitaevskii equation (hereafter also called nonlinear Schrödinger equation, NLS)


 2σ t i
ψ˙ t = H0 ψ t +  ψ t  ψ t , ψ˙ t = ∂ψ ∂t ,  ψ t (x)t=0 = ψ 0 (x) ∈ L 2 (R), ψ 0  L 2 = 1,

(1)

 This work is partially supported by the INdAM project Mathematical modeling and numerical analysis of quantum systems with applications to nanosciences. DB was also supported by MIUR under the project COFIN2005 Sistemi dinamici nonlineari ed applicazioni fisiche. AS was also supported by MIUR under the project COFIN2005 Sistemi dinamici classici, quantistici e stocastici.

2

D. Bambusi, A. Sacchetti

Fig. 1. Plot of a trapping potential with N wells

where σ = 1, 2, . . . , is a positive integer number and H0 = −
2

d2 + V, x ∈ R dx2

(2)

is the linear Hamiltonian operator and V (x) a N –well potential. By this we mean that V has N nondegenerate distinct minima x1 , ..., x N , where the potential has essentially the same behavior (e.g. one can assume that its first r derivatives are equal at all the minima, for some positive integer r ≥ 4). We also assume that the potential is trapping, i.e. V tends to infinity as |x| → ∞ (see Fig. 1). Equation (1) with a multiple well potential describes particular phenomena associated with wave propagation in nonlinear multiple quantum well waveguides consisting of N unit cells where each of them is formed by linear films sandwiched between two nonlinear ones [26, 28]. Another situation we have in mind is that of a weakly interacting Bose Einstein condensate trapped in N cells of an optical array created by two counter propagating laser beams (see, e.g. [1], where the condensate was trapped in N ∼ 30 wells and where each well contained approximatively 1000 condensate atoms; see also [11, 27]). In such a case the parameter  can be thought of as a measure of the number of particles in the 2 condensate, that is  = N pσ 4πm
a , where m is the atomic mass (in Eq. (1), for the sake of definiteness, we choose the units such that 2m = 1), N p is the total number of particles and a is the wave scattering length. Consider first the linearized problem. The fundamental state of H0 is approximatively degenerate in the sense that, denoting by λ1 < λ2 < · · · < λ N < ...

Normal Forms and NLS

3

the eigenvalues of H0 , one has λ N − λ1  λ N +1 − λ N in the semiclassical limit, i.e.
 1. Then the most interesting situation occurs when the normalized eigenfunctions ϕ1 , ..., ϕ N , corresponding to λ1 , ..., λ N , are delocalized among the wells. Indeed, in such a case a solution in 0 :=span(ϕ1 , ..., ϕ N ) performs a quasiperiodic motion and the probability of finding the particle in any fixed well undergoes great changes over a time scale of the order of T = π
/ω with ω = (λ N − λ1 )/2. In the case of double well potential such a phenomenon is usually known as beating motion and the beating period is given by T := π
/ω, with ω = (λ2 − λ1 )/2. The main question is the behavior of the system when the nonlinearity is restored. In the case of a double well potential the problem was tackled in a series of papers [2, 8, 12, 13, 20, 23, 24, 29, 30, 32]; in particular, it was shown that, up to times of order T , the dynamics is well described by an Hamiltonian integrable system with two degrees of freedom obtained by restricting the Hamiltonian, i.e. the energy of the system, to 0 . In particular, this result has been used in order to show that the beating motion is generic for values of the nonlinearity strength  below a certain threshold value, while new localized states appear for larger nonlinearity strength (i.e. as the number of particles of a Bose-Einstein condensate increase) and for even larger values of the nonlinearity strength the beating motion disappears. In the case of a multiple well potential the situation was studied e.g. in the paper [27] where the authors deduced (formally) the discrete nonlinear Schrödinger as an effective equation for the dynamics in 0 and used it in order to study some of the features of the model. We are not aware of rigorous results in the case of a multiple well potential. In the present paper we study the nonlinear dynamics using the methods of Hamiltonian perturbation theory for PDEs [4–6, 15]. This is used in order to greatly improve the times over which the dynamics is described. Indeed, in the above quoted paper the dynamics close to 0 is described over a time scale of order T , i.e. the inverse of the perturbative parameter of our problem. Here we obtain a description of the dynamics valid over a much longer time scale, i.e. over a time scale exponentially long with T . The difference is analogous to the difference existing in classical Hamiltonian systems between the averaging theory of the 19th century and the Nekhoroshev theory of the ’70s. In this connection it is worth to remark that typically the development of Nekhoroshev’s theory requires the control of resonance relations among the frequencies, while here we obtain a theory valid for any potential, i.e. without any assumption on the frequencies of the system. We think that this is the most interesting aspect of the paper (see below for further comments on this point). We come now to a more precise description of our main result (Theorem 2). It consists in proving that the manifold 0 , which is invariant for the linear dynamics, is only slightly deformed by the nonlinearity into a new manifold M which is approximatively invariant for the complete dynamics. By this statement we mean that solutions starting on M, or close to it, will remain close to M for times which are exponentially long with T (Remark 4). Moreover we show (see Lemma 1) that the dynamics on M is described by a system which is a perturbation of a discrete nonlinear Schrödinger equation  2σ i ψ˙ j = δ j ψ j + j ψ j+1 + j−1 ψ j−1 + ηψ j ψ j  , j = 1, ..., N , (3) where ψ0 = ψ N +1 := 0, j , δ j , η are suitable constants and where, having in mind the case of Bose-Einstein condensates, |ψ j |2 represents the fraction of particles in the j th

4

D. Bambusi, A. Sacchetti

well for j = 1, ..., N (see Sect. 2.4 for their precise definition). In particular, it is quite easy to study the discrete NLS (3) from the anticontinuum limit [17] obtaining that when η is large enough then solutions corresponding to almost all initial data have the property that |ψ j |2 is essentially a constant of motion. This allows to show (see Corollary 4) that a similar property is enjoyed by the solutions of the original NLS, but only over an exponentially long time scale, a time scale, that, using the words of Littlewood “while not eternity, this is a considerable slice of it” [16]. In our treatment the number N of wells does not play a particular role; in fact, for the sake of definiteness, we’ll give the detailed proof of our main result (Theorem 2) in the case N = 2 since the proof for a generic number of wells N follows in the same way. Furthermore, in the particular case of the double well potential, we get that the dynamics on M is (up to an exponentially small error) that of an integrable Hamiltonian system with two degrees of freedom. This allows us to control also the trajectories of the solutions on M showing that for small η the beating phenomenon persists for exponentially long times, while for higher values of η only motions which are essentially localized in one of the two wells exist, at least for exponentially long times. From the technical point of view the main result (Theorem 2) is quite surprising since, as we recalled at the beginning of the introduction, the application of canonical perturbation theory is typically possible only when some non-resonance conditions are satisfied. On the contrary, here the result is valid for any multiple well potential, whose eigenvalues might fulfill arbitrary resonance conditions (i.e. the eigenvalues can be linearly dependent over the relative integers). This is possible since NLS is an infinite dimensional Gauge invariant Hamiltonian system. To explain how this property is exploited we recall that canonical perturbation theory allows us to remove from the Hamiltonian all non-resonant monomials. In particular, given an arbitrary monomial it can be eliminated if it is non-resonant. However, in NLS only Gauge invariant monomials appear, and we will show that, for a Gauge invariant monomial, the non-resonance condition is (almost) trivially fulfilled. Actually, in order to avoid any restriction on the potential we have to use a resonant construction which is not self evident a priori (see Subsect. 3.2). We think that it should also be possible to prove a KAM-like theorem showing that for most values of the parameters the system admits an exact invariant manifold which corresponds to the approximatively invariant 0 , but, in order to obtain this result, it is not obvious how to deal with the small denominators appearing in KAM theory. Our approach is based on the analyticity of the Hamiltonian, a property which holds only when σ is an integer. We think that these ideas could be useful in the study of Hamiltonian systems with symmetry, and possibly also for the investigation of further dynamical properties of NLS. A further technical ingredient which is fundamental for the proof is the use of Sobolevlike spaces constructed as the domains of the powers of H0 . To use such spaces one has to show that they form Banach algebras under the pointwise multiplication. Here we give a detailed proof of such a property that we think could be useful in further investigations of NLS. The paper is organized as follows. In Sect. 2 we state our main results (Theorem 2, Theorem 3 and their corollaries). Section 2 is divided into 5 subsections: in Subsect. 2.1 we review some known facts about the structure of the low-lying eigenvalues of the linear Schrödinger equation with an N well potential; in Subsect. 2.2 we introduce the Sobolev like spaces in which NLS equation will be studied and give their main properties; in Subsect. 2.3 we will state the result on the approximate invariant manifold;

Normal Forms and NLS

5

in Subsect. 2.4 we will give the effective equation on the approximatively invariant manifold and deduce the localization properties of the solutions; finally in Subsect. 2.5 we will study the particular case of a double well potential. In Sect. 3 we prove our main results. This section is also divided into six subsections that correspond to the different parts of the proof. The proof of Theorem 1 (algebra property of Sobolev like spaces) and of some technical lemmas are left to Appendixes A and B respectively. 2. Main Results 2.1. Linear theory. Hypothesis 1. The potential V (x) ∈ C ∞ (R) is a real valued function such that: i. V (x) admits N minima at x1 < x2 < ... < x N such that V (x) > Vmin = V (x j ) = 1, ∀x ∈ R, x = x j , j = 1, ..., N ; ii. There exists a constant C > 1 such that C −1 x2 ≤ V (x), x =



(4)

1 + x 2;

iii. There exists a positive m ≥ 2 such that for any k ∈ N,  k   d V (x)  m−k    d x k  ≤ Ck x for some positive constant Ck ; iv. The minima are nondegenerate and d 2 V (x j ) =C >0 dx2

(5)

with C independent of j; v. The shape of the potential at the bottom of the minimum x j is approximatively independent of j; precisely: there exists r ≥ 4 such that dk V dk V (x ) = (xi ), ∀i, j = 1, 2, . . . , N , ∀k = 2, · · · , r. j dxk dxk

(6)

Hereafter, C will always denote a positive, typically large, constant whose value changes from line to line, and which is independent of
,  and t. The operator H0 formally defined by (2) admits a self-adjoint realization (still denoted by H0 ) on L 2 (R) (Theorem III.1.1 in [10]) with purely discrete spectrum. Let λk , k ∈ N, be the non degenerate eigenvalues of H0 , k↑∞

λ1 < λ2 < λ3 < λ4 < · · · < λk < · · · , λk → ∞, with associated normalized (in L 2 ) eigenvectors ϕk (x); the set {ϕk (x)}∞ k=1 is an ortho2 normal base of L . The lowest part of the spectrum can be studied in the semiclassical limit using the construction of [14], that we shortly recall.

6

D. Bambusi, A. Sacchetti

Having fixed a positive constant a > 1 = Vmin we consider the set V −1 ((−∞, a)) and we assume that a is such that this set is the union of N disjoint open sets U j with x j ∈ U j . Having fixed j we consider the operator formally defined on L 2 as H j = −
2

d2 + Vj, dx2

where V j is a modified potential defined as  V (x) V j (x) = max[a, V (x)]

for x ∈ U j . for x ∈ U j

(7)

Let λˆ j be the lowest eigenvalue of H j with associated normalized eigenvector ϕˆ j . Using the semiclassical construction of the eigenvalues close to the bottom of a well (see e.g. [7, 25]) one has that assumption Hyp. 1, v., implies that λˆ j = 1 + O(
), |λˆ j − λˆ i | ≤ C
r/2 and ϕˆ j (x) − ϕˆi (x + xi − x j ) L 2 ≤ C
r/2 , for any i, j = 1, 2, . . . , N . Let  x j+1  j = V (x) − 1 d x, j = 1, 2, . . . , N − 1, xj

be the Agmon distance among the two minima x j and x j+1 . Let  be any fixed positive real number such that  < min j  j . Then, from the theory of [14] there exist some constants ci j such that for any i and j,   n      ck j ϕˆk  ≤ Ce−/
, (8) ϕ j −   2 k=1 L   ϕˆ j ϕˆi  ∞ ≤ Ce−/
, i = j, (9) L |λi j − λˆ j | ≤ Ce−/


for some i j ∈ {1, ..., N } .

(10)

ˆ on ˆ 0 :=span(ϕˆ j ) is a bijection among 0 :=span(ϕ j ) and Moreover, the projector  ˆ 0 itself. Remark also that one can choose the functions ϕˆ j to be real valued. Hence, the lowest N eigenvalues of H0 fulfill C −1
< λ j − 1 < C
,

j = 1, ..., N ,

for some C > 1 and ω=

1 (λ N − λ1 ) ≤ C
r/2 . 2

(11)

Furthermore, making use of the same arguments, it follows also that inf

inf

[λ − λ j ] ≥ C −1
.

j=1,...,N λ∈σ (H0 )−{λ1 ,...,λ N }

(12)

Normal Forms and NLS

7

We also use the following projectors: =

N 

ϕ j , ·ϕ j and c = I −  and 0 = L 2 .

j=1

Remark 1. As already emphasized in the introduction the most interesting situation occurs when the true eigenfunctions ϕ j are delocalized between the wells, a property that occurs if the potential is exactly periodic in some region or more generally if the order of magnitude of some of the off diagonal elements of the matrix formed by the constants ci j is of the same order of magnitude as the difference between the approximate eigenvalues λˆ j , a property that is quite difficult to ensure in a general situation. For this reason, essentially in order to fix ideas, we decided to state our assumptions in the form (i–v) above. 2.2. The nonlinear system: Analytic framework and well posedness. s/2

Definition 1. For any integer s ≥ 0 define the Hilbert space X s := D(H0 ) endowed by the graph norm; more precisely in X s we will use the following norm equivalent to the graph norm:      s/2 2 φ2s :=  H0 φ  2 ≡ H0s φ, φ L 2 = φ¯ H0s φdx, φ ∈ X s . (13) R

L

The main step for the proof that the spaces X s form a Banach algebra under the pointwise multiplication is the following theorem. Theorem 1. Let s be any positive integer number. For small enough
the two norms φ2s and (−
2 )s/2 φ2L 2 + V s/2 φ2L 2

(14)

are equivalent with an
independent constant. The proof, which is a semiclassical variant of the proof of Lemma 7.2 of [31], is deferred to Appendix A. Remark 2. In particular one has that a function φ is in X s if and only if it belongs to the Sobolev space H s and it decays at infinity so fast that |φ|2 V s is integrable. In the spaces X s with s ≥ 1 the system (1) is semilinear, since, using (14) and Gagliardo-Nirenberg inequality one has Corollary 1. For any integer s ≥ 1 there exists a positive constant Cs independent of
, such that φ1 φ2 s ≤ Cs
−1/2 φ1 s φ2 s , ∀φ1 , φ2 ∈ X s ,
 1.

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Moreover, the map ¯ → |φ|2σ φ ∈ X s X s × X s  (φ, φ) is an entire analytic map for any
> 0 and fulfills    2σ  +1 . |φ| φ  ≤ Cs
−σ φ2σ s s

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8

D. Bambusi, A. Sacchetti

Remark 3. In dimension d > 1 this result remains valid provided s > d/2. Then by standard Segal theory (see e.g. [19]) the system (1) is locally well posed in all the spaces X s with s ≥ 1. Actually s > d/2 is enough, but our proof only applies to integer values of s; in [31] a Strichartz inequality argument was used to show that it is also locally well posed (LWP) in X s with some s smaller than d/2. From now on we assume that the index s of the space is a fixed positive integer number and fulfills the condition s ≥ 1. In the following, in order to fix ideas, one can just think of the case s = 1. We will denote by d(.; .) the distance in the norm of X s . 2.3. The nonlinear system: Approximatively invariant manifold. In order to state our main result we assume that the size of the nonlinearity is small enough, i.e. || 
σ , and we introduce the small parameter. µ := ω +

||  1,
σ

(17)

where ω was defined in (11). Theorem 2. Consider the system (1) and fix a positive s ≥ 1. There exists a positive µ∗ such that, if µ < µ∗
3/2 , then there exists a manifold M (dependent on all the parameters of the system) with the following properties: i. M is close to 0 , i.e. d( 0 , M) ≤ C

µ ,
3/2

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where d( 0 , M) = sup inf ψ − ϕs , ψ∈ 0 ϕ∈M

and where ϕs is the norm (13). ii. Let d0 = d(ψ 0 , M) = inf ψ 0 − ϕs ϕ∈M

be the initial distance from M and let 

 µ∗
3/2 . δ = max d0 , exp − 2µ

(19)

Then for all times t fulfilling |t| ≤

1 C
µδ

(20)

one has d(ψ t , M) ≤ Cδ,    c ψ t  ≤ C µ . s
3/2

(21) (22)

Normal Forms and NLS

9

Such a manifold M is called an approximatively invariant manifold. Remark 4. The most interesting cases are when

 µ∗
3/2 µ δ = exp − or δ = C 3/2 . 2µ
Indeed in the first case the time of validity of all the estimates is exponentially long, while in the second case it is easy to obtain the following corollary.   Corollary 2. Assume that c ψ 0 s ≤ Cµ
3/2 then, up to the times |t| ≤


3/2 Cµ2

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the estimate (22) holds. Remark 5. This corollary is a direct extension of the results of [13, 23, 24] in which the estimate (22) has been proved (for the double well potential) for a time scale of order T =

π

≤ C  µ−2
3/2 . ω µ

The improvement is due to the fact that our construction implies that M is linearly stable up to an exponentially small error.

2.4. The nonlinear system: discrete NLS and suppression of tunneling. To start with we remark that NLS is a Hamiltonian system (see Subsect. 3.1 for a precise description) with Hamiltonian function given by ¯ + P0 (ψ, ψ), ¯ ¯ := E0 (ψ, ψ) E(ψ, ψ) where ¯ := E0 (ψ, ψ) and ¯ := P0 (ψ, ψ)

 R

(24)

¯ ψ(x)H 0 ψ(x)d x

 1 1 ψ σ +1 2L 2 := ψ¯ σ +1 ψ σ +1 d x. σ +1 σ +1 R

The general idea is that for initial data close to 0 the system should be well described ˆ 0 which is close to 0 . Denote by the Hamiltonian system obtained by restricting E to ψ=

N  j=1

then we have the following

ˆ0 ψ j ϕˆ j ∈

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D. Bambusi, A. Sacchetti

ˆ 0 takes the form Lemma 1. The restriction of (24) to E| ˆ 0 =

N N N   

 ( + ν j )|ψ j |2 + c |ψ j |2σ +2 + c j ψ¯ j ψ j−1 + ψ j ψ¯ j−1 j=1

+ O(e

j=1 −2/


) + O(e

−/


j=2

),

where  was introduced before Eq. (8) and  N 1 = λ j , ν j := ϕ¯ˆ j H0 ϕˆ j d x −  = O(
r/2 ) n R j=1 and c = c(
) :=

N   1 1 +2 −σ/2 ϕˆ j 2σ 2σ +2 = O
L σ +1n j=1

and  c j :=

R

ϕ¯ˆ j H0 ϕˆ j−1 d x = O(e−/
).

Proof. Indeed, we have that E| ˆ 0 = E0 | ˆ 0 +  P0 | ˆ 0 . The first term takes the form E0 | ˆ 0 =

N 

|ψ j |2 d j j +



ψ¯ i ψ j di j +

|i− j|=1

j=1



ψ¯ i ψ j di j

|i− j|≥1

with  di j =

R

ϕ¯ˆi H0 ϕˆ j d x,

and where (see, e.g., [14, 24]) di, j = O(e−/
) when |i − j| = 1 and di, j = O(e−2/
) when |i − j| > 1. For what concerns the second term, following [24] and making use of (9), we have that 1  +2 ϕˆ j 2σ |ψ j |2σ +2 + O(e−/
), L 2σ +2 σ +1 N

P0 | ˆ 0 =

j=1

  +2 = O
−σ/2 is approximatively independent of j. Remark also that, where ϕˆ j 2σ L 2σ +2 since the functions ϕˆ j are real valued, the quantities c j turn out to be real.  

Normal Forms and NLS

11

Hence, up to higher order terms, to the Gauge transformation ψ → eit/
ψ and to ˆ 0 is given by a rescaling of time t → ωt/
the restriction of (24) to K0 :=

N  j=1

where

δ j |ψ j |2 + η

N 

|ψ j |2σ +2 +

j=1

N +1 

j (ψ¯ j ψ j−1 + ψ j ψ¯ j−1 ),

(25)

j=1

    νj cj e−/
|| c , δ j := =O = O(1), j := =O η= , ω ω
σ/2 ω ω ω

and the equation of motion of (25) is given by ψ˙ j = −i

∂K0 , with j = 1, ..., N . ∂ ψ¯ j

The system (25) has an integral of motion (the restriction of the square of the L 2 norm ˆ 0 ) given by to I :=

N 

|ψ j |2 .

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j=1

We analyze now the consequences of our theorem for the dynamics. From the proof of Theorem 2 we will be able to exactly describe the restriction of the system to M. Actually, we can state the following Theorem 3, which is a result of Theorem 6 stated below. Theorem 3. Under the same assumptions of Theorem 2 there exists an analytic canonical transformation T : U → X s , with U an open neighborhood of 0 with size independent of µ and
, such that ˜ E ◦ T = I + ωK + R, where ˆ = K, i.e. K depends only on the variables ψ1 , ..., ψ N . K◦ {I, K} ≡ 0, i.e. I is an integral of motion for the system with Hamiltonian K. 3/2 ). K = K0+ O(µ/
 3/2    ˜ = O exp − µ∗
R + O c ψ2s and similar estimates hold for its vector 2µ field. v. T = I + O(µ/
3/2 ) i.e. the transformation is close to identity.

i. ii. iii. iv.

Remark 6. In this framework the manifold M turns out to simply be M = T ( 0 ). In particular it follows that the dynamics on M is, up to a small error, the same of a Hamiltonian system with Hamiltonian function close to K0 with an integral of motion given by I. Thus, it is possible to deduce the following corollary of Theorem 7 below, which is particularly relevant in the double well case.

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D. Bambusi, A. Sacchetti

Corollary 3. Under the same assumptions of Theorem 2 one also has µ µ |I(t) − I(0)| ≤ C 3/2 , |K0 (t) − K0 (0)| ≤ C 3/2

up to the times (20).

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We focus now on the exponentially long time scale, thus we assume that the quantity δ of Theorem 2 is given by 

µ∗
3/2 , δ = exp − 2µ µ∗
3/2

˜ fulfills and from (20) it follows that up to times of order e 2µ the vector field X R˜ of R the a priori estimate

3/2     X ˜  ≤ C exp − µ∗
. R s µ Thus, up to an exponentially small drift, a Gauge transformation and a rescaling of time the dynamics on M ≡ T ( 0 ) is that of the N –dimensional Hamiltonian system K, which is a perturbation of K0 , i.e. of the discrete NLS (25). It is worth mentioning that when η = 0 the dynamics of K0 is that of N decoupled harmonic oscillators corresponding to the normal modes of the linearized system. If j  δi , for any i and j, then the normal modes are localized, i.e. each normal mode essentially involves only one of the ψ j ; on the contrary, in the much more interesting case where the δ j are of the same order of magnitude of the j , typically the normal modes are collective motions of the system and, correspondingly, in typical solutions the term |ψ j (t)|2 undergoes great changes for each j. In the opposite limit η → ∞ (anticontinuum limit, see [17]), K0 becomes a system of decoupled anharmonic oscillators. Correspondingly |ψ j (t)|2 is a constant of motion. One can use KAM or Nekhoroshev theory in order to study the dynamics of K when 1  η < ∞ and to deduce results on the dynamics of the complete NLS equation. Here we will state a result that can be obtained in this way. To this end, for any ρ > 0 fixed, we will denote ⎧ ⎫ N ⎨ ⎬  SρN := (ψ1 , ..., ψ N ) ∈ C N : |ψ j |2 = 1 and |ψ j | > ρ , ∀ j , ⎩ ⎭ j=1

  and we will denote by S N  its Lebesgue measure. ρ

Theorem 4. (KAM theorem K) Consider the Hamiltonian system K, then there exists a constant η , such that, for any |η| > η there exists a set Sη ⊂ SρN with Lebesgue measure estimated by    N  Sη  ≥ S  − Cη−1/2 (28) ρ with the property that, if the initial datum is in Sη then the solution of the Hamiltonian system with Hamiltonian function K is quasiperiodic and fulfills  2  2   (29)  ψ j (t) − ψ j (0)  < Cη−1/2 for any t.

Normal Forms and NLS

13

To state a corresponding result for the NLS equation, consider the set L of the ψ ∈ Xs having L 2 norm equal to 1 and fulfilling 

µ∗
3/2 . d(ψ, M) ≤ C exp − 2µ For ψ ∈ L, denote

N

˜

j=1 ψ j ϕˆ j

ˆ and = ψ

! S˜ρN := ψ ∈ L : |ψ˜ j | > ρ , ∀ j .

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Corollary 4. Consider the NLS equation (1), under the same assumptions of Theorem 2 assume also η large enough; then there exists a set S˜η ⊂ S˜ρN whose “measure” is estimated by      ˆ ˜   ˆ ˜N  (31) Sη  ≥ Sρ  − Cη−1/2 such that, if ψ 0 ∈ S˜η then along the corresponding solution one has  2  2   ψ˜ j (t) − ψ˜ j (0)  < Cη−1/2  

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for all the times t fulfilling (20). Remark 7. All the constants in the above Theorem 4 depend on N , in [3, 9] one can find some N independent statements.

2.5. The double well potential. In the particular case of a double well potential one can get much more precise results, both for the linear and for the nonlinear system. By a double well potential we will mean here a potential V (x) ∈ C ∞ (R) fulfilling assumptions (i-v) of Sect. 2.1 (with N = 2) and which moreover is symmetric with respect to spacial reflection V (−x) = V (x). It is well known that the splitting ω between the two lowest eigenvalues fulfills the asymptotic estimate ω=

1 (λ2 − λ1 ) ≤ Ce−/
, 2

(33)

for any  < 1 where 1 is the Agmon distance between the two wells and C is a positive constant (depending on ), and moreover the normalized eigenvectors ϕ1,2 associated to λ1,2 can be chosen to be real-valued functions such that ϕ1 and ϕ2 are respectively of even and odd-parity, thus, defining the single well states 1 1 ϕ R = √ [ϕ1 + ϕ2 ] and ϕ L = √ [ϕ1 − ϕ2 ] , 2 2 they essentially coincide with the functions ϕˆ j used in the multiple well case. In particular they fulfill ϕ R ϕ L  L ∞ ≤ Ce−/
,  > 0.

(34)

14

D. Bambusi, A. Sacchetti

Thus, for ψ ∈ 0 , in this section we will write ψ = ψ1 ϕ L + ψ2 ϕ R .

(35)

Here the tunneling gives rise to the so-called phenomenon of beating: for almost any initial datum ψ 0 ∈ 0 the expectation value of the position   2 t

x = x ψ t (x) d x (36) R

periodically oscillates between positive and negative values with a period given by T := π
/ω. In this case the restricted approximate Hamiltonian K0 takes the form [24]   (37) K0 := ψ¯ 2 ψ1 + ψ2 ψ¯ 1 + η |ψ1 |2σ +2 + |ψ2 |2σ +2 with the same definition of η as in the previous subsections. Then Theorems 2 and 3 hold together with their corollaries. The main improvement that one can obtain in the double well case is due to the fact that since the system K is now a system with two degrees of freedom with an integral of motion independent of the Hamiltonian (namely I), then it is integrable. This allows us to describe in a very precise way the trajectories of the system K which are just the intersection of the level surfaces of the functions K and I. This is possible since K is close to K0 . To be definite, from now on in this section, we will restrict to the case of quintic nonlinearity, i.e. σ = 2. In fact, σ = 2 in dimension 1 is the most interesting case, which corresponds to the critical nonlinearity model exhibiting bifurcation phenomenon [2, 20]. The system K0 , cf. (37), has been already studied in [13] (see also [20, 29, 30]) obtaining that, for |η| < 2 almost all solutions perform beating motions, while at η = ±2 a bifurcation occurs and new equilibria, localized close to the minima of the Hamiltonian function, appear. As η increase the domain of stability of such solutions increase its size, so that, for η large enough essentially only localized motions exist. Concerning the complete system we can state that if η is not at a bifurcation point, then non-homoclinic trajectories associated to the Hamiltonian K0 approximate the solution ψ t for times of the order (20). Corollary 5. Under the same assumptions as in Theorem 2, assume also η = ±2; consider an initial datum such that ψ1 (0), ψ2 (0) = 0, and η − 1 if η > 0, 2 η K0 (0) = 1 − if η < 0, 2

K0 (0) =

and also such that δ ≤ Cµ
3/2 ; then, there exists a positive constant µ , depending only on how much the above quantities differ from the considered values, such that, provided µ < µ
3/2 , there exists a solution of the Hamiltonian system (25) with trajectory γ such that µ (38) d(ψ t , γ ) ≤ C 3/2
for the times (23).

Normal Forms and NLS

15

Remark 8. Homoclinic trajectories are absent when |η| ≤ 2; initial conditions such that K0 (0) = η2 − 1, for η > 2, and K(0) = 1 − η2 , for η < −2 corresponds to an homoclinic trajectory. Remark 9. The topology of the trajectory γ is determined by the condition d(ψ 0 , γ ) ≤ C

µ
3/2

(39)

in the sense that all curves fulfilling this condition have the same topology if the assumptions of the corollary are fulfilled. Remark 10. Thus one has that also for the true system beats are present for |η| < 2 while their importance decreases as |η| increases above 2. In particular for large values of η only motions localized close to one well are present [30] at least for the time scales controlled by our theorems. 3. Proof of the Main Results In order to simplify the notations all the proofs will be carried out in the case of a potential with only 2 wells. In fact, in the general case of N ≥ 2 wells then the elements u and z in (56) will be defined as u = (u 1 , u 2 , . . . , u N ) and z = (z N +1 , . . .) and similarly K , L , k, l, m, n, . . . will be redefined. Also the terms H0 and  in (64) will be redefined too. However, the techniques are the same as in the case N = 2; for this reason we give a detailed proof only in the case of N = 2 wells. 3.1. Hamiltonian Formalism. First consider the real Hilbert space XRs := D((H0,R )s ), where H0,R is the operator H0 restricted to real valued functions. We make XRs ⊕ XRs a symplectic space by introducing the semiclassical symplectic form  

   α ( p, q); ( p  , q  ) :=
p (x)q(x) − p(x)q  (x) dx. R

Given a smooth real valued function H( p, q), then we define its Hamiltonian vector field X H ∈ XRs ⊕ XRs by the property α(X H , h) = dHh

(40)

for any h = (h p , h q ) ∈ XRs ⊕ XRs , where dH denotes the differential of H. It is well known that X H is in general defined only on a subset of XRs ⊕ XRs . Define also the L 2 gradient ∇ p H of H with respect to p by   ∇ p H(x)h p (x)d x = d p Hh p , ∀h p ∈ XRs , (41) ∇ p H, h p L 2 := R

and similarly we introduce the quantity ∇q H. Then X H =
−1 (−∇q H, ∇ p H),

16

D. Bambusi, A. Sacchetti

and thus the Hamilton equations of H are given by   d 1 1 p˙ = − ∇q H , q˙ = ∇ p H . ( p, q) = X H ( p, q) ⇐⇒ dt

The Poisson brackets between two functions H and K are defined as 

 1 {H, K} := −α(X H , X K ) = ∇ p H∇q K − ∇q H∇ p K dx
R

(42)

(43)

which in general is only defined on a subdomain of XRs ⊕ XRs . We shall use complex coordinates in XRs ⊕XRs identifying this space with X s , through 1 ( p, q) → ψ = √ (q + i p). 2 Therefore, we set 1 1 ∇ψ = √ (∇q − i∇ p ) and ∇ψ¯ = √ (∇q + i∇ p ) 2 2

(44)

¯ is a smooth real valued function, we have the identification so that, if H = H(ψ, ψ) i ¯ = − ∇ψ¯ H(ψ, ψ), ¯ X H (ψ, ψ)
and in complex coordinates the Poisson brackets are computed by    i {H, K} := ∇ψ H∇ψ¯ K − ∇ψ¯ H∇ψ K dx.
R

(45)

(46)

With such a notation then the NLS (1) can be written in the form of a Hamiltonian system, the corresponding Hamiltonian function being the energy eq. (24). Remark 11. Such a Hamiltonian is invariant under the action of the Gauge group ψ(x) → ψ(x)e−iβ ,

(47)

for any β independent of x. The corresponding conserved quantity is the L 2 norm  ¯ = |ψ(x)|2 dx. (48) N (ψ, ψ) R

Equivalently one has {E, N } ≡ 0.

(49)

∞ Let {λk }∞ k=1 and {ϕk (x)}k=1 be the eigenvalues and the normalized eigenvectors of H0 , let

ψ(x) =

∞  k=1

ζk ϕk (x),

(50)

Normal Forms and NLS

17

and define the Hilbert spaces 2s of the complex sequences such that ζ 2s :=



λsk |ζk |2 < ∞.

(51)

k≥1

In such a way we have defined the correspondence ψ ∈ X s ↔ U(ψ) := ζ ≡ (ζ1 , ζ2 ...ζ j , ...) ∈ 2s which is a unitary isomorphism. In particular, if E is the Hamiltonian (24), then E ◦ U −1 (still denoted by E) is the Hamiltonian of the same system written in terms of the new variables ζ . In terms of these variables the quadratic part E0 of the Hamiltonian is given by E0 =



 2 λ j ζ j  ,

(52)

j≥1

and the Poisson brackets can be written as {H, K} =

 ∞

i  ∂H ∂K ∂H ∂K . −
∂ζk ∂ ζ¯k ∂ ζ¯k ∂ζk k=1

Remark 12. From now on we will work in the space 2s and moreover, in order to simplify, we rescale time by the transformation t → t/
, thus the Poisson brackets take the form {H, K} = i

∞

 ∂H ∂K k=1

 ∂H ∂K . − ∂ζk ∂ ζ¯k ∂ ζ¯k ∂ζk

(53)

3.2. Non coupling monomial. It is useful to introduce also a different notation for the first two variables (here we recall that we are working in the double well case) and for the remaining ones, thus let us denote u 1 := ζ1 , u 2 := ζ2 , z j := ζ j ,

j ≥ 3.

(54)

Consider now a monomial of the form ζ K ζ¯ L = u k u¯ l z m z¯ n ,

(55)

where we used the notations K = (k, m), k = (k1 , k2 ), u k ≡ u k11 u k22 ,

L = (l, n), m = (m 3 , m 4 , m 5 , ....), l = (l1 , l2 ), n = (n 3 , n 4 , n 5 , ....), m z m ≡ z 3m 3 z 4m 4 . . . z q q . . . .

(56)

18

D. Bambusi, A. Sacchetti

Remark 13. A monomial of the form (55) is Gauge invariant, i.e. invariant under the transformation ψ → eiβ ψ that is ζk → eiβ ζk if and only if |K | = |L|, where |K | =



|K j |.

j

In fact (55) is Gauge invariant if, and only if, ⎛ ⎞ !  K L 0 = N (ζ, ζ¯ ), ζ ζ¯ = i ⎝ (K j − L j )⎠ ζ K ζ¯ L ,

(57)

j

where N (ζ, ζ¯ ) =



j≥1 |ζ j |

2

is the L 2 norm; that is |K | = |L| again.

Due to our assumption Hyp.1 on the potential one has Lemma 2. Let σ (H0 ) be the spectrum of the linear operator H0 and let
be small enough. There exists a sequence of (not necessarily continuous with respect to
) functions {E γ (
)}γ ∈N , 0 < E 0 < 1 and there exists a positive constant C > 1, independent of
and γ , such that: i. [E γ − C −1
, E γ + C −1
] ∩ σ (H0 ) = ∅; ii. 1 < E γ − E γ −1 < 3 for all
and all γ ≥ 1; iii. For any
fixed, we consider the sets of indexes ' & Jγ (
) = Jγ := j ∈ N : E γ −1 < λ j < E γ , then the cardinality of these sets is estimated as # Jγ (
) ≤

C .


Proof. The proof is an immediate consequence of the following result (see Theorem (V-11) in [21], see also Theorem (XIII-81) in [22]):  ! 1   N[α,β] =  (x, p) ∈ R2 : α ≤ p 2 + V (x) ≤ β  + O(
) , 2π
where N[α,β] is the number of eigenvalues of H0 contained in the interval [α, β] and where |.| denotes the Lebesgue measure of a set. Indeed, let us consider the intervals [2γ , 2γ + 1], γ ∈ N, then the number of eigenvalues of H0 belonging to these intervals is given by N (γ ) := N[2γ ,2γ +1]  C ! 1   =  (x, p) ∈ R2 : 2γ ≤ p 2 + V (x) ≤ 2γ + 1  + O(
) ≤ 2π

since V (x) ≥ C x2 , for some positive constant C independent of
and γ . From this estimate it follows that there exists at least one value E γ ∈ (2γ , 2γ + 1) and C > 1 satisfying i. Furthermore, conditions ii. and iii. immediately follow since 1 = (2γ ) − [2(γ − 1) + 1] ≤ E γ − E γ −1 ≤ (2γ + 1) − 2(γ − 1) = 3

Normal Forms and NLS

19

and # Jγ ≤ N (γ ) + N (γ − 1) ≤

2C .


  We fix a sequence with such properties. Hereafter, all the perturbative construction will involve only Gauge invariant monomials (hence |K | = |L|). Keeping this in mind we give the following Definition 2. A monomial of the form (55) will be called coupling if the indexes (K , L) fulfill the following conditions: i. |K | = |L|, ii. |m| + |n| = 1, 2, iii. if m i = 1 and n j = 1 then i ∈ Jγ and j ∈ Jγ  with γ = γ  . A monomial which is not coupling will be called non-coupling. A polynomial containing only coupling (resp. non-coupling) monomial will be called coupling (resp. non-coupling). Remark 14. For Gauge invariant monomials the condition i. is always fulfilled. Furthermore, in terms of the indexes k, l, m, n condition i. reads |k| − |l| = |n| − |m|.

(58)

Remark 15. Any Gauge invariant analytic function can be uniquely decomposed into the sum of a coupling and a noncoupling part. We recall that, as in finite dimensional spaces, an analytic function is a function whose Taylor series is convergent (see e.g. [18]). We also define a new (
dependent) norm in the space X s as follows: Definition 3. Denote N E (z, z¯ ) :=

 γ

E γs



|z j |2

(59)

j∈Jγ , j≥3

then the quantity (u, z)2E ≡ ζ 2E := |u 1 |2 + |u 2 |2 + N E (z, z¯ )

(60)

will be called the E-norm of ζ . Denote by 2E the space of sequences z = {z j } j≥3 equipped with the norm N E (z, z¯ ). By abuse of notation sometime we will also write z2E := N E (z, z¯ ). Remark 16. The E-norm is equivalent to the standard norm of X s with an
independent constant. The proof is a trivial computation and is left to the reader. Lemma 3. Let ζ K ζ¯ L be a non coupling Gauge invariant monomial of degree at most two in z, z¯ (i.e. |n| + |m| ≤ 2), then one has ! ! N E , ζ K ζ¯ L = 0 and |u 1 |2 + |u 2 |2 , ζ K ζ¯ L = 0. (61)

20

D. Bambusi, A. Sacchetti

Proof. First remark that if u k u¯ l z n z¯ m is noncoupling of degree at most two in z, z¯ then one has |k| = |l| and thus ! u 1 u¯ 1 + u 2 u¯ 2 , u k u¯ l z n z¯ m = i(|k| − |l|)u k u¯ l z n z¯ m = 0. (62) ¯ j˜ ∈ Jγ¯ such that Furthermore, either it is of degree zero in z, z¯ or there exists γ¯ and j, n j¯ = 1 and m j˜ = 1. In the first case one has that ! ! N E , ζ K ζ¯ L = N E (z, z¯ ), u k u¯ l = 0. In the second case, then

! !   N E (z, z¯ ), u k u¯ l z n z¯ m = u k u¯ l z j z¯ j , u k u¯ l z n z¯ m E γs γ

= u u¯ z z¯

k l n m

j∈Jγ

E γs¯



i(n j − m j ) = 0.

j∈Jγ¯

  Thus the E–norm is invariant under the dynamics of a noncoupling Hamiltonian of degree at most 2 in z, z¯ . In the more general case one has Corollary 6. Let Z be a non coupling polynomial. Assume that it has a smooth vector field, then there exists C such that |{N E , Z} (u, u, ¯ z, z¯ )| ≤ C [N E (z, z¯ )]3/2 .

(63)

¯ as follows: 3.3. Normal form construction. Let us rewrite E(ψ, ψ) E = H0 + P , where ¯ := (|ζ1 |2 + |ζ2 |2 ) + H0 (ψ, ψ)

 j≥3

 2 λ1 + λ2 , λ j ζ j  ,  := 2

(64)

and ¯ := P (ψ, ψ)

ω ¯ (|ζ2 |2 − |ζ1 |2 ) + P0 (ψ, ψ). 

(65)

We are going to prove that there exists a canonical transformation T which gives the Hamiltonian the form E ◦ T = H0 + Z + R,

(66)

where Z is a non coupling polynomial and R has a smooth vector which is exponentially small with  −1 . The construction will be recursive. To this end we assume one has been able to construct a canonical transformation Tr putting the Hamiltonian in the form E ◦ Tr ≡ E (r ) = H0 + Z (r ) () +  r +1 R(r ) ()

(67)

Normal Forms and NLS

21

with Z (r ) being a non-coupling polynomial and where R(r ) () has a vector field which is bounded, uniformly with respect to  (Z (0) = 0 and R(0) = P ). We look for an auxiliary Hamiltonian Gr +1 such that considering the corresponding Hamilton equations ζ˙ = X Gr +1 (ζ ) and the corresponding flow φrt +1 , one has that E (r ) ◦ φr+1 is in the form (67) with r + 1 in place of r . Explicitly one has r +1

E (r ) ◦ φr+1 = H0 + Z (r )   +  r +1 R(r ) + {H0 , Gr +1 } r +1

+ H0 ◦ φr+1 − H0 −  r +1 {H0 , Gr +1 }   r +1 +  Z (r ) ◦ φr+1 − Z (r )   r +1 +  r +1 R(r ) ◦ φr+1 − R(r ) . r +1

(68) (69) (70) (71) (72)

Now, it is quite easy to understand that (70–72) are higher order terms (they will be estimated later), while (69) is the term of order  r +1 . Thus, if one is able to choose Gr +1 so that R(r ) + {H0 , Gr +1 }

(73)

is non-coupling, then the coupling terms are pushed to order r + 2, and   Z r +1 = Z r +  r R(r ) + {H0 , Gr +1 } is non-coupling. The main step of the proof is the construction and the estimate of such a function G fulfilling the homological equation associated to (73). 3.4. Framework and notations. Before proceeding to the construction and the estimation of such a G it is useful to extend the setting in which we will work. Indeed, since we are working with analytic functions it is useful to work in the complexification of our space. More precisely, we consider now a phase space in which the variables v := u¯ and w := z¯ are independent of u, z. Actually this is equivalent to complexify the space in which the variables p, q of Sect. 3.1 vary. With such an extension the Poisson bracket will take the form   2

∞

  ∂H ∂K ∂H ∂K ∂K ∂H ∂K ∂H {H, K} = i +i . − − ∂u k ∂vk ∂u k ∂vk ∂z j ∂w j ∂z j ∂w j k=1

j=3

Such a phase space will be denoted by XCs . In XCs we will use the norm ' & "(u, v, z, w)# := max (u, v)C2 , (z, w) E

(74)

with (z, w) E defined by (z, w)2E := z2E + w2E , where  · 2E = N E (·, ¯·).

(75)

22

D. Bambusi, A. Sacchetti

Sometimes it is also useful to use more compact notations for the phase space variables, thus we will also write ζ := (u, z), η := (v, w), ξ := (ζ, η). Moreover, given R > 0 we will denote by B R the ball of radius R and center 0 in the phase space: & ' B R = ξ ∈ XCs : "ξ # ≤ R . Given an analytic function H with Hamiltonian vector X H analytic as a map from B R to XCs we will denote |X H | R := sup "X H (ξ )# and |H| R := sup |H(ξ )|, ξ ∈B R

ξ ∈B R

(76)

with norm defined by (74). Remark 17. By Corollary 1, our initial perturbation P , defined by (65), is such that    X P  ≤ ω R + C R 2σ +1 .  R 
σ

(77)

3.5. Solution of the Homological equation. In this section we will construct and estimate the solution of the Homological equation associated to (73); that is we look for Gr +1 such that (73) is non coupling. Precisely, consider the Gauge invariant Hamiltonian function H and decompose it (Remark (15)) as H = F + Z with Z non coupling and F coupling, and consider the equation {H0 , G} = F, F = coupling part of H.

(78)

Then we are going to prove the following. Theorem 5. If H has a Hamiltonian vector X H analytic as a map from B R to XCs , then there exists a solution G of (78) which is coupling and has a vector field X G with the same analyticity properties; moreover there exists a positive α = α(
) such that the following inequality holds:    X G  ≤ 1 |X H | R , R α

(79)

moreover α(
) ≥ C −1
3/2 . Before starting the proof we need some preparation. Since F is coupling then it is of degree at most 2 with respect to z and w. Therefore, we can decompose F as follows: F := F 10 + F 01 + F 20 + F 11 + F 02 , with

( ) ( ) F 10 = F 10 (u, v); z , F 01 = F 01 (u, v); w , ( ) ( ) F 20 = F 20 (u, v)z; z , F 02 = F 02 (u, v)w; w , ( ) F 11 = F 11 (u, v)z; w ,

(80)

(81) (82) (83)

Normal Forms and NLS

23

and let .; . be the scalar product of the real Hilbert space 2 defined by ( )  (1) (2) z (1) ; z (2) := zj zj

(84)

j≥3

and F 01 , F 10 which are XCs valued functions of (u, v) and F 20 , F 02 , F 11 functions taking values in suitable spaces of linear operators. The term F r s (u, v) is of degree r with respect to z and with degree s with respect to w. We will denote by F j01 (u, v) 02 the components of F 01 ; we will denote by Fi02 j (u, v) the matrix elements of F (u, v), and similarly for the other quantities. Furthermore, by Definition 2-iii it follows that F j11j := 0. Lemma 4. The Homological equation (78) has a formal solution G with the same structure as F (cf. (80)–(83)), but with the quantities G defined by G 10 j (u, v) := G 20 jl (u, v) :=

F j10 (u, v) i(λ j − )

, G 01 j (u, v) :=

F jl20 (u, v) i(λ j + λl − 2)

F j01 (u, v) −i(λ j − )

, G 02 jl (u, v) :=

11 G 11 j j (u, v) := 0, G jl (u, v) :=

F jl11 (u, v) i(λ j − λl )

,

(85)

F jl02 (u, v) −i(λ j + λl − 2)

, j = l.

,

(86) (87)

Remark 18. By definitions (85)–(87) and since F is a coupling term then the solution G is coupling too. Proof. We consider explicitly only the term G 01 , since all the other terms can be studied exactly in the same way. First define H(1) (u, v) := (u 1 v1 + u 2 v2 ), H(∞) (z, w) :=



λjz jwj,

j≥3

so that H0 = H(1) + H(∞) . Now one has ! ! ! H0 , G 01 = H(1) , G 01 + H(∞) , G 01 !   = H(1) , G 01 − iλ j G 01 j j wj,

(88)

j

' & where we simply computed explicitly H(∞) , G 01 . Thus to ask ! H0 , G 01 = F 01

(89)

! 01 H(1) , G 01 − iλ j G 01 j j = Fj .

(90)

is equivalent to ask

24

D. Bambusi, A. Sacchetti

We compute now the above Poisson Bracket. To this end decompose G 01 j in Taylor series. One has  k l G 01 G 01 (91) j (u, v) = j,kl u v , kl

from which

!  k l H(1) , G 01 = G 01 j j,kl i(|l| − |k|)u v ,

(92)

kl

but, assuming that G 01 is coupling (which will be verified in a while), due to the limitation (58), one has |k| − |l| = |m| − |n| = −1 (due to the fact that G 01 is linear in w), and therefore ! = iG 01 H(1) , G 01 (93) j j . Inserting this expression in (90) one gets 01 01 iG 01 j − iλ j G j = F j

which shows that the function G 01 of (85) actually fulfills Eq. (89) and is coupling. All the other terms can be studied in the same way; a detailed proof is omitted.   End of the Proof of Theorem 5. We will explicitly prove only the estimate of the norms of the vector fields of X G 01 and X G 11 , the other being similar and simpler. We start with X G 01 . Consider first the vector field of F 01 , whose components have the form * 01 + ∂F ul ;w , (94) X F 01 = i ∂vl z 01 . XF 01 = i F

(95)

Since F is the coupling part of H then u u XF 01 = dw X H (u, v, 0, 0)w

and therefore, adding also the v components, the Cauchy inequality implies that    ∂ F 01  1 1  (u,v)  (u,v)   (u, v) ≤ sup"X H (u, v, z, w)# = X H  ,  ∂(u, v)  R R BR R L(X s ,C4 )

(96)

(97)

C

where the norm at l.h.s. is the norm as a linear operator from XCs to C4 . Using this (u,v) inequality and making use of (85) it is very easy to estimate X G 01 : to this end define w˜ j := w j /i(λ j − ) and let C > 1 be such that C −1
:= inf |λ j − |. j≥3

Then one has w ˜ E ≤ C w E /


(98)

Normal Forms and NLS

25

and, using the definition of G 01 , cf. (85), one has *

u

X G 01

∂ F 01 =i ; w˜ ∂v

+ (99)

and similarly for the v components, and thus   C  (u,v)   (u,v)   X G 01  ≤  X H  R R


(100)

for some C > 1. The estimate of the z components of the vector field is simpler (due to the simpler form of the z component) and collecting the two one gets    X G 01  ≤ C |X H | R . R


(101)

We come now to the estimate of the vector field of G 11 . Preliminary to this estimate we remark that the components of the vector field of F 11 are given by (u,v) (u,v) 2 (u, v, 0, 0)[z, w], XF 11 = dzw X H w w 11 XF z, 11 = dz X H (u, v, 0, 0)z = F

(102) (103)

thus in particular, by Cauchy inequality, sup F 11 (u, v)L(2 ,2 ) ≤ E

BR

E

1  w  X , R H R

(104)

where the norm at l.h.s. is the norm as a linear operator from 2E to 2E . In order to estimate X Gz 11 we have just to estimate the norm of the operator (from 2E to itself) whose matrix is defined in (87). To this end remark that the boundedness of G 11 as an operator from 2E to itself (or to its dual) is equivalent to the boundedness of 2 the operator with matrix elements tl G 11 jl s j as an operator from  to itself, where tl , s j 2 2 are suitable positive numbers (in fact,  and  E are spaces with inequivalent norms). Thus Lemma 8 of the Appendix ensures that sup G 11 (u, v)L(2 ,2 ) ≤ BR

E

E

1  w  X , α ≥ C −1
3/2 , α H R

(105)

for some C > 1, since |λ j − λl | ≥ C −1
if j and l belong to different sets Jγ . Working in a similar way for the other components and the other parts of the vector field of the function G, one gets the result.   From the proof (especially (96), (102) and (103)) we get also the following useful lemma: Lemma 5. The following estimates hold: |X F | R ≤ 5 |X H | R , |X Z | R ≤ 6 |X H | R , |Z| R ≤ 6 |H| R .

(106)

26

D. Bambusi, A. Sacchetti

3.6. Quantitative estimates. First we fix a positive R in such a way that X P is analytic on B R , and choose constants P, P ∗ (that depend on
and on all the other parameters) such that    X P  ≤ P, |P | R ≤ P ∗ . (107)  R All along this section we fix a small value of
, and will make explicit estimates so that at the end it will be possible to insert the dependence on
of the final estimate. Lemma 6 (Iterative Lemma). Consider a Gauge invariant Hamiltonian of the form E (r ) = H0 + Z (r ) +  r +1 R(r ) ,

(108)

with Z (r ) non coupling and where R(0) = P and Z (0) = 0. Fix δ < R/(r + 1), assume that the Hamiltonian vector fields of Z (r ) and of R(r ) are analytic on B R−r δ , and that ⎧ 0 if r = 0 ⎨    i r −1  X (r )    ≤ , (109) Z R−r δ if r ≥ 1 ⎩6P  0 i=0 ⎧ 0 if r = 0  ⎨   i  (r )  r −1  Z , (110) ≤   if r ≥ 1 ⎩6P ∗ R−r δ i=0 0     P P∗  (r )   X (r )  ≤ , ≤ , (111)  R R R−r δ R−r δ 0r 0r with 0 :=

αδ , where α = α(
) ≥ C −1
3/2 , 75P

(112)

  < 1/2 there exists a Hamiltonian function Gr +1 analytic on B R−δr gener0 r +1 r +1 ating the canonical transformation φr+1 such that E (r ) ◦ φr+1 has the form (108) and satisfies the estimates (109)–(111) with r +1 in place of r , moreover the new Hamiltonian is Gauge invariant and the canonical transformation fulfills  r , P  r +1 . (113) sup ξ − φ  (ξ ) ≤  0 α "ξ #≤R−(r +1)δ 

then, if

Proof. Decompose R(r ) into its coupling part Fr and its non-coupling part Zr . Define  Z (r +1) := Z (r ) +  r Zr and use Lemma 5 to estimate  X Z (r +1)  R−δ(r +1) . Use Theorem 2 to construct Gr +1 as the solution of {H0 ; Gr +1 } = −Fr , then  XG

r +1

 

R−r δ

≤

 1  P X R(r )  R−r δ ≤ r . α α0

(114)

Normal Forms and NLS

27

The Hamiltonian E (r ) ◦φ  was computed in Subsect. 3.2 and is given by Eqs. (68)–(72), which has the form (108) provided one defines r +1

(r +1)

R(r +1) = Ra(r +1) + Rb

+ Rc(r +1) ,

where (r +1)

Ra(r +1) =  −(r +2) (70), Rb

=  −(r +2) (71) and Rc(r +1) =  −(r +2) (72).

Then, from Lemma 11, with µ =  (r +1) , it follows that       5   ≤ r +1 µ  X Z (r )  R−r δ  X Gr +1  R−r δ X R(r +1)  b δ (R−r δ)−δ . r −1   /     i P 5 ≤ 6P δ 0 α0r i=0

60P 2 , ≤ αδ0r since  ≤ 21 0 . Similarly     X R(r +1)  c

(R−r δ)−δ

   5  µ X R(r )  R−r δ  X Gr +1  R−r δ δ      P 5P 2  r 5 r P ≤ ≤ δ 0r α0r αδ0r 0 ≤

≤

5P 2 . 2r αδ0r

Furthermore, from Lemma 12, with H = R(r ) and µ =  (r +1) , it follows that       25   ≤ r +2 µ2  X R(r )  R−r δ  X Gr +1  R−r δ X R(r +1)  a (R−r δ)−δ δ      25P 2  r P 25 r P ≤ ≤ δ 0r α0r αδ0r 0 ≤

25P 2 . 2r αδ0r

Hence  X

 (r +1) 

R

(R−r δ)−δ



P 25P 60P 5P ≤ + r + r r αδ 2r 0r 0 2   0

 30 P P 60 + r . ≤ αδ 0r 2

(115)

If r = 0 then the second term is not present and the square bracket in (115) is equal to 30P; if r ≥ 1, one has that the square bracket in (115) is less than 75P/0r and thus the thesis on the vector fields follows. The estimates of the moduli are obtained in a similar way from Lemma 5.  

28

D. Bambusi, A. Sacchetti

It is clear that the Hamiltonian (24) of the NLS fulfills the assumptions of the lemma with r = 0 and thus the lemma allows us to put our Hamiltonian in normal form up to any order r . To obtain the exponentially small estimate of the remainder take δ = R/2r in order to fix the domain of definition of the final Hamiltonian, and then choose an optimal value of r . This can be done by minimizing the estimate of the remainder or simply choosing 

Rα r∗ := 150e P (with [.] denoting the integer part). One thus obtains the following theorem which is the main technical result of the paper, where Z = Z (r ) and R =  r +1 R(r ) . Theorem 6. Consider the Hamiltonian E (cf. (24)), define ∗ :=

αR , 150e P

(116)

and assume || < ∗ /2. Then there exists an analytic canonical transformation T such that E ◦ T = H0 + Z + R,

(117)

with Z non-coupling; both Z and R have a vector field which is analytic in B R/2 and fulfill the estimate     ∗ |X Z | R/2 ≤ 12 P , |X R | R/2 ≤  P exp − , (118)      ∗ |Z| R/2 ≤ 12 P ∗ , |R| R/2 ≤  P ∗ exp − . (119)  Moreover the transformed Hamiltonian is Gauge invariant and the canonical transformation fulfills sup "ξ − T (ξ )# ≤ 2

"ξ #≤R/2

P . α

(120)

It is also important to reformulate the theorem computing P in terms of the original quantities ω, ,
, i.e. using (77), thus one easily gets the following Corollary 7. Fix a positive R, consider the Hamiltonian E (cf. (24)), and define the small parameter µ := ω +

|| .
σ

(121)

Then there exists µ∗ > 0 independent of
, ω, , such that, if µ < µ∗
3/2 /2 then there exists an analytic canonical transformation T which transforms E into (117), where 

µ∗
3/2 |X Z | R/2 ≤ Cµ , |X R | R/2 ≤ Cµ exp − , (122) µ 

µ∗
3/2 |Z| R/2 ≤ Cµ, |R| R/2 ≤ Cµ exp − . (123) µ

Normal Forms and NLS

29

Moreover the transformed Hamiltonian is Gauge invariant and the canonical transformation fulfills µ sup "ξ − T (ξ )# ≤ C 3/2 . (124)
"ξ #≤R/2 Theorem 3 is a direct corollary of Corollary 7. The manifold z = 0 is approximately invariant for the dynamics of the system (117) and on such a manifold the dynamics is that of a Hamiltonian system with a Hamiltonian function which is an exponentially small perturbation of K(u, u) ¯ := Z(u, u, ¯ 0, 0).

(125)

Such a system has the additional integral of motion  I(u, u) ¯ = |u j |2 .

(126)

j

In the true nonlinear system one has Theorem 7. Assume µ < µ∗
3/2 and consider the Cauchy problem for the system (117) which is equivalent to NLS. Define  

µ∗
3/2 , z 0 2s δ 2 = max exp − µ (z 0 being the initial datum for z), then one has z(t)s ≤ Cδ

(127)

|I(t) − I(0)| ≤ Cδ 2 , |K(t) − K(0)| ≤ Cµδ 2

(128)

and for all times t fulfilling 1 . (129) Cµδ Proof. First remark that due to the equivalence of the E norm and the s norm one has  N E (ζ0 ) = z 0  E ≤ Cδ. (130) |t| ≤

Thus, by (63) and (123) one has



dN E µ∗
3/2  ≤ Cµ(N E )3/2 + Cµ exp − N E ≤ Cµ(N E )3/2 dt µ

(131)

which can be solved giving the estimate N E (t) ≤ 2δ 2

(132)

for the times (129). The estimate of z(t)s follows from equivalence of the norms. To obtain the estimate of the diffusion of I simply remark that N = I + N E is an exact integral of motion and use the estimate (132). Finally, to estimate the diffusion of K remark that the total Hamiltonian is an integral of motion, but one has  

µ∗
3/2 2 |E ◦ T − H0 + K| ≤ Cµ z(t)s + exp − (133) µ which using the previous estimates implies the thesis.

 

30

D. Bambusi, A. Sacchetti

Proof of Theorem 2. Define first M := T ( 0 ). In order to distinguish between the original coordinates of the NLS equation and the new coordinates introduced by the transformation T we will denote the new variables by adding a prime, i.e. we will write ζ = T (ζ  ). Remark that since T is Lipschitz together with its inverse, then for any couple of points ζ, ζ˜ one has             C −1 ζ − ζ˜  ≤ ζ  − ζ˜   ≤ C ζ − ζ˜  , (134) s

and therefore

s

s

  d(ψ 0 , M) ≤ δ =⇒ d(ζ 0 , 0 ) ≤ Cδ ⇐⇒ z 0 s ≤ Cδ,

(135)

where by a slight abuse of notation we wrote ψ 0 = T (ζ  ). Thus Eq. (127) implies (21) for the considered times. To get (22), just remark that (again with a slight abuse of notation)  µ  c ψ = z = z  + O 3/2 ,
and thus (127) implies (22).   The proof of Corollary 3 is also a direct consequence of Eqs. (128) when one considers the deformation induced by the change of variables. A detailed proof is omitted. Appendix A. Proof of Theorem 1 For possible future reference, in this section we will work in Rd . We start by recalling some notations and definitions from Robert [21] that will be used in the following Definition 4. A function m : R2d → [0, +∞) is called a tempered weight if there exist C0 , N0 > 0 such that m(X ) ≤ C0 m(X 1 )(1 + |X − X 1 |) N0 , ∀X, X 1 ∈ R2d .

(136)

Definition 5. A function a ∈ C ∞ (R2d ) is called a semiclassical symbol with weight (m, ρ), with ρ ≥ 0, if for any multi index α there exists a constant Cα such that  α  ∂ a(X ) ≤ Cα

m(X ) , ∀X ∈ R2d . (1 + |X |)ρ|α|

(137)

In this case we write a ∈ ρm . In the following we will also need an extension to
dependent families of symbols. Definition 6. A smooth map
→ a(
) ∈ ρm is called an
-admissible symbol of class (m, ρ), if for any integer N large enough one has a(
) = a0 +
a1 +
2 a2 + · · · +
N a N + r N +1 m,−2 j

m,−2(N +1)

and r N +1 (
) bounded in ρ with a j ∈ ρ symbols with weight   m (1 + |X |) Kρ , ρ .

(138)

. Here ρm,K is the class of

Normal Forms and NLS

31

Given an
-admissible symbol a of class (m, ρ), one defines the corresponding Weyl operator acting on L 2 by   x + y ) 1 w i (x−y,ξ 2 , ξ,
ψ(y)dd ydd ξ. e a (139) O p (a)ψ(x) := (2π
)d R2d 2 By the theory of [21] one has that, for any
> 0, such an operator is well defined on the Schwartz space. Under suitable conditions it extends to a selfadjoint operator on L 2 . Definition 7. A strongly admissible operator of weight (m, ρ) is a C ∞ application A : (0,
∗ ) → L(S(Rd ), L 2 (Rd )) such that there exists an admissible symbol a ∈ ρm such that A(
) = O p w (a(
)). One of the most important properties of strongly admissible operators is that given by the following theorem. Theorem 8. (Theorem II-32 of [21]) Let A = O p w (a) and B = O p w (b) be two strongly admissible operators of weights (m, ρ) and (n, ρ) respectively; then AB isa strongly admissible operator with symbol c(
) and weight (mn, ρ). Moreover, if a ∼ j≥0
j a j   and b ∼ j≥0
j b j , then c ∼ j≥0
j c j with 

cj =

|α|+|β|+k+l= j

1 α!β!

 |α|  |β| 1 1 β (∂ξα Dxβ ak )(∂ξ Dxα bl ), 2 2

(140)

where Dx = −i∂x . Furthermore, for any N ≥ 0, one has c=

N 


j c j +
N +1 δ N +1 ,

(141)

j=0

and, for any α, β there exists a positive finite M such that the following estimate holds (for simplicity we restrict to the case ρ = 0)     α β (142) ∂x ∂ξ δ N +1 (x, ξ ) ≤ Cm(x, ξ )n(x, ξ ) ⎡ N    ×⎣ pm M (a j ) pn M (r N +1 (b)) + pn M (b j ) pm M (r N +1 (a)) (143) j=0

+

⎤



pm M (a j ) pq M (bk ) + pm M (r N +1 (a)) pn M (r N +1 (b))⎦ ,

(144)

N ≤ j+k≤2N

where we denoted by r N +1 (a) the remainder of the asymptotic expansion of a truncated at order N and we denoted β

pm M (a) =

sup

|α|+|β|≤M, (x,ξ )

|∂xα ∂ξ a(x, ξ )| m(x, ξ )

.

(145)

32

D. Bambusi, A. Sacchetti

Fix a positive integer s ≥ 1, denote A = H0s (where H0 is the operator (2)). From Theorem 8 it follows that A = O p w (a) with a suitable a having principal symbol a0 = (ξ 2 + V )s . Denote also b := ξ 2s + V s and B = O p w (b). Since V (x) ≥ 1 then one has 1 1 ≤ b(x, ξ ) ≤ a0 (x, ξ ) ≤ Cb(x, ξ ), ∀(x, ξ ) ∈ R2d . (146) C C Lemma 7. One has Aψ L 2 ≤ C Bψ L 2 , ∀ψ ∈ D(B), Bψ L 2 ≤ C  Aψ L 2 , ∀ψ ∈ D(A).

(147) (148)

−1

Proof. Consider b−1 , then b−1 ∈ 0b . Denote B := O p w (b−1 ). By Theorem 8 one has a  0 +
O p w (δ1 ), (149) AB = O p w b w BB = I +
O p (δ2 ), (150) with δ1,2 estimated by (142)–(144). Since ab0 is bounded together with its derivatives, it follows that 1 := O p w (δ1 ) is bounded. Similarly 2 := O p w (δ2 ) is bounded. Thus, using Neumann’s is formula one gets that the operator I +
2 is invertible provided
is small enough. So, from (150) one has (I +
2 )−1 BB = I

B −1 = (I +
2 )−1 B.

⇐⇒

(151)

Finally one has

        Aψ L 2 =  ABB −1 ψ  2 ≤ ABL(L 2 ,L 2 ) B −1 ψ  2 L L     −1 ≤ C (I +
2 ) Bψ  2 ≤ C Bψ L 2 .

(152)

L

  Appendix B. Technical Lemmas We start with a lemma which is needed for the estimate of the solution of the homological equation. In its statement we will denote by 2 the real Hilbert space of the sequences (z j ) j≥3 endowed by the scalar product   

z; z   := z j z j = z j z j . j≥3

γ

j≥3, j∈Jγ

Lemma 8. Let F : 2 → 2 be a bounded linear operator, assume that the corresponding matrix elements F jl are different from zero only if j ∈ Jγ and l ∈ Jγ  with γ = γ  . Define a new linear operator G with matrix G jl :=

F jl . i(λl − λ j )

(153)

Then there exists a positive C such that the following estimate holds: GL(2 ,2 ) ≤

C FL(2 ,2 ) .
3/2

(154)

Normal Forms and NLS

33

Proof. First we recall that |λ j − λl | ≥ C −1
if j ∈ Jγ and l ∈ Jγ  for γ = γ  , and that # Jγ ≤ C/
. Fix l ∈ Jγ  . First remark that  |F jl |2 ≤ F2L(2 ,2 ) , (155) j

then, by Schwartz inequality, ⎛ ⎞1/2 ⎛   |G jl | ≤ ⎝ |F jl |2 ⎠ ⎝ j

 j∈Jγ ,γ =γ 

j

⎞1/2 1 ⎠ . |λl − λ j |2

Fix l ∈ Jγ  and estimate  j

  1 1 = 2 |λl − λ j | |λl − λ j |2 γ =γ  j∈Jγ ⎞⎛ ⎛  −2 γ    ⎠⎝ + + =⎝ γ =γ  ±1

γ =1

γ ≥γ  +2

j∈Jγ

(156) ⎞ 1 ⎠, |λl − λ j |2

but, due to the choice of the numbers E γ one has ⎧ −1 if γ = γ  ± 1   ⎨ C
λ j − λl  ≥ E  − E if γ ≤ γ  − 2 . ⎩ E γ − Eγ  if γ ≥ γ  + 2 γ −1 γ Thus (156) is estimated by ⎞  ⎛  −2 γ 2  2C 1 1 ⎠. sup # Jγ ⎝ 2 + +
(E γ  − E γ )2 (E γ −1 − E γ  )2 γ  γ =1

(157)

(158)

(159)

γ ≥γ +2

Since the sums in (159) are convergent due to our choice of the sequence E γ one has  j

1 C ≤ 3, 2 |λl − λ j |


which gives 

C FL(2 ,2 ) .
3/2

|G jl | ≤

j

From this one has Gz22

 2            2      G jl G jl |zl | = G jl zl  ≤    j l j l l     C F2L(2 ,2 )      2     z22 . G jl G jl |zl | ≤ =
3 l

 

j

l

(160)

34

D. Bambusi, A. Sacchetti

We report now some lemmas from [4] which are needed for the proof of Lemma 6. Here µ will be a small parameter (not the small parameter used in the main part of the text) that in applications will be replaced by  r with some r . Lemma 9. Let G : Bρ → C, ρ > 0 be a function whose Hamiltonian vector field is   analytic as map from Bρ → XCs ; fix a positive δ < ρ. Assume µ  X G ρ < δ and consider the flow φ t of the corresponding Hamiltonian vector field. Then, for |t| ≤ µ, one has   (161) |φ t − I|ρ−δ ≤ µ  X G ρ . Proof. It is just an application of the equality  t  t dξ ξ(t) − ξ(0) = X G (ξ(s))ds. (s)ds = 0 dt 0   Lemma 10. Consider G as above and let H be an analytic function with vector field analytic in Bρ , and fix 0 < δ < ρ assume µ  X G ρ ≤ δ/3. Then, for |t| ≤ µ, one has     3    X H◦φ t  X µ ≤ 1 + G ρ |X H |ρ . ρ−δ δ Proof. First remark that, since φ t is a canonical transformation one has X H◦φ t (ξ ) = dφ −t (φ t (ξ ))X H (φ t (ξ )) from which

(162)

  X H◦φ t (ξ ) = dφ −t (φ t (ξ )) − I X H (φ t (ξ )) + X H (φ t (ξ )).

To estimate the first term fix δ1 := δ/3; we have sup

"ξ #≤ρ−3δ1

"dφ −t (φ t (ξ )) − I# ≤ ≤

 

sup

"ξ #≤ρ−2δ1

1 δ1

sup

"dφ −t (ξ ) − I#

"ξ #≤ρ−δ1

"φ −t (ξ ) − ξ # ≤

µ   XG ρ . δ1

Going back to δ and adding the trivial estimate of the second term one has the thesis.

  Lemma 11. Let G and H be as above, fix 0 < δ < ρ, and assume µ  X G ρ < δ/3. Then, for |t| ≤ µ one has    X H◦φ t −H 

ρ−δ

≤

  5 |X H |ρ µ  X G ρ . δ

Proof. One has

 

 X H◦φ t (ξ ) − X H (ξ ) = dφ −t (φ t (ξ )) − I X H (φ t (ξ )) + X H (φ t (ξ )) − X H (ξ ) .

The norm of the square bracket is easily estimated using the Lagrange Theorem and the Cauchy inequality in order to bound d X H . The other term was already estimated in Lemma 10, so we have the thesis.  

Normal Forms and NLS

35

Although H0 has an unbounded vector field the vector field of H0 ◦ φ µ − H0 is bounded, more precisely we have Lemma 12. Let H, H0 , F and G as in § 2.3; that is G is the solution of the Homological equation (78); denote by φ t the flow of the corresponding Hamiltonian vector field and (ξ ) := H0 (φ µ (ξ )) − H0 (ξ ) − µ {H0 , G} , then the vector field of  is analytic and, for any δ < ρ, satisfies |X  |ρ−δ ≤ µ2

25   X G ρ |X H |ρ . δ

Proof. One has H0 (φ µ (ξ )) − H0 (ξ ) =

 0

µ

d H0 (φ t (ξ ))dt = − dt



µ

F(φ t (ξ ))dt,

0

where we used the homological equation (78) to calculate {H0 , G}. Then one has  µ

 F(φ t (ξ )) − F(ξ ) dt. (ξ ) = 0

Using Lemma 5 and Lemma 11 one gets the thesis.

 

Acknowledgements. DB would like to thank Panos Kevrekidis for pointing his attention to [27] and the connection between the Gross-Pitaevskii and the discrete NLS equations.

References 1. Anderson, B.P., Kasevich, M.A.: Macroscopic quantum interference from atomic tunnel arrays. Science 282, 1686–1689 (1998) 2. Aschbacher, W.H., Fröhlich, J., Graf, G.M., Schnee, K., Troyer, M.: Symmetry breaking regime in the nonlinear Hartree equation. J. Math. Phys. 43, 3879–3891 (2002) 3. Bambusi, D.: Exponential stability of breathers in Hamiltonian networks of weakly coupled oscillators. Nonlinearity 9, 433–457 (1996) 4. Bambusi, D.: Nekhoroshev Theorem for small amplitude solutions in nonlinear Schrödinger equations. Math. Z. 130, 345–387 (1999) 5. Bambusi, D.: On long time stability in Hamiltonian perturbations of non-resonant linear PDEs. Nonlinearity 12, 823–850 (1999) 6. Bambusi, D., Grebert, B.: Birkhoff normal form for PDEs with tame modulus. Duke Math. J. 135, 507–567 (2006) 7. Bambusi, D., Graffi, S., Paul, T.: Normal form and quantization formulae. Commun. Math. Phys. 207, 173–195 (1999) 8. Bambusi, D., Sacchetti, A.: Stability of spectral eigenspaces in nonlinear Schrödinger equations. http://arxiv.org/list/math-ph/0608010, 2006 9. Bambusi, D., Vella, D.: Quasi periodic breathers in Hamiltonian lattices with symmetries. DCDSB 2, 389–399 (2002) 10. Berezin, F.A., Shubin, M.A.: The Schrödinger equation. Amsterdam: Kluwer Ac. Publ., 1991 11. Burger, S., et al.: Superfluid and Dissipative Dynamics of a Bose-Einstein Condensate in a Periodic Optical Potential. Phys. Rev. Lett. 86, 4447–4450 (2001) 12. Grecchi, V., Martinez, A.: Non-linear Stark effect and molecular localization. Commun. Math. Phys. 166, 533–548 (1995) 13. Grecchi, V., Martinez, A., Sacchetti, A.: Destruction of the beating effect for a non-linear Schrödinger equation. Commun. Math. Phys. 227, 191–209 (2002) 14. Helffer, B.: Semi-classical analysis for the Schrödinger operator and applications. Lecture Notes in Mathematics 1336, Berlin-Heidelberg: Springer-Verlag, 1988

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15. Kuksin, S.B.: Nearly integrable infinite-dimensional Hamiltonian systems. Lecture Notes in Mathematics 1556, Berlin: Springer-Verlag, 1993 16. Littlewood, J.E.: On the equilateral configuration in the restricted problem of three bodies. Proc. London Math. Soc. 3(9), 343–372 (1959) 17. MacKay, R.S., Aubry, S.: Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators. Nonlinearity 7, 1623–1643 (1994) 18. Mujica, J.: Complex analysis in Banach spaces. Mathematical Studies 120, Amsterdam: North Holland, 1986 19. Pazy, A.: Semigroups of linear operators and applications to partial differential equations. New York: Springer-Verlag, 1983 20. Raghavan, S., Smerzi, A., Fantoni, S., Shenoy, S.R.: Coherent oscillations between two weakly coupled Bose-Einstein condensates: Josephson effects, π oscillations, and macroscopic quantum self-trapping. Phys. Rev. A 59, 620–633 (1999) 21. Robert, D.: Autour de l’approximation semi-classique. Progress in Mathematics 68, Boston: Birkhäuser, 1987 22. Reed, M., Simon, B.: Methods of modern mathematical physics: IV Analysis of operators. New-York: Academic Press, 1972 23. Sacchetti, A.: Nonlinear time-dependent one-dimensional Schrödinger equation with double well potential. SIAM: J. Math. Anal. 35, 1160–1176 (2004) 24. Sacchetti, A.: Nonlinear double well Schrödinger equations in the semiclassical limit. J. Stat. Phys. 119, 1347–1382 (2005) 25. Sjöstrand, J.: Semi-excited levels in non-degenerate potential wells. Asymptotic Analysis 6, 29–43 (1992) 26. Selleri, S., Zoboli, M.: Stability analysis in nonlinear TE polarized waves in multiple-quantum-well waveguides. IEEE J. Quant. Elect. 31, 1785–1789 (1995) 27. Trombettoni, A., Smerzi, A.: Discrete solitons and breathers with diluite Bose–Einstein condensates. Phys. Rev. Lett. 86, 2353–2356 (2001) 28. Trutschel, U., Lederer, F., Golz, M.: Nonlinear guided waves in multylayer systems. IEEE J. Quant. Elect. 25, 194–200 (1989) 29. Vardi, A., Anglin, J.R.: Bose-Einstein condensates beyond mean field theory: quantum back reaction as decoherence. Phys. Rev. Lett. 86, 568–571 (2001) 30. Vardi, A., Anglin, J.R.: Dynamics of a two-mode Bose-Einstein condensate beyond mean field theory. Phys. Rev. A 64, 013605 (2001) 31. Yajima, K., Zhang, G.: Local smoothing property and Strichartz inequality for Schrödinger equations with potentials superquadratic at infinity. J. Diff. Eq. 202, 81–101 (2004) 32. Zhang, J.: Stability of attractive Bose-Einstein condensates. J. Stat. Phys. 101, 731–746 (2000) Communicated by G. Gallavotti

Commun. Math. Phys. 275, 37–95 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0289-0

Communications in

Mathematical Physics

Properties of Generalized Univariate Hypergeometric Functions F. J. van de Bult1 , E. M. Rains2,
, J. V. Stokman1 1 KdV Institute for Mathematics, Universiteit van Amsterdam, Plantage Muidergracht 24, 1018 TV

Amsterdam, The Netherlands. E-mail: [email protected]; [email protected]

2 Department of Mathematics, University of California, Davis, USA. E-mail: [email protected]

Received: 4 August 2006 / Accepted: 4 January 2007 Published online: 24 July 2007 – © Springer-Verlag 2007

Abstract: Based on Spiridonov’s analysis of elliptic generalizations of the Gauss hypergeometric function, we develop a common framework for 7-parameter families of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric functions. In each case we derive the symmetries of the generalized hypergeometric function under the Weyl group of type E 7 (elliptic, hyperbolic) and of type E 6 (trigonometric) using the appropriate versions of the Nassrallah-Rahman beta integral, and we derive contiguous relations using fundamental addition formulas for theta and sine functions. The top level degenerations of the hyperbolic and trigonometric hypergeometric functions are identified with Ruijsenaars’ relativistic hypergeometric function and the Askey-Wilson function, respectively. We show that the degeneration process yields various new and known identities for hyperbolic and trigonometric special functions. We also describe an intimate connection between the hyperbolic and trigonometric theory, which yields an expression of the hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric hypergeometric functions. 1. Introduction The Gauss hypergeometric function, one of the cornerstones in the theory of classical univariate special functions, has been generalized in various fundamental directions. A theory on multivariate root system analogues of the Gauss hypergeometric function, due to Heckman and Opdam, has emerged, forming the basic tools to solve trigonometric and hyperbolic quantum many particle systems of Calogero-Moser type and generalizing the Harish-Chandra theory of spherical functions on Riemannian symmetric spaces (see [8] and references therein). A further important development has been the generalization to q-special functions, leading to the theory of Macdonald polynomials [16],
Current address: Department of Mathematics, California Institute of Technology, Davis, USA. E-mail: [email protected]

38

F. J. van de Bult, E. M. Rains, J. V. Stokman

which play a fundamental role in the theory of relativistic analogues of the trigonometric quantum Calogero-Moser systems (see e.g. [25]) and in harmonic analysis on quantum compact symmetric spaces (see e.g. [18, 14]). In this paper, we focus on far-reaching generalizations of the Gauss hypergeometric function within the classes of elliptic, hyperbolic and trigonometric univariate special functions. Inspired by results on integrable systems, Ruijsenaars [24] defined gamma functions of rational, trigonometric, hyperbolic and elliptic type. Correspondingly there are four types of special function theories, with the rational (resp. trigonometric) theory being the standard theory on hypergeometric (resp. q-hypergeometric) special functions, while the hyperbolic theory is well suited to deal with unimodular base q. The theory of elliptic special functions, initiated by Frenkel and Turaev in [4], is currently in rapid development. The starting point of our analysis is the definition of the various generalized hypergeometric functions as an explicit hypergeometric integral of elliptic, hyperbolic and trigonometric type depending on seven auxiliary parameters (besides the bases). The elliptic and hyperbolic analogue of the hypergeometric function are due to Spiridonov [33], while the trigonometric analogue of the hypergeometric function is essentially an integral representation of the function  introduced and studied extensively by Gupta and Masson in [7]. Under a suitable parameter discretization, the three classes of generalized hypergeometric functions reduce to Rahman’s [20] (trigonometric), Spiridonov’s [33] (hyperbolic), and Spiridonov’s and Zhedanov’s [35, 33] (elliptic) families of biorthogonal rational functions. Spiridonov [33] gave an elementary derivation of the symmetry of the elliptic hypergeometric function with respect to a twisted action of the Weyl group of type E 7 on the parameters using the elliptic analogue [31] of the Nassrallah-Rahman [17] beta integral. In this paper we follow the same approach to establish the E 6 -symmetry (respectively E 7 -symmetry) of the trigonometric (respectively hyperbolic) hypergeometric function, using now the Nassrallah-Rahman beta integral (respectively its hyperbolic analogue from [37]). The E 6 -symmetry of  has recently been established in [15] by different methods. Spiridonov [33] also gave elementary derivations of contiguous relations for the elliptic hypergeometric function using the fundamental addition formula for theta functions (see (3.6)), entailing a natural elliptic analogue of the Gauss hypergeometric differential equation. Following the same approach we establish contiguous relations and generalized Gauss hypergeometric equations for the hyperbolic and trigonometric hypergeometric function. For  it again leads to simple proofs of various results from [7]. Although the elliptic hypergeometric function is the most general amongst the generalized hypergeometric functions under consideration (rigorous limits between the different classes of special functions have been obtained in the recent paper [23] of the second author), it is also the most rigid in its class, in the sense that it does not admit natural degenerations within the class of elliptic special functions itself (there is no preferred limit point on an elliptic curve). On the other hand, for the hyperbolic and trigonometric hypergeometric functions various interesting degenerations within their classes are possible, as we point out in this paper. It leads to many nontrivial identities and results, some of which are new and some are well known. In any case, it provides new insight in identities, e.g. as being natural consequences of symmetry breaking in the degeneration process, and it places many identities and classes of univariate special functions in a larger framework. For instance, viewing the trigonometric hypergeometric function as a degeneration of the elliptic hypergeometric function, we show that the breaking of symmetry (from E 7 to E 6 ) leads to a second important integral representation of .

Properties of Generalized Univariate Hypergeometric Functions

39

Moreover we show that Ruijsenaars’ [26] relativistic analogue R of the hypergeometric function is a degeneration of the hyperbolic hypergeometric function, and that the D4 -symmetry [28] of R and the four Askey-Wilson second-order difference equations [26] satisfied by R are direct consequences of the E 7 -symmetry and the contiguous relations of the hyperbolic hypergeometric function. Similarly, the Askey-Wilson function [11] is shown to be a degeneration of the trigonometric hypergeometric function. In this paper we aim at deriving the symmetries of (degenerate) hyperbolic and trigonometric hypergeometric functions directly from appropriate hyperbolic and trigonometric beta integral evaluations using the above mentioned techniques of Spiridonov [33]. The rational level, in which case the Wilson function [6] appears as a degeneration, will be discussed in a subsequent paper of the first author. We hope that the general framework proposed in this paper will shed light on the fundamental, common structures underlying various quantum relativistic Calogero-Moser systems and various quantum noncompact homogeneous spaces. In the present univariate setting, degenerations and specializations of the generalized hypergeometric functions play a key role in solving rank one cases of quantum relativistic integrable CalogeroMoser systems and in harmonic analysis on various quantum SL2 groups. On the elliptic level, the elliptic hypergeometric function provides solutions of particular cases of van Diejen’s [2] very general quantum relativistic Calogero-Moser systems of elliptic type (see e.g. [33]), while elliptic biorthogonal rational functions have been identified with matrix coefficients of the elliptic quantum SL2 group in [12]. On the hyperbolic level, the Ruijsenaars’ R-function solves the rank one case of a quantum relativistic Calogero-Moser system of hyperbolic type (see [29]) and arises as a matrix coefficient of the modular double of the quantum SL2 group (see [1]). On the trigonometric level, similar results are known for the Askey-Wilson function, which is a degeneration of the trigonometric hypergeometric function (see [11] and [10]). For higher rank only partial results are known, see e.g. [13, 21] (elliptic) and [36] (trigonometric). The outline of the paper is as follows. In Sect. 2 we discuss the general pattern of symmetry breaking when integrals with E 7 -symmetry are degenerated. In Sect. 3 we introduce Spiridonov’s [33] elliptic hypergeometric function. We shortly recall Spiridonov’s [33] techniques to derive the E 7 -symmetry and the contiguous relations for the elliptic hypergeometric function. In Sect. 4 these techniques are applied for the hyperbolic hypergeometric function and its top level degenerations. We show that a reparametrization of the top level degeneration of the hyperbolic hypergeometric function is Ruijsenaars’ [26] relativistic hypergeometric function R. Key properties of R, such as a new integral representation, follow from the symmetries and contiguous relations of the hyperbolic hypergeometric function. In Sect. 5 these techniques are considered on the trigonometric level. We link the top level degeneration of the trigonometric hypergeometric function to the Askey-Wilson function. Moreover, we show that the techniques lead to elementary derivations of series representations and three term recurrence relations of the various trigonometric integrals. The trigonometric integrals are contour integrals over indented unit circles in the complex plane, which can be re-expressed as integrals over the real line with indentations by “unfolding” the trigonometric integral. We show that this provides a link with Agarwal type integral representations of basic hypergeometric series (see [5, Chap. 4]). Finally, in Sect. 6 we extend the techniques from [37] to connect the hyperbolic and trigonometric theory. It leads to an explicit expression of the hyperbolic hypergeometric function as a bilinear sum of trigonometric hypergeometric functions. In the top level degeneration, it explicitly relates Ruijsenaars’ relativistic hypergeometric function to the Askey-Wilson function.

40

F. J. van de Bult, E. M. Rains, J. V. Stokman

1.1. Notation. We denote positive values on R>0 .

√ 1 · for the branch of the square root z → z 2 on C \ R<0 with

2. Weyl Groups and Symmetry Breaking The root system of type E 7 and its parabolic root sub-systems plays an important role in this article. In this section we describe our specific choice of realization of the root systems and Weyl groups, and we explain the general pattern of symmetry breaking which arises from degenerating integrals with Weyl group symmetries. Degeneration of integrals with Weyl group symmetries in general causes symmetry breaking since the direction of degeneration in parameter space is not invariant under the symmetry group. All degenerations we consider are of the following form. For a basis  of a given irreducible, finite root system R in Euclidean space (V, ·, ·) with associated Weyl group W we denote V + () = {v ∈ V | v, α ≥ 0 ∀ α ∈ } for the associated positive Weyl chamber. We will study integrals I (u) meromorphically depending on a parameter u ∈ G. The parameter space will be some complex hyperplane G canonically isomorphic to the complexification VC of V , from which it inherits a W -action. The integrals under consideration will be W -invariant under an associated twisted W -action. We degenerate such integrals by taking limits in parameter space along distinguished directions v ∈ V + (). The resulting degenerate integrals will thus inherit symmetries with respect to the isotropy subgroup Wv = {σ ∈ W | σ v = v}, which is a standard parabolic subgroup of W with respect to the given basis , generated by the simple reflections sα , α ∈  ∩ v ⊥ (since v ∈ V + ()). All symmetry groups we will encounter are parabolic subgroups of the Weyl group W of type E 8 . We use in this article the following explicit realization of the root system R(E 8 ) of type E 8 . Let k be the k th element of the standard orthonormal basis of V = R8 , with corresponding scalar product denoted by ·, ·. We also denote ·, · for its complex bilinear extension to C8 . We write δ = 21 (1 + 2 + · · · + 8 ). We realize the root system R(E 8 ) of type E 8 in R8 as R(E 8 ) = {v =

8


c j  j + cδ | v, v = 2, c j , c ∈ Z and

j=1

8


c j even}.

j=1

For later purposes, it is convenient to have explicit notations for the roots in R(E 8 ). The roots are ±α +jk (1 ≤ j < k ≤ 8), α − jk (1 ≤ j = k ≤ 8), β jklm (1 ≤ j < k < l < m ≤ 8), ±γ jk (1 ≤ j < k ≤ 8) and ±δ, where α +jk =  j + k , α− jk =  j − k , 1 ( j + k + l + m − n −  p − q − r ), 2 1 = (− j − k + l + m + n +  p + q + r ), 2

β jklm = γ jk

and with ( j, k, l, m, n, p, q, r ) a permutation of (1, 2, 3, 4, 5, 6, 7, 8).

Properties of Generalized Univariate Hypergeometric Functions

41

The canonical action of the associated Weyl group W on C8 is determined by the reflections sγ u = u − u, γ γ for u ∈ C8 and γ ∈ R(E 8 ). It is convenient to work with two different choices 1 , 2 of bases for R(E 8 ), namely − − − − − − + 1 = {α76 , β1234 , α65 , α54 , α43 , α32 , α21 , α18 },

− − − − − − 2 = {α23 , α56 , α34 , α45 , β5678 , α18 , α87 , γ18 },

with corresponding (affine) Dynkin diagrams − − − + α18 α21 α32 α43 −δ ◦ • • • •

− α54 •

− α65 •

− α76 •

• β1234 and −δ ◦

γ18 •

− α87 •

− α18 •

β5678 •

− α45 •

− α34 •

(2.1) − α23 •

− (2.2) • α56 respectively, where the open node corresponds to the simple affine root, which we have labeled by minus the highest root of R(E 8 ) with respect to the given basis (which in both cases is given by δ ∈ V + ( j )). The reason for considering two different bases is the following: we will see that degenerating an elliptic hypergeometric integral with W (E 7 )+ ∈  , symmetry to the trigonometric level in the direction of the basis element α18 1 respectively the basis element γ18 ∈ 2 , leads to two essentially different trigonometric hypergeometric integrals with W (E 6 )-symmetry. The two integrals can be easily related since they arise as a degeneration of the same elliptic hypergeometric integral. This leads directly to highly nontrivial trigonometric identities, see Sect. 5 for details. This remark in fact touches on the basic philosophy of this paper: it is the symmetry breaking in the degeneration of hypergeometric integrals which lead to various nontrivial identities. It forms an explanation why there are so many more nontrivial identities on the hyperbolic, trigonometric and rational level when compared to the elliptic level. Returning to the precise description of the relevant symmetry groups, we will mainly encounter stabilizer subgroups of the isotropy subgroup W−δ . Observe that W−δ is a standard parabolic subgroup of W with respect to both bases  j since −δ ∈ V + ( j ) + }, respectively ( j = 1, 2), with associated simple reflections sα , α ∈ 1 := 1 \ {α18 sα , α ∈ 2 := 2 \ {γ18 }. Hence W−δ is isomorphic to the Weyl group of type E 7 , and we accordingly write

W (E 7 ) := W−δ . We realize the corresponding standard parabolic root system R(E 7 ) ⊂ R(E 8 ) as R(E 7 ) = R(E 8 ) ∩ δ ⊥ ⊆ δ ⊥ ⊂ R8 . Both 1 and 2 form a basis of R(E 7 ), and the associated (affine) Dynkin diagrams are given by − − − − − − − α54 α65 α21 α32 α43 α76 −α78 • • • • • • ◦ • β1234

(2.3)

42

F. J. van de Bult, E. M. Rains, J. V. Stokman

and − α87 •

− α18 •

β5678 •

− α45 •

− α34 •

− α23 −β1278 • ◦

− • α56

(2.4)

− respectively (where we have used that α78 , respectively β1278 , is the highest root of R(E 7 ) with respect to the basis 1 , respectively 2 ). Note that the root system R(E 7 ) consists of the roots of the form α − jk and β jklm . The top level univariate hypergeometric integrals which we will consider in this article depend meromorphically on a parameter u ∈ Gc with Gc ⊂ VC = C8 (c ∈ C) the complex hyperplane

Gc =

8
c u j = 2c}. δ + δ ⊥ = {u = (u 1 , u 2 , . . . , u 8 ) ∈ C8 | 2 j=1

The action on C8 of the isotropy subgroup W (E 7 ) = W−δ ⊂ W preserves the hyperplane δ ⊥ and fixes δ, hence it canonically acts on Gc . We extend it to an action of the associated affine Weyl group Wa (E 7 ) of R(E 7 ) as follows. Denote L for the (W (E 7 )invariant) root lattice L ⊂ δ ⊥ of R(E 7 ), defined as the Z-span of all R(E 7 )-roots. The affine Weyl group Wa (E 7 ) is the semi-direct product group Wa (E 7 ) = W (E 7 )
L. The action of W (E 7 ) on Gc can then be extended to an action of the affine Weyl group Wa (E 7 ) depending on an extra parameter z ∈ C by letting γ ∈ L act as the shift τγz u = u − zγ ,

u ∈ Gc .

We suppress the dependence on z whenever its value is implicitly clear from context. We also use a multiplicative version of the W (E 7 )-action on Gc . Consider the action of the group C2 of order two on C8 , with the non-unit element of C2 acting by multiplication by −1 of each coordinate. We define the parameter space Hc for a parameter c ∈ C× = C \ {0} as Hc = {t = (t1 , . . . , t8 ) ∈ C8 |

8 

t j = c2 }/C2 .

j=1

 Note that this is well defined because if t satisfies ti = c2 , then so does −t. We sometimes abuse notation by simply writing t = (t1 , . . . , t8 ) for the element ±t in Hc if no confusion can arise. We view the parameters t ∈ Hexp(c) as the exponential parameters associated to u ∈ Gc . Modding out by the action of the 2-group C2 allows us to put a Wa (E 7 )-action on Hexp(c) , which is compatible to the Wa (E 7 )-action on Gc as defined above. Concretely, consider the surjective map ψc : Gc → Hexp(c) defined by ψc (u) = ±(exp(u 1 ), . . . , exp(u 8 )),

u ∈ Gc .

For u ∈ Gc we have ψc−1 (ψc (u)) = u + 2πi L, where L is the root lattice of R(E 7 ) as defined above. Since L is W (E 7 )-invariant, we can now define the action of Wa (E 7 ) on Hexp(c) by σ ψc (u) = ψc (σ u), σ ∈ Wa (for any auxiliary parameter z ∈ C).

Properties of Generalized Univariate Hypergeometric Functions

43

Regardless of whether we view the action of the affine Weyl group additively or multiplicatively, we will use the abbreviated notations s jk = sα − , w = sβ1234 and jk

z = ταz − throughout the article. Note that s jk ( j = k) acts by interchanging the j th and τ jk jk

k th coordinate. Furthermore, W (E 7 ) is generated by the simple reflections sα (α ∈ 1 ), which are the simple permutations s j, j+1 ( j = 1, . . . , 6) and w. The multiplicative action of w on Hc is explicitly given by w(±t) = ±(st1 , st2 , st3 , st4 , s −1 t5 , s −1 t6 , s −1 t7 , s −1 t8 ), where s 2 = c/t1 t2 t3 t4 = t5 t6 t7 t8 /c. Finally, note that the longest element v of the Weyl group W (E 7 ) acts by multiplication with −1 on the root system R(E 7 ), and hence it 1 1 acts by vu = c/2 − u on Gc and by v(±t) = ±(c 2 /t1 , . . . , c 2 /t8 ) on Hc . 3. The Univariate Elliptic Hypergeometric Function 3.1. The elliptic gamma function. We will use notations which are consistent with [5]. We fix throughout this section two bases p, q ∈ C satisfying | p|, |q| < 1. The q-shifted factorial is defined by ∞    a; q ∞ = (1 − aq j ). j=0

       We write a1 , . . . , am ; q ∞ = mj=1 a j ; q ∞ , (az ±1 ; q ∞ = (az, az −1 ; q ∞ etc. as shorthand notations for products of q-shifted factorials. The renormalized Jacobi thetafunction is defined by   θ (a; q) = a, q/a; q ∞ . The elliptic gamma function [24], defined by the infinite product e (z; p, q) =

∞  1 − z −1 p j+1 q k+1 , 1 − zp j q k

j,k=0

is a meromorphic function in z ∈

C×

= C \ {0} which satisfies the difference equation

e (qz; p, q) = θ (z; p)e (z; p, q),

(3.1)

satisfies the reflection equation e (z; p, q) = 1/ e ( pq/z; p, q), and is symmetric in p and q, e (z; p, q) = e (z; q, p). For products of theta-functions and elliptic gamma functions we use the same shorthand notations as for the q-shifted factorial, e.g. e (a1 , . . . , am ; p, q) =

m 

e (a j ; p, q).

j=1

In this section we call a sequence of points a downward (respectively upward) sequence of points if it is of the form ap j q k (respectively ap − j q −k ) with j, k ∈ Z≥0 for some a ∈ C. Observe that the elliptic gamma function e (az; p, q), considered as a meromorphic function in z, has poles at the upward sequence a −1 p − j q −k ( j, k ∈ Z≥0 ) of points and has zeros at the downward sequence a −1 p j+1 q k+1 ( j, k ∈ Z≥0 ) of points.

44

F. J. van de Bult, E. M. Rains, J. V. Stokman

3.2. Symmetries of the elliptic hypergeometric function. The fundamental starting point of our investigations is Spiridonov’s [31] elliptic analogue of the classical beta integral,  6 ±1  (q; q)∞ ( p; p)∞ j=1 e (t j z ; p, q) dz = e (t j tk ; p, q) (3.2) 2 e (z ±2 ; p, q) 2πi z C 1≤ j
 for generic parameters t ∈ C6 satisfying the balancing condition 6j=1 t j = pq, where the contour C is chosen as a deformation of the positively oriented unit circle T separating the downward sequences t j p Z≥0 q Z≥0 ( j = 1, . . . , 6) of poles of the integrand Z≤0 q Z≤0 ( j = 1, . . . , 6). Note here that the factor from the upward sequences t −1 j p ±2 1/ e (z ; p, q) of the integrand is analytic on C× . Moreover, observe that we can take the positively oriented unit circle T as contour if the parameters satisfy |t j | < 1 ( j = 1, . . . , 6). Several elementary proofs of (3.2) are now known, see e.g. [31, 32 and 21]. We define the integrand Ie (t; z) = Ie (t; z; p, q) for the univariate elliptic hypergeometric function as 8 ±1 j=1 e (t j z ; p, q) , Ie (t; z; p, q) = e (z ±2 ; p, q)  8  where t = (t1 , t2 , . . . , t8 ) ∈ C× . For parameters t ∈ C8 with 8j=1 t j = p 2 q 2 and ti t j ∈ p Z≤0 q Z≤0 for 1 ≤ i, j ≤ 8 (possibly equal), we can define the elliptic hypergeometric function Se (t) = Se (t; p, q) by  dz Se (t; p, q) = , Ie (t; z; p, q) 2πi z C where the contour C is a deformation of T which separates the downward sequences Z≤0 q Z≤0 t j p Z≥0 q Z≥0 ( j = 1, . . . , 8) of poles of Ie (t; ·) from the upward sequences t −1 j p ( j = 1, . . . , 8). If the parameters satisfy |t j | < 1 this contour can again be taken as the positively oriented unit circle T. The elliptichypergeometric function Se extends uniquely to a meromorphic function on {t ∈ C8 : t j = p 2 q 2 }. In fact, for a particular value τ of the parameters for which the integral is not defined, we first deform for t in a small open neighborhood of τ the contour C to include those upward poles which collide at t = τ with downward poles. The resulting expression is an integral which is analytic at an open neighborhood of τ plus a sum of residues depending meromorphically on the parameters t. This expression yields the desired meromorphic extension of Se (t) at τ . For further detailed analysis of meromorphic dependencies of integrals like Se , see e.g. [26 and 21]. Since Ie (t; −z) = Ie (−t; z), where −t = (−t1 , . . . , −t8 ), we have Se (t) = Se (−t), hence we can and will view Se as a meromorphic function Se : H pq → C. Furthermore, Se (t) is the special case II 1BC of Rains’ [21] multivariate elliptic hypergeometric integrals II m BC , and it coincides with Spiridonov’s [33, §5] elliptic analogue V (·) of the Gauss hypergeometric function. Remark 3.1. Note that Se (t; p, q) reduces to the elliptic beta integral (3.2) when e.g. t1 t6 = pq. More generally, for e.g. t1 t6 = p m+1 q n+1 (m, n ∈ Z≥0 ) it follows from [34, Thm. 11] that Se (t; p, q) essentially coincides with the two-index elliptic biorthogonal

Properties of Generalized Univariate Hypergeometric Functions

45

rational function Rnm of Spiridonov [34, App. A], which is the product of two verywell-poised terminating elliptic hypergeometric 12 V11 series (the second one with the role of the bases p and q reversed). Next we determine the explicit W (E 7 )-symmetries of Se (t) in terms of the W (E 7 ) action on t ∈ H pq from Sect. 2. This result was previously obtained by Rains [21] and by Spiridonov [33]. We give here a proof which is similar to Spiridonov’s [33, §5] proof. Theorem 3.2. The elliptic hypergeometric function Se (t) (t ∈ H pq ) is invariant under permutations of (t1 , . . . , t8 ) and it satisfies   e (t j tk ; p, q) e (t j tk ; p, q) (3.3) Se (t; p, q) = Se (wt; p, q) 1≤ j
5≤ j
as meromorphic functions in t ∈ H pq , where (recall) w = sβ1234 . Proof. The permutation symmetry is trivial. To prove (3.3) we first prove it for param eters t ∈ C8 satisfying 8j=1 t j = p 2 q 2 and satisfying the additional restraints |t j | < 1  1 ( j = 1, . . . , 8), |t j | > | pq| 3 ( j = 5, . . . , 8) and | 8j=5 t j | < | pq| (which defines a  non-empty open subset of parameters of {t ∈ C | 8j=1 t j = p 2 q 2 } since | p|, |q| < 1). For these special values of the parameters we consider the double integral 4



j=1 e (t j z

±1 ;

p, q)e (sx ±1 z ±1 ; p, q) e

T2

(z ±2 , x ±2 ;

8

j=5 e (t j s

−1 x ±1 ;

p, q)

p, q) dz d x , 2πi z 2πi x

 where s is chosen to balance both the z as the x integral, so s 2 4j=1 t j = pq =  s −2 8j=5 t j . By the additional parameter restraints we have |s| < 1 and |t j /s| < 1 for j = 5, . . . , 8, hence the integration contour T separates the downward pole sequences of the integrand from the upward ones for both integration variables. Using the elliptic beta integral (3.2) to integrate this double integral either first over the variable z, or first over the variable x, now yields (3.3). Analytic continuation then implies the identity (3.3) as meromorphic functions on H pq .   An interesting equation for Se (t) arises from Theorem 3.2 by considering the action of the longest element v of W (E 7 ), using its decomposition v = s45 s36 s48 s37 s34 s12 ws37 s48 ws35 s46 w

(3.4)

as products of permutations and w. Corollary 3.3. We have Se (t; p, q) = Se (vt; p, q)



e (t j tk ; p, q)

(3.5)

1≤ j
as meromorphic functions in t ∈ H pq . Remark 3.4. Corollary 3.3 is the special case n = m = 1 of [21, Thm. 3.1], see also [33, §5, (iii)] for a proof close to our present derivation.

46

F. J. van de Bult, E. M. Rains, J. V. Stokman

3.3. Contiguous relations. For sake of completeness we recall here Spiridonov’s [33, §6] derivation of certain contiguous relations cq. difference equations for the elliptic hypergeometric function Se (t) (most notably, Spiridonov’s elliptic hypergeometric equation). The starting point is the fundamental theta function identity [5, Exercise 2.16], 1 1 1 θ (ux ±1 , yz ±1 ; p) + θ (uy ±1 , zx ±1 ; p) + θ (uz ±1 , x y ±1 ; p) = 0, y z x

(3.6)

which holds for arbitrary u, x, y, z ∈ C× . For the Wa (E 7 )-action on H pq we take in − log(q) , which multiplies ti by q and divides t j by q. Note that this subsection τi j = τi j  the q-difference operators τi j are already well defined on {t ∈ C8 | 8j=1 t j = p 2 q 2 }. Using the difference equation (3.1) of the elliptic gamma function and using (3.6), we have θ (q −1 t8 t7±1 ; p) θ (t6 t7±1 ; p)

Ie (τ68 t; z) + (t6 ↔ t7 ) = Ie (t; z),

where (t6 ↔ t7 ) means the same term with t6 and t7 interchanged. For generic t ∈ C8  with 8j=1 t j = p 2 q 2 we integrate this equality over z ∈ C, with C a deformation of T which separates the upward and downward pole sequences of all three integrands at the same time. We obtain θ (q −1 t8 t7±1 ; p) θ (t6 t7±1 ; p)

Se (τ68 t) + (t6 ↔ t7 ) = Se (t)

(3.7)

as meromorphic functions in t ∈ H pq . This equation is also the n = 1 instance of [22, Thm. 3.1]. Note that in both terms on the left-hand side the same parameter t8 is divided by q, while two different parameters (t6 and t7 ) are multiplied by q. We can obtain a different equation (i.e. not obtainable by applying an S8 symmetry to (3.7)) by substituting the parameters vt in (3.7), where v ∈ W (E 7 ) is the longest Weyl group element, and by using (3.5). The crux is that τ68 vt = vτ86 t. We obtain 5 5  θ (t7 /qt8 ; p)  θ (t j t6 /q; p)Se (τ86 t) + (t6 ↔ t7 ) = θ (t j t8 ; p)Se (t) θ (t7 /t6 ; p) j=1

(3.8)

j=1

for t ∈ H pq . We arrive at Spiridonov’s [33, §6] elliptic hypergeometric equation for Se (t). Theorem 3.5. ([31]) We have A(t)Se (τ87 t; p, q) + (t7 ↔ t8 ) = B(t)Se (t; p, q)

(3.9)

as meromorphic functions in t ∈ H pq , where A and B are defined by A(t) =

6  1 θ (t j t7 /q; p), t8 θ (t7 /qt8 , t8 /t7 ; p) j=1

B(t) =

5 5   θ (t7 t8 /q; p) θ (t6 /t8 , t6 t8 ; p) θ (t j t6 ; p)− θ(t j t7 /q; p) t6 θ (t7 /qt6 , t8 /qt6 ; p) t6 θ (t7 /qt6 , t7 /qt8 , t8 /t7 ; p) j=1

−

θ (t6 /t7 , t6 t7 ; p) t6 θ (t7 /t8 , t8 /qt6 , t8 /qt7 ; p)

j=1

5  j=1

θ (t j t8 /q; p).

Properties of Generalized Univariate Hypergeometric Functions

47

Remark 3.6. Note that B has an S6 -symmetry in (t1 , t2 , . . . , t6 ) even though it is not directly apparent from its explicit representation. Proof. This follows by taking an appropriate combination of three contiguous relations for Se (t). Specifically, the three contiguous relations are (3.7) and (3.8) with t6 and t8 interchanged, and (3.7) with t7 and t8 interchanged.   By combining contiguous relations for Se (t) and exploring the W (E 7 )-symmetry of Se (t), one can obtain various other contiguous relations involving Se (τx t), Se (τ y t), and Se (τz t) for suitable root lattice vectors x, y, z ∈ L. A detailed analysis of such procedures is undertaken for three term transformation formulas on the trigonometric setting by Lievens and Van der Jeugt [15] (see also Sect. 5). Remark 3.7. Interchanging the role of the bases p and q and using the symmetry of Se (t; p, q) in p and q, we obtain contiguous relations for Se (t; p, q) with respect to multiplicative p-shifts in the parameters. 4. Hyperbolic Hypergeometric Integrals 4.1. The hyperbolic gamma function. We fix throughout this section ω1 , ω2 ∈ C satisfying (ω1 ), (ω2 ) > 0, and we write ω1 + ω2 . ω= 2 Ruijsenaars’ [24] hyperbolic gamma function is defined by     ∞ sin(2zt) z dt G(z; ω1 , ω2 ) = exp i − 2 sinh(ω t) sinh(ω t) ω ω t t 1 2 1 2 0 for z ∈ C satisfying |(z)| < (ω). There exists a unique meromorphic extension of G(ω1 , ω2 ; z) to z ∈ C satisfying G(z; ω1 , ω2 ) = G(z; ω2 , ω1 ), G(z; ω1 , ω2 ) = G(−z; ω1 , ω2 )−1 , G(z + iω1 ; ω1 , ω2 ; ) = −2is((z + iω)/ω2 )G(z; ω1 , ω2 ),

(4.1)

where we use the shorthand notation s(z) = sinh(π z). The second equation here is called the reflection equation. In this section we will omit the ω1 , ω2 dependence of G if no confusion is possible, and we formulate all results only with respect to iω1 -shifts. We use similar notations for products of hyperbolic gamma functions as for q-shifted factorials and elliptic gamma functions, e.g. G(z 1 , . . . , z n ; ω1 , ω2 ) =

n 

G(z j ; ω1 , ω2 ).

j=1

The hyperbolic gamma function G is a degeneration of the elliptic gamma function e ,     π(z − iω) lim e exp(2πir z); exp(−2π ω1r ), exp(−2π ω2 r ) exp r 0 6ir ω1 ω2 (4.2) = G(z − iω; ω1 , ω2 ) for ω1 , ω2 > 0, see [24, Prop. III.12].

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In this section we call a sequence of points a downward (respectively upward) sequence of points if it is of the form a + iZ≤0 ω1 + iZ≤0 ω2 (respectively a + iZ≥0 ω1 + iZ≥0 ω2 ) for some a ∈ C. Recall from [24] that the hyperbolic gamma function G(ω1 , ω2 ; z), viewed as a meromorphic function in z ∈ C, has poles at the downward sequence −iω + iZ≤0 ω1 + iZ≤0 ω2 of points and has zeros at the upward sequence iω + iZ≥0 ω1 + iZ≥0 ω2 of points. The pole of G(z; ω1 , ω2 ) at z = −iω is simple and   i √ ω1 ω2 . Res G(z; ω1 , ω2 ) = 2π

z=−iω

(4.3)

All contours in this section will be chosen as deformations of the real line R separating the upward pole sequences of the integrand from the downward ones. We will also need to know the asymptotic behavior of G(z) as (z) → ±∞ (uniformly for (z) in compacta of R). For our purposes it is sufficient to know that for any a, b ∈ C we have     πi z G(z − a; ω1 , ω2 ) πi exp lim (b − a) = exp (b2 − a 2 ) , (4.4) (z)→∞ G(z − b; ω1 , ω2 ) ω1 ω2 2ω1 ω2 where the corresponding o((z))-tail as (z) → ∞ can be estimated uniformly for (z) in compacta of R, and that for periods satisfying ω1 ω2 ∈ R>0 ,   u

|G(u + x; ω1 , ω2 )| ≤ M exp π  |x| , ∀x ∈ R (4.5) ω1 ω2 for some constant M > 0, provided that the line u + R does not hit a pole of G. See [26, App. A] for details and for more precise asymptotic estimates. 4.2. Symmetries of the hyperbolic hypergeometric function. The univariate hyperbolic beta integral [37, (1.10)] is   √ G(iω ± 2z; ω1 , ω2 ) dz = 2 ω1 ω2 G(iω − u j − u k ; ω1 , ω2 ) (4.6) 6 C j=1 G(u j ± z; ω1 , ω2 ) 1≤ j 0 due to the imposed conditions (ω j ) > 0 on the periods ω j ( j = 1, 2). We define now the integrand of the hyperbolic hypergeometric function Ih (u; z) = Ih (u; z; ω1 , ω2 ) as G(iω ± 2z; ω1 , ω2 ) Ih (u; z; ω1 , ω2 ) = 8 j=1 G(u j ± z; ω1 , ω2 ) for parameters u ∈ C8 . The hyperbolic hypergeometric function Sh (u) = Sh (u; ω1 , ω2 ) is defined by  Sh (u; ω1 , ω2 ) = Ih (u; z; ω1 , ω2 )dz C

Properties of Generalized Univariate Hypergeometric Functions

49

for generic parameters u ∈ G2iω (see Sect. 2 for the definition of G2iω ). The asymptotic behaviour of Ih (u; z) at z = ±∞ is again O(exp(−4π |z|ω/ω1 ω2 )), so the integral absolutely converges. It follows from (4.3) and the analytic difference equations for the hyperbolic gamma function that Sh (u) has a unique meromorphic extension to u ∈ G2iω , cf. the analysis for the elliptic hypergeometric function Se (t). We thus can and will view Sh (u) as a meromorphic function in u ∈ G2iω . Note furthermore that the real line can be chosen as an integration contour in the definition of Sh (u) if u ∈ G2iω satisfies (u j − iω) < 0 for all j. The hyperbolic hypergeometric function Sh (u) (u ∈ G2iω ) coincides with Spiridonov’s [33, §5] hyperbolic analogue s(·) of the Gauss hypergeometric function. Using (4.2) and the reflection equation of G, we can obtain the hyperbolic hypergeometric function Sh (vu; ω1 , ω2 ) = Sh (iω − u 1 , . . . , iω − u 8 ; ω1 , ω2 ) (u ∈ G2iω ) formally as the degeneration r ↓ 0 of the elliptic hypergeometric function Se (t; p, q) with p = exp(−2π ω1r ), q = exp(−2π ω2 r ) and t = ψ2iω (2πir u) ∈ Hexp(−4πr ω) = H pq . This degeneration, which turns out to preserve the W (E 7 )-symmetry (see below), can be proven rigorously, see [23]. This entails in particular a derivation of the hyperbolic beta integral (4.6) as a rigorous degeneration of the elliptic beta integral (3.2) (see [37, §5.4] for the formal analysis). Next we give the explicit W (E 7 ) symmetries of Sh (u) in terms of the W (E 7 ) action on u ∈ G2iω from Sect. 2. Theorem 4.1. The hyperbolic hypergeometric function Sh (u) (u ∈ G2iω ) is invariant under permutations of (u 1 , . . . , u 8 ) and it satisfies  Sh (u; ω1 , ω2 ) = Sh (wu; ω1 , ω2 ) G(iω − u j − u k ; ω1 , ω2 ) ×



1≤ j
G(iω − u j − u k ; ω1 , ω2 )

5≤ j
as meromorphic functions in u ∈ G2iω . Proof. The proof is analogous to the proof in the elliptic case (Theorem 3.2). For the w-symmetry we consider for suitable u ∈ G2iω the double integral  G(iω ± 2z, iω ± 2x) dzd x 4 8 R2 j=1 G(u j ± z)G(iω + s ± x ± z) k=5 G(u k − s ± x) with s = iω − 21 (u 1 + u 2 + u 3 + u 4 ) = −iω + 21 (u 5 + u 6 + u 7 + u 8 ). We impose the conditions (s) < 0 and   u j − iω) < 0 ( j = 1, . . . , 4), (u k − iω) < (s) (k = 5, . . . , 8) (4.7) on u ∈ G2iω to ensure that the upward and downward pole sequences of the integrand of the double integral are separated by R. Next we show that the parameter restraints  s   ω  < <0 (4.8) − ω1 ω2 ω1 ω2 on u ∈ G2iω suffice to ensure absolute convergence of the double integral. Using the reflection equation and asymptotics (4.5) of G we obtain the estimate   s + iω   1 ≤ M exp −2π  |z + x| + |z − x| , ∀ (x, z) ∈ R2 |G(iω + s ± x ± z)| ω1 ω2

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F. J. van de Bult, E. M. Rains, J. V. Stokman

−1 for some constant M > 0. It follows that the  factor G(iω + s ± x ± z) of the   integrand is absolutely and uniformly bounded if  (iω + s)/ω ω ) ≥ 0, i.e. if  s/ω1 ω2 ≥ 1 2   − ω/ω1 ω2 . The asymptotic behaviour of the remaining factors of the integrand (which breaks up in factors only depending on x or on z) can easily be determined by (4.5), leading finally to the parameter restraints (4.8) for the absolute convergence of the double integral. It is easy to verify that the parameter subset of G2iω defined by the additional restraints (s) < 0, (4.7) and (4.8) is non-empty (by e.g. constructing parameters u ∈ G2iω with small associated balancing parameter s). Using Fubini’s Theorem and the hyperbolic beta integral (4.6), we now reduce the double integral to a single integral by either evaluating the integral over x, or by evaluating the integral over z. Using furthermore that

wu = (u 1 + s, u 2 + s, u 3 + s, u 4 + s, u 5 − s, u 6 − s, u 7 − s, u 8 − s) for u ∈ G2iω , it follows that the resulting identity is the desired w-symmetry of Sh for the restricted parameter domain. Analytic continuation now completes the proof.   The symmetry of Sh (u) (u ∈ G2iω ) with respect to the action of the longest Weyl group element v ∈ W (E 7 ) is as follows. Corollary 4.2. The hyperbolic hypergeometric function Sh satisfies  G(iω − u j − u k ; ω1 , ω2 ) Sh (u; ω1 , ω2 ) = Sh (vu; ω1 , ω2 )

(4.9)

1≤ j
as meromorphic functions in u ∈ G2iω . Proof. This follows from Theorem 4.1 and (3.4).

 

4.3. Contiguous relations. Contiguous relations for the hyperbolic hypergeometric function Sh can be derived in nearly exactly the same manner as we did for the elliptic hypergeometric function Se (see Sect. 3.3 and [33, §6]). We therefore only indicate the main steps. Using the p = 0 case of (3.6) we have s(x ± v)s(y ± z) + s(x ± y)s(z ± v) + s(x ± z)s(v ± y) = 0, 1 where s(x ± v) = s(x + v)s(x − v). In this subsection we write τ jk = τ iω jk (1 ≤ j = k ≤ 8), which acts on u ∈ G2iω by subtracting iω1 from u j and adding iω1 to u k . We now obtain in analogy to the elliptic case the difference equation

s((u 8 + iω ± (u 7 − iω))/ω2 ) Sh (τ68 u) + (u 6 ↔ u 7 ) = Sh (u) s((u 6 − iω ± (u 7 − iω))/ω2 ) as meromorphic functions in u ∈ G2iω . Using (4.9) we subsequently obtain 5 s((u 7 − u 8 + 2iω)/ω2 )  s((u j + u 6 )/ω2 )Sh (τ86 u) + (u 6 ↔ u 7 ) s((u 7 − u 6 )/ω2 ) j=1

=

5  j=1

s((u j + u 8 − 2iω)/ω2 )Sh (u)

Properties of Generalized Univariate Hypergeometric Functions

51

as meromorphic functions in u ∈ G2iω . Combining these contiguous relations and simplifying we obtain A(u)Sh (τ87 u) − (u 7 ↔ u 8 ) = B(u)Sh (u),

u ∈ G2iω ,

(4.10)

where A(u) = s((2iω − u 7 + u 8 )/ω2 )

6 

s((u j + u 7 )/ω2 ),

j=1

B(u) =

s((u 8 ± u 7 )/ω2 )s((2iω + u 8 − u 7 )/ω2 )s((2iω − u 8 + u 7 )/ω2 ) s((2iω + u 8 − u 6 )/ω2 )s((2iω + u 7 − u 6 )/ω2 ) ×

5 

s((−2iω + u j + u 6 )/ω2 )

j=1

− ×

s((2iω − u 8 + u 7 )/ω2 )s((u 7 − u 6 )/ω2 )s((−2iω + u 6 + u 7 )/ω2 ) s((2iω + u 8 − u 6 )/ω2 ) 5 

s((u j + u 8 )/ω2 )

j=1

+

s((2iω + u 8 − u 7 )/ω2 )s((u 8 − u 6 )/ω2 )s((−2iω + u 6 + u 8 )/ω2 ) s((2iω + u 7 − u 6 )/ω2 )

×

5 

s((u j + u 7 )/ω2 ).

j=1

This leads to the following theorem. Theorem 4.3. We have A(u)(Sh (τ87 u) − Sh (u)) − (u 7 ↔ u 8 ) = B2 (u)Sh (u)

(4.11)

as meromorphic functions in u ∈ G2iω , where A(u) is as above and with B2 (u) defined by B2 (u) =

s((u 7 ± u 8 )/ω2 )s((u 7 − u 8 ± 2iω)/ω2 ) 4 ⎞ ⎛ 8 6

s(2(iω + u j )/ω2 ) − s(2(iω − u j )/ω2 )⎠ . ×⎝ j=7

(4.12)

j=1

Proof. It follows from (4.10) that (4.11) holds with B2 (u) = B(u) − A(u) − A(s78 u). The alternative expression (4.12) for B2 was obtained by Mathematica. Observe though that part of the zero locus of B2 (u) (u ∈ G2iω ) can be predicted in advance. Indeed, the left-hand side of (4.11) vanishes if u 7 = u 8 (both terms then cancel each other), and it vanishes if u 7 = u 8 ± iω (one term vanishes due to an s-factor, while the other term vanishes since either Sh (τ87 u) = Sh (u) or Sh (τ78 u) = Sh (u)). The zero of B2 (u) at u 7 = −u 8 can be predicted from the fact that all hyperbolic hypergeometric functions Sh in (4.11) can be evaluated for u 7 = −u 8 using the hyperbolic beta integral (4.6).  

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F. J. van de Bult, E. M. Rains, J. V. Stokman

4.4. The degeneration to the hyperbolic Barnes integral. In this subsection we degenerate the hyperbolic hypergeometric function Sh (u) (u ∈ G2iω ) along the highest root β1278 of R(E 7 ) with respect to the basis 2 of R(E 7 ) (see (2.4) for the associated Dynkin diagram). The resulting degenerate integral Bh (u) thus inherits symmetries with respect to the standard maximal parabolic subgroup W2 (D6 ) := W (E 7 )β1278 ⊂ W (E 7 ), which is isomorphic to the Weyl group of type D6 and is generated by the simple reflec− tions sα (α ∈ 2 \ {α23 }). The corresponding Dynkin diagram is − − − − − α18 α34 α23 α87 β5678 α45 • • • • ◦ • − • α56 we define Bh (u) = Bh (u; ω1 , ω2 ) by

Concretely, for generic parameters u ∈ G2iω 6  j=3 G(z − u j ; ω1 , ω2 )  dz. Bh (u; ω1 , ω2 ) = 2 C j=1,2,7,8 G(z + u j ; ω1 , ω2 )

This integral converges absolutely since the asymptotic behaviour of the integrand at z = ±∞ is exp(−4π ω|z|/ω1 ω2 ). We may take the real line as integration contour if u ∈ G2iω satisfies (u j − iω) < 0 for all j. Observe that the integral Bh (u) has a unique meromorphic extension to u ∈ G2iω . We call Bh (u) the hyperbolic Barnes integral since it is essentially Ruijsenaars’ [26] hyperbolic generalization of the Barnes integral representation of the Gauss hypergeometric function, see Subsect. 4.6. Remark 4.4. The parameter space of the hyperbolic Barnes integral Bh is in fact the quotient space G2iω /Cβ1278 . Indeed, for ξ ∈ C we have Bh (u + ξβ1278 ) = Bh (u) as meromorphic functions in u ∈ G2iω , which follows by an easy application of (4.4) and Cauchy’s Theorem. Proposition 4.5. For u ∈ G2iω satisfying (u j − iω) < 0 ( j = 1, . . . , 8) we have  



6  2πr ω πi 2 2 uj − uj lim Sh (u − rβ1278 ) exp ω1 ω2 exp 2ω1 ω2 = Bh (u). r →∞

j=1,2,7,8

j=3

Proof. The conditions on the parameters u ∈ G2iω allow us to choose the real line as an integration contour in the integral expression of Sh (u − rβ1278 ) (r ∈ R) as well as in the integral expression of Bh (u). Using that the integrand Ih (u; z) of Sh (u) is even in x, using the reflection equation for the hyperbolic gamma function, and by a change of integration variable, we have  ∞ 2πr ω 2πr ω G(iω ± 2z) Sh (u − rβ1278 )e ω1 ω2 = e ω1 ω2 dz  6 r r −∞ j=3 G(u j + 2 ± z) j=1,2,7,8 G(u j − 2 ± z)  ∞ 2πr ω G(iω ± 2z) ω ω 1 2 = 2e dz  6 r r 0 j=3 G(u j + 2 ± z) j=1,2,7,8 G(u j − 2 ± z)  ∞ =2 k1 (2z + r )k2 (z + r )L(z)dz, − r2

Properties of Generalized Univariate Hypergeometric Functions

53

where 6 j=3

L(z) = 

G(z − u j )

j=1,2,7,8

G(z + u j )

,

   G(z + iω) − ω2π ωωz e 1 2 = 1 − e−2π z/ω1 1 − e−2π z/ω2 , G(z − iω)  j=1,2,7,8 G(z − u j ) ω4π ωωz k2 (z) = 6 e 1 2. j=3 G(z + u j ) k1 (z) =

Here the second expression of k1 follows from the analytic difference equations satisfied by G. The pointwise limits of k1 and k2 are lim k1 (z) = 1

z→∞

πi

lim k2 (z) = e 2ω1 ω2

(

6

2  2 j=3 u j − j=1,2,7,8 u j )

z→∞

.

Moreover, observe that k1 (z) is uniformly bounded for z ∈ R≥0 by 4, and that k2 (z), being a continuous function on R≥0 with finite limit at infinity, is also uniformly bounded for z ∈ R≥0 . Denote by χ(−r/2,∞) (z) the indicator function of the interval (−r/2, ∞). By Lebesgue’s theorem of dominated convergence we now conclude that lim Sh (u − rβ1278 )e

r →∞

2πr ω ω1 ω2

 = 2 lim

∞

r →∞ − r 2  ∞

k1 (2z + r )k2 (z + r )L(z)dz

lim χ(−r/2,∞) (z)k1 (2z + s)k2 (z + s)L(z)dz  ∞  u 2 − j=1,2,7,8 u 2j ) = 2e 2ω1 ω2 j=3 j L(z)dz =2

=e as desired.

−∞ r →∞ 6 πi

6 πi 2  2 2ω1 ω2 ( j=3 u j − j=1,2,7,8 u j )

−∞

Bh (u),

 

In the following corollary we use Proposition 4.5 to degenerate the hyperbolic beta integral (4.6). The resulting integral evaluation formula is an hyperbolic analogue of the nonterminating Saalschütz formula [5, (2.10.12)], see Subsect. 5.4. Corollary 4.6. For generic u ∈ C6 satisfying 

6

j=1 u j

= 4iω we have

 √ G(z − u 4 , z − u 5 , z − u 6 ) dz = ω1 ω2 G(iω − u j − u k ). C G(z + u 1 , z + u 2 , z + u 3 ) j=1 k=4 3

6

(4.13)

Proof. Substitute the parameters u  = (u 1 , u 2 , u 4 , u 5 , u 6 , 0, u 3 , 0) in Proposition 4.5  with u j ∈ C satisfying (u j − iω) < 0 and 6j=1 u j = 4iω. Then Bh (u  ) is the lefthand side of (4.13), multiplied by 2. On the other hand, by Proposition 4.5 and (4.6) we have

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F. J. van de Bult, E. M. Rains, J. V. Stokman

⎛

⎞ 3 6

2πr ω πi Bh (u  ) = lim Sh (u  − rβ1278 ) exp ⎝ + ( u 2j − u 2j )⎠ r →∞ ω1 ω2 2ω1 ω2 j=1

√ = 2 ω1 ω2

3  6 

j=4

G(iω − u j − u k )

j=1 k=4

⎛ ⎛ ⎞⎞ 3 6

G(iω − u − u + r ) j k 2πr ω πi 1≤ j
j=1

√ = 2 ω1 ω2

3  6 

j=4

G(iω − u j − u k ),

j=1 k=4

where the last equality follows from a straightforward but tedious computation using  (4.4). The result for arbitrary generic parameters u ∈ C6 satisfying 6j=1 u j = 4iω now follows by analytic continuation.   Next we determine the explicit W2 (D6 )-symmetries of Bh (u). Proposition 4.7. The hyperbolic Barnes integral Bh (u) (u ∈ G2iω ) is invariant under permutations of (u 1 , u 2 , u 7 , u 8 ) and of (u 3 , u 4 , u 5 , u 6 ) and it satisfies Bh (u) = Bh (wu)

 

G(iω − u j − u k )

j=1,2 k=3,4

 

G(iω − u j − u k ) (4.14)

j=5,6 k=7,8

as meromorphic functions in u ∈ G2iω . Proof. The permutation symmetry is trivial. The symmetry (4.14) can be proven by degenerating the corresponding symmetry of Sh , see Theorem 4.1. We prove here the w-symmetry by considering the double integral  R2

G(z − u 3 , z − u 4 , x − u 5 + s, x − u 6 + s, z − x − iω − s) dzd x G(z + u 1 , z + u 2 , x + u 7 − s, x + u 8 − s, z − x + iω + s)

with s = iω − 21 (u 1 + u 2 + u 3 + u 4 ) = −iω + 21 (u 5 + u 6 + u 7 + u 8 ), where we impose on u ∈ G2iω the additional conditions  ω   s   ω  − < < ω1 ω2 ω1 ω2 ω1 ω2

(4.15)

to ensure the absolute convergence of the double integral (this condition is milder than the corresponding condition (4.8) for Sh due to the missing factors G(iω ± 2z, iω ± 2x) in the numerator of the integrand), and the conditions (4.7) to ensure that the upward and downward pole sequences are separated by R. Using Fubini’s Theorem and the hyperbolic Saalschütz summation (4.13), similarly as in the proof of Theorem 4.1, yields (4.14).  

Properties of Generalized Univariate Hypergeometric Functions

55

4.5. The degeneration to the hyperbolic Euler integral. In this subsection we degenerate − the hyperbolic hypergeometric function Sh (u) (u ∈ G2iω ) along the highest root α78 of R(E 7 ) with respect to the basis 1 of R(E 7 ) (see (2.3) for the associated Dynkin diagram). The resulting degenerate integral E h (u) thus inherits symmetries with respect to the standard maximal parabolic subgroup W1 (D6 ) := W (E 7 )α − ⊂ W (E 7 ), 78

which is isomorphic to the Weyl group of type D6 and is generated by the simple reflec− tions sα (α ∈ 1 \ {α76 }). The corresponding Dynkin diagram is − α21 •

− α32 •

− α43 •

− α54 •

− α65 •

− α76 ◦

• β1234

  By the conditions (ω j ) > 0 on the periods ω j ( j = 1, 2) we have that  ω1ωω2 > 0. For generic parameters u = (u 1 , . . . , u 6 ) ∈ C6 satisfying 

6   2ω   1
uj >  ω1 ω2 ω1 ω2

(4.16)

j=1

we now define E h (u) = E h (u; ω1 , ω2 ) by  E h (u; ω1 , ω2 ) =

G(iω ± 2z; ω1 , ω2 ) dz. 6 C j=1 G(u j ± z; ω1 , ω2 )

It follows from the asymptotics (4.4) of the hyperbolic gamma function that the condition (4.16) on the parameters ensures the absolute convergence of E h (u). Furthermore, E h (u) admits a unique meromorphic continuation to parameters u ∈ C6 satisfying (4.16) (in fact, it will be shown later that E h (u) has a unique meromorphic continuation to u ∈ C6 by relating E h to the hyperbolic Barnes integral Bh ). Observe furthermore that E h (u) reduces to the hyperbolic beta integral (4.6) when the parameters u ∈ C6 satisfy the  balancing condition 6j=1 u j = 4iω. We call E h (u) the hyperbolic Euler integral since its trigonometric analogue is a natural generalization of the Euler integral representation of the Gauss hypergeometric function, see Subsect. 5.4 and [5, §6.3].  Proposition  4.8. For u ∈ G2iω satisfying (u j − iω) < 0 ( j = 1, . . . , 8),  (u 7 + u 8 )/ω1 ω2 ≥ 0 and (4.16), we have   πi − lim Sh (u − r α78 ) exp − (u 7 + u 8 )(2r − u 7 + u 8 ) = E h (u 1 , u 2 , u 3 , u 4 , u 5 , u 6 ). r →∞ ω1 ω2 (4.17)

Remark 4.9. Proposition 4.8 is trivial when u 7 = −u 8 due to the reflection equation for G. The resulting limit is the hyperbolic beta integral (4.6) (since the balancing condition  reduces to 6j=1 u j = 4iω).

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F. J. van de Bult, E. M. Rains, J. V. Stokman

Proof. The assumptions on the parameters ensure that the integration contours in Sh and E h can be chosen as the real line. We denote the integrand of the Euler integral by G(iw ± 2z) J (z) = 6 , j=1 G(u j ± z) and we set

  πi z G(z − u 7 ) exp − (u 7 + u 8 ) . H (z) = G(z + u 8 ) ω1 ω2

This allows us to write Ih (u

− − r α78 ; z) exp

  2πir − (u 7 + u 8 ) = J (z)H (r + z)H (r − z), ω1 ω2

where (recall) Ih (u; z) is the integrand of the hyperbolic hypergeometric function Sh (u). Observe that H is a continuous function on R satisfying   πi 2 2 lim H (z) = exp (u − u 7 ) , z→∞ 2ω1 ω2 8     2πi z πi (u 7 + u 8 ) = exp (u 27 − u 28 ) lim H (z) exp z→−∞ ω1 ω2 2ω1 ω2 by (4.4) and by the reflection equation for the hyperbolic gamma function. Moreover,  H is uniformly bounded on R in view of the parameter condition  u 7 + u 8 /ω1 ω2 ≥ 0 on the parameters, and we have   πi 2 2 lim H (r + z)H (r − z) = exp (u − u 7 ) r →∞ ω1 ω2 8 for fixed z ∈ R. By Lebesgue’s theorem of dominated convergence we conclude that     2πir − exp − lim Sh u − r α78 (u 7 + u 8 ) = lim J (z)H (r + z)H (r − z)dz r →∞ r →∞ R ω1 ω2  = J (z) lim H (r + z)H (r − z)dz r →∞ R   πi 2 2 (u − u 7 ) , = E h (u 1 , . . . , u 6 ) exp ω1 ω2 8 as desired.

 

As a corollary of Proposition 4.8 we obtain the hyperbolic beta integral of AskeyWilson type, initially independently proved in [29] and in [37].    Corollary 4.10. For generic u = (u 1 , u 2 , u 3 , u 4 ) ∈ C4 satisfying  ω11ω2 4j=1 u j >  2ω   ω1 ω2 we have  √ G(iω ± 2z) dz = 2 ω1 ω2 G(u 1 + u 2 + u 3 + u 4 − 3iω) 4 C j=1 G(u j ± z)  × G(iω − u j − u k ). (4.18) 1≤ j
Properties of Generalized Univariate Hypergeometric Functions

57

Proof. Apply Proposition 4.8 under the additional condition u 5 = −u 6 on the associated parameters u ∈ G2iω . Using the reflection equation for the hyperbolic gamma function we see that the right-hand side of (4.17) becomes the hyperbolic Askey-Wilson integral. − On the other hand, Sh (u − r α78 ) can be evaluated by the hyperbolic beta integral (4.6), resulting in   √ G(iω ± 2z) dz = 2 ω1 ω2 G(iω − u 7 − u 8 ) G(iω − u j − u k ) 4 C j=1 G(u j ± z) 1≤ j
4  G(iω − u j − u 7 + r ) G(−iω + u j + u 8 + r ) j=1

√

= 2 ω1 ω2 G(u 1 + u 2 + u 3 +u 4 − 3iω)



G(iω − u j − u k ),

1≤ j
where we used the balancing condition on u and the asymptotics (4.4) of the hyperbolic gamma function to obtain the last equality. The additional parameter restrictions which we have imposed in order to be able to apply Proposition 4.8 can now be removed by analytic continuation.   Since both the Euler and Barnes integrals are limits of the hyperbolic hypergeometric function we can connect them according to the following theorem. Theorem 4.11. We have Bh (u) = E h (u 2 − s, u 7 − s, u 8 − s, u 3 + s, u 4 + s, u 5 + s) ×

5 

G(iω − u 1 − u j )

j=3



G(iω − u 6 − u j )

(4.19)

j=2,7,8

    as meromorphic functions in {u ∈ G2iω |  (u 1 + u 6 )/ω1 ω2 <  2ω/ω1 ω2 }, where s=

1 1 (u 2 + u 6 + u 7 + u 8 ) − iω = iω − (u 1 + u 3 + u 4 + u 5 ). 2 2

Proof. This theorem can be proved by degenerating a suitable E 7 -symmetry of Sh using Proposition 4.5 and Proposition 4.8. We prove the theorem here directly by analyzing the double integral   G(iω ± 2z) 5j=3 G(x − u j ) 1  dzd x √ ω1 ω2 R2 G(iω + s + x ± z)G(x + u 1 ) j=2,7,8 G(u j − s ± z) for (ω1 ), (ω2 ) > 0, u ∈ G2iω and s = 21 (u 2 + u 6 + u 7 + u 8 ) − iω, where we impose the additional parameter restraints ω1 ω2 ∈ R>0 and |(s)| < (ω),

(u 6 + s) < 0

58

F. J. van de Bult, E. M. Rains, J. V. Stokman

to ensure absolute convergence of the double integral (which follows from a straightforward analysis of the integrand using (4.4) and (4.5), cf. the proof of Theorem 4.1), and (s) < 0,

(iω − u j ) > 0 ( j = 1, 3, 4, 5),

(iω − u k + s) > 0 (k = 2, 7, 8)

to ensure pole sequence separation by the integration contours. Note that these parameter restraints imply the parameter condition (u 1 + u 6 ) < 2(ω) needed for the hyperbolic Euler integral in the right-hand side of (4.19) to be defined. Integrating the double integral first over x and using the integral evaluation formula (4.13) of Barnes type, we obtain an expression of the double integral as a multiple of E h (u 2 − s, u 3 + s, u 4 + s, u 5 + s, u 7 − s, u 8 − s). Integrating first over z and using the hyperbolic Askey-Wilson integral (4.18), we obtain an expression of the double integral as a multiple of Bh (u). The resulting identity is (4.19) for a restricted parameter domain. Analytic continuation now completes the proof.   Corollary 4.12. The hyperbolic Euler integral E h (u) has a unique meromorphic continuation to u ∈ C6 (which we also denote by E h (u)). From the degeneration from Sh to E h (see Proposition 4.8) it is natural to interpret − the parameter domain C6 as G2iω /Cα78 via the bijection 6
  − C6  u → u 1 , . . . , u 6 , 2iω − u j , 0 + Cα78 .

(4.20)

j=1

We use this identification to transfer the natural W1 (D6 ) = W (E 7 )α − -action on G2iω / 78

− to the parameter space C6 of the hyperbolic Euler integral. It is generated by perCα78 mutations of (u 1 , . . . , u 6 ) and by the action of w ∈ W1 (D6 ), which is given explicitly by  w u) = (u 1 + s, u 2 + s, u 3 + s, u 4 + s, u 5 − s, u 6 − s), u ∈ C6 , (4.21)

where s = iω − 21 (u 1 +u 2 +u 3 +u 4 ). An interesting feature of W1 (D6 )-symmetries of the hyperbolic Euler integral (to be derived in Corollary 4.14), is the fact that the nontrivial w-symmetry of E h generalizes to the following explicit integral transformation for E h . Proposition 4.13. For periods ω1 , ω2 ∈ C with (ω1 ), (ω2 ) > 0 and ω1 ω2 ∈ R>0 and for parameters s ∈ C and u = (u 1 , . . . , u 6 ) ∈ C6 satisfying 

 ω   s  > − , ω1 ω2 ω1 ω2



 u 1 + u 2 + u 3 + u 4   u 5 + u 6 − 2s   2ω  , > ω1 ω2 ω1 ω2 ω1 ω2 (4.22)

and (u j − iω) < 0 ( j = 1, . . . , 4),

(u k − iω) < (s) < 0 (k = 5, 6), (4.23)

we have  R

E h (u 1 , u 2 , u 3 , u 4 , iω + s + x, iω + s − x)

√ = 2 ω1 ω2

G(iω ± 2x) dx G(u 5 − s ± x, u 6 − s ± x)

G(iω − u 5 − u 6 + 2s) E h (u). G(iω − u 5 − u 6 , iω + 2s)

(4.24)

Properties of Generalized Univariate Hypergeometric Functions

59

Proof. Observe that the requirement ω1 ω2 ∈ R>0 ensures the existence of parameters u ∈ C6 and s ∈ C satisfying the restraints (4.22) and (4.23). Furthermore, (4.22) ensures that 

6 4  1 
  1 
  2ω  , uj , u j + 2iω + 2s >  ω1 ω2 ω1 ω2 ω1 ω2 j=1

j=1

hence both hyperbolic Euler integrals in (4.24) are defined. We derive the integral transformation (4.24) by considering the double integral  G(iω ± 2z, iω ± 2x) dzd x, 4 6 R2 G(iω + s ± x ± z) j=1 G(u j ± z) k=5 G(u k − s ± x) which absolutely converges by (4.22). Integrating the double integral first over x using the hyperbolic Askey-Wilson integral (4.18) yields the right-hand side of (4.24). Integrating first over z results in the left-hand side of (4.24).   Corollary 4.14. The hyperbolic Euler integral E h (u) (u ∈ C6 ) is symmetric in (u 1 , . . . , u 6 ) and it satisfies E h (u) = E h (wu)G(iω − u 5 − u 6 )G(

6




u j − 3iω)

j=1

G(iω − u j − u k )

1≤ j
(4.25) as meromorphic functions in u ∈ C6 . Proof. The permutation symmetry is trivial. For (4.25) we apply Proposition 4.13 with s = iω − 21 (u 1 + u 2 + u 3 + u 4 ). The hyperbolic Euler integral in the left-hand side of the integral transformation (4.24) can now be evaluated by the hyperbolic beta integral (4.6). The remaining integral is an explicit multiple of E h (wu). The resulting identity yields (4.25) for a restricted parameter domain. Analytic continuation completes the proof.   Remark 4.15. The w-symmetry (4.25) of E h can also be proved by degenerating the w-symmetry of Sh , or by relating (4.25) to a W2 (D6 )-symmetry of Bh using Theorem 4.11. The longest Weyl group element v1 ∈ W1 (D6 ) and the longest Weyl group element − v ∈ W (E 7 ) have the same action on G2iω /Cα78 . Consequently, under the identification 6 (4.20), v1 acts on C by v1 (u) = (iω − u 1 , . . . , iω − u 6 ),

u ∈ C6 .

Corollary 4.16. The symmetry of the hyperbolic Euler integral E h (u) with respect to the longest Weyl group element v1 ∈ W1 (D6 ) is E h (u) = E h (v1 u)G(−3iω +

6
j=1

as meromorphic functions in u ∈ C6 .

u j)

 1≤ j
G(iω − u j − u k )

60

F. J. van de Bult, E. M. Rains, J. V. Stokman

Proof. For parameters u ∈ G2iω such that both u and vu satisfy the parameter restraints of Proposition 4.8, we degenerate the v-symmetry (4.9) of Sh using (4.17). Analytic continuation completes the proof.   The contiguous relations for Sh degenerate to the following contiguous relations for Eh . Lemma 4.17. We have 4 j=1 s((u j + u 5 )/ω2 ) s((u 5 − u 6 + 2iω)/ω2 )

iω1 (E h (τ65 u) − E h (u)) − (u 5 ↔ u 6 )

= s((u 5 ± u 6 )/ω2 )s((2iω −

6


u j )/ω2 )E h (u)

(4.26)

j=1

as meromorphic functions in u ∈ C6 . Proof. Use Proposition 4.8 to degenerate the contiguous relation (4.11) for the hyperbolic hypergeometric function Sh to E h .   4.6. Ruijsenaars’ R-function. Motivated by the theory of quantum integrable, relativistic particle systems on the line, Ruijsenaars [26, 28, 29] introduced and studied a generalized hypergeometric R-function R, which is essentially the hyperbolic Barnes integral Bh (u) with respect to a suitable reparametrization (and re-interpretation) of the parameters u ∈ G2iω . The new parameters will be denoted by (γ , x, λ) ∈ C6 with γ = (γ1 , . . . , γ4 )T ∈ C4 , where x (respectively λ) is viewed as the geometric (respectively spectral) parameter, while the four parameters γ j are viewed as coupling constants. As a consequence of the results derived in the previous subsections, we will re-derive many of the properties of the generalized hypergeometric R-function, and we obtain a new integral representation of R in terms of the hyperbolic Euler integral E h . Set N (γ ) =

3 

G(iγ0 + iγ j + iω).

j=1

Ruijsenaars’ [26] generalized hypergeometric function R(γ ; x, λ; ω1 , ω2 ) = R(γ , λ) is defined by 1 N (γ ) R(γ ; x, λ) = √ Bh (u), 2 ω1 ω2 G(iγ0 ± x, i γˆ0 ± λ)

(4.27)

where u ∈ G2iω /Cβ1278 with u 1 = iω, u 5 = −i γˆ0 + λ,

u 2 = iω + iγ0 + iγ1 , u 3 = −iγ0 + x, u 6 = −i γˆ0 − λ, u 7 = iω + iγ0 + iγ2 ,

u 4 = −iγ0 − x, u 8 = iω + iγ0 + iγ3 . (4.28)

Note that R(γ ; x, λ; ω1 , ω2 ) is invariant under permuting the role of the two periods ω1 and ω2 . Observe furthermore that the map (γ , x, λ) → u + Cβ1278 , with u given by ∼ (4.28), defines a bijection C6 → G2iω /Cβ1278 .

Properties of Generalized Univariate Hypergeometric Functions

We define the dual parameters γˆ by ⎛ 1 1 ⎜1 γˆ = ⎝ 2 1 1

1 1 −1 −1

1 −1 1 −1

⎞ 1 −1 ⎟ γ. −1 ⎠ 1

61

(4.29)

We will need the following auxiliary function: 3 1 c(γ ; y) = G(2y+iω) j=0 G(y − iγ j ). The following proposition was derived by different methods in [28]. Proposition 4.18. R is even in x and λ and self-dual, i.e. R(γ ; x, λ) = R(γ ; −x, λ) = R(γ ; x, −λ) = R(γˆ ; λ, x). Furthermore, for an element σ ∈ W (D4 ), where W (D4 ) is the Weyl-group of type D4 acting on the parameters γ by permutations and even numbers of sign flips, we have R(γ ; x, λ) R(σ γ ; x, λ) = . c(γ ; x)c(γˆ ; λ)N (γ ) c(σ γ ; x)c(σ γ ; λ)N (σ γ ) Proof. These symmetries are all direct consequences of the W2 (D6 )-symmetries of the hyperbolic Barnes integral Bh (see Proposition 4.7). Concretely, note that the W2 (D6 )action on C6  G2iω /Cβ1278 is given by s78 (γ , x, λ) = (γ0 , γ1 , γ3 , γ2 , x, λ), s18 (γ , x, λ) = (−γ3 , γ1 , γ2 , −γ0 , x, λ), w(γ , x, λ) = (γ1 , γ0 , γ2 , γ3 , x, λ),  1 1 1 i i i (γ0 + γˆ0 ) + (x + λ), (γ1 + γˆ1 ) − (x + λ), (γ2 + γˆ2 ) − (x + λ), s45 (γ , x, λ) = 2 2 2 2 2 2  1 i i i 1 1 (γ3 + γˆ3 )− (x +λ), (γˆ0 − γ0 )+ (x −λ), (γˆ0 −γ0 ) + (λ−x) , 2 2 2 2 2 2 s34 (γ , x, λ) = (γ , −x, λ), s56 (γ , x, λ) = (γ , x, −λ). The fact that R(γ ; x, λ) is even in x and λ follows now from the s34 ∈ W2 (D6 ) and s56 ∈ W2 (D6 ) symmetry of Bh , respectively (see Proposition 4.7). Similarly, the duality is obtained from the action of s35 s46 and using that γ0 + γi = γˆ0 + γˆi (i = 1, 2, 3), while the W (D4 )-symmetry in γ follows from considering the action of s27 ∈ W2 (D6 ) (which interchanges γ1 ↔ γ2 ), s78 , s18 and w.   Remark 4.19. Corollary 4.6 implies the explicit evaluation formula R(γ ; iω + iγ3 , λ; ω1 , ω2 ) =

2  j=1

G(iω + iγ0 + iγ j ) . G(iω + iγ j + iγ3 )G(i γˆ j ± λ)

Using the W (D4 )-symmetry of R, this implies R(γ ; iω + iγ0 , λ; ω1 , ω2 ) = 1, in accordance with [26, (3.26)].

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F. J. van de Bult, E. M. Rains, J. V. Stokman

Using Proposition 4.7 and Theorem 4.11 we can derive several different integral representations of the R-function. First we derive the integral representation of R which was previously derived in [1] by relating R to matrix coefficients of representations of the modular double of the quantum group Uq (sl2 (C)). Proposition 4.20. We have N (γ ) G(x − iγ0 , x − iγ1 , λ − i γˆ0 , λ − i γˆ1 ) Bh (υ), R(γ ; x, λ) = √ 2 ω1 ω2 G(x + iγ2 , x + iγ3 , λ + i γˆ2 , λ + i γˆ3 ) where i λ iω + ± (γ0 − γ1 ), 2 2 2 i λ iω ± (−γ0 − γ1 ), = + 2 2 2

i λ iω + ± (γ3 − γ2 ), 2 2 2 i λ iω ± (γ2 + γ3 ), = + 2 2 2

υ1/2 = x −

υ3/4 = −x −

υ5/6

υ7/8

and υ j/k = α ± β means υ j = α + β and υk = α − β. Proof. Express Bh (s36 ws35 s28 ws18 u) in terms of Bh (u) using the W2 (D6 )-symmetries of the hyperbolic Barnes integral Bh (see Proposition 4.7) and specialize u as in (4.28). This gives the desired equality.   Moreover we can express R in terms of the hyperbolic Euler integral E h , which leads to a previously unknown integral representation. Theorem 4.21. We have 1 R(γ ; x, λ) = √ 2 ω1 ω2 1 = √ 2 ω1 ω2

3 j=1

3

G(iγ0 + iγ j + iω, λ − i γˆ j ) G(λ + i γˆ0 )

E h (u)

G(iγ0 + iγ j + iω, λ − i γˆ j ) E h (υ),  G(λ + i γˆ0 ) 3j=0 G(iγ j ± x) j=1

where u ∈ C6 is given by uj =

iω i γˆ0 λ + iγ j−1 − + , 2 2 2

( j = 1, . . . , 4),

u 5/6 =

i γˆ0 λ iω ±x+ − , 2 2 2

( j = 1, . . . , 4),

υ5/6 =

i γˆ0 λ iω ±x− − . 2 2 2

and υ ∈ C6 is given by υj =

iω i γˆ0 λ − iγ j−1 + + , 2 2 2

Proof. To prove the first equation, express R(γ ; x, λ) in terms of R(−γ3 , γ1 , γ2 , −γ0 ; x, λ) using the W (D4 )-symmetry of R (see Proposition 4.18). Subsequently use the identity relating Bh to E h , see Theorem 4.11. To obtain the second equation, apply the symmetry of E h with respect to the longest Weyl-group element v1 ∈ W1 (D6 ) (see Corollary 4.16) in the first equation and use that R is even in λ.   The contiguous relation for E h (Lemma 4.17) now becomes the following result.

Properties of Generalized Univariate Hypergeometric Functions

63

Proposition 4.22. ([26]) Ruijsenaars’ R-function satisfies the Askey-Wilson second order difference equation A(γ ; x; ω1 , ω2 )(R(γ ; x + iω1 , λ) − R(γ ; x, λ)) + (x ↔ −x) = B(γ ; λ; ω1 , ω2 )R(γ ; x, λ), where

3

j=0 s((iω

(4.30)

+ x + iγ j )/ω2 )

, s(2x/ω2 )s(2(x + iω)/ω2 ) B(γ ; λ; ω1 , ω2 ) = s((λ − iω − i γˆ0 )/ω2 )s((λ + iω + i γˆ0 )/ω2 ). A(γ ; x; ω1 , ω2 ) =

Remark 4.23. As is emphasized in [26], R satisfies four Askey-Wilson second order difference equations; two equations acting on the geometric variable x (namely (4.30), and (4.30) with the role of ω1 and ω2 interchanged), as well as two equations acting on the spectral parameter λ by exploring the duality of R (see Proposition 4.18). For later purposes, it is convenient to rewrite (4.30) as the eigenvalue equation  ω ,ω  Lγ 1 2 R(γ ; · , λ; ω1 , ω2 ) (x) = B(γ ; λ; ω1 , ω2 )R(γ ; x, λ; ω1 , ω2 ) for the Askey-Wilson second order difference operator     ω,ω  Lγ 2 f (x) := A(γ ; x; ω1 , ω2 ) f (x + iω1 ) − f (x) + x ↔ −x).

(4.31)

5. Trigonometric Hypergeometric Integrals 5.1. Basic hypergeometric series. In this section we assume that the base q satisfies 0 < |q| < 1. The trigonometric gamma function [24] is essentially the q-gamma function q (x), see [5]. For ease of presentation we express all the results in terms of the q-shifted factorial z; q ∞ , which are related to q (x) by   q; q ∞ q (x) =  x  (1 − q)1−x q ;q ∞ (with a proper interpretation of the right-hand side). The q-shifted factorial is the p = 0 degeneration of the elliptic gamma function, e (z; 0, q) = 

1  , z; q ∞

(5.1)

while the role of the first order analytic difference equation is taken over by     z; q ∞ = (1 − z) qz; q ∞ . However there is no reflection equation anymore; its role is taken over by the product formula for Jacobi’s (renormalized) theta function   θ (z; q) = z, q/z; q ∞ .   As a function of z the q-shifted factorial z; q ∞ is entire with zeros at z = q −n for n ∈ Z≥0 . In this section we call a sequence of the form aq −n (n ∈ Z≥0 ) an upward

64

F. J. van de Bult, E. M. Rains, J. V. Stokman

sequence (since they diverge to infinity for large n) and a sequence of the form aq n (n ∈ Z≥0 ) a downward sequence (as the elements converge to zero for large n). We will use standard notations for basic hypergeometric series from [5]. In particular, the r +1 φr basic hypergeometric series is   
∞  a1 , . . . , ar +1 ; q n n a1 , . . . , ar +1   z , ; q, z = |z| < 1, r +1 φr b1 , . . . br q, b1 , . . . , br ; q n=0

n

   j where a; q n = n−1 j=0 (1 − aq ) and with the usual convention regarding products of such expressions. The very-well-poised r +1 φr basic hypergeometric series is ⎞ ⎛ 1 1 2 2   , qa , −qa , a , . . . , a a 1 4 r +1 1 1 ; q, z ⎠ . 1 r +1 Wr a1 ; a4 , a5 , . . . , ar +1 ; q, z = r +1 φr ⎝ 1 2 2 a1 , −a1 , qa1 /a4 , . . . , qa1 /ar +1 Finally, the bilateral basic hypergeometric series r ψr is defined as  
 ∞  a1 , a2 , . . . , ar ; q n n a1 , a2 , . . . , ar   z ; q, z = r ψr b1 , b2 , . . . , br b1 , b2 , . . . , br ; q n n=0    ∞ 
q/b1 , q/b2 , . . . , q/br ; q n b1 · · · br n   + , q/a1 , q/a2 , . . . , q/ar ; q n a1 · · · ar z n=1 provided that |b1 · · · br /a1 · · · ar | < |z| < 1 to ensure absolute and uniform convergence. We end this introductory subsection by an elementary lemma which will enable us to rewrite trigonometric integrals with compact integration cycle in terms of trigonometric integrals with noncompact integration cycle. Let H+ be the upper half plane in C. In this section we choose τ ∈ H+ such that q = e(τ ) once and for all, where e(x) is a shorthand notation for exp(2πi x). We furthermore write  = Z + Zτ . Lemma 5.1. Let u, v ∈ C be such that u ∈ v + . There exists an η = η(u, v) ∈ C, unique up to -translates, such that   θ e(u + v − η − x), e(x − η); q   θ e(u − η), e(v − η); q     e((v − u)/τ ) − 1 θ e(x − u), e(v − x); q     = τ q, q; q ∞ θ e(v − u); q ×

∞


n=−∞



1  . 1 − e((v − x − n)/τ ) e((x + n − u)/τ ) − 1

(5.2)

Proof. Set  q = e(−1/τ ). The bilateral sum ∞


1   1 − e((v − x − n)/τ ) e((x + n − u)/τ ) − 1 n=−∞   1 e((v − x)/τ ), e((u − x)/τ ) = ψ ;  q ,  q 2 2  q e((v − x)/τ ),  q e((u − x)/τ ) (1 − e((v − x)/τ ))(e((x − u)/τ ) − 1)

f (x) =



Properties of Generalized Univariate Hypergeometric Functions

65

defines an elliptic function on C/, with possible poles at most simple and located at u +  and at v + . Hence there exists a η ∈ C (unique up to -translates) and a constant Cη ∈ C such that   θ e(u + v − η − x), e(x − η); q   f (x) = Cη . θ e(x − u), e(v − x); q We now compute the residue of f at u in two different ways: Res( f ) = x=u

1 τ   2πi 1 − e((v − u)/τ )

from the bilateral series expression of f , and   Cη θ e(u − η), e(v − η); q     Res( f ) = − x=u 2πi q, q; q ∞ θ e(v − u); q from the expression of f as a quotient of theta-functions. Combining both identities yields an explicit expression of the constant Cη in terms of η, resulting in the formula       τ q, q; q ∞ θ e(v − u); q θ e(u + v − η − x), e(x − η); q     f (x) = e((v − u)/τ ) − 1 θ e(x − u), e(v − x), e(u − η), e(v − η); q for f . Rewriting this identity yields the desired result.

 

5.2. Trigonometric hypergeometric integrals with E 6 symmetries. We consider trigonometric degenerations of Se (t) (t ∈ H pq ) along root vectors α ∈ R(E 8 ) lying in the W (E 7 ) = W (E 8 )δ -orbit  + = {α +jk , γ jk | 1 ≤ j < k ≤ 8}, (5.3) O := W (E 7 ) α18 cf. Sect. 2. The degenerations relate to the explicit bijection ∼

G0 −→ Glog( pq) ,

(u 1 , . . . , u 8 ) → (u 1 , . . . , u 8 ) + log( pq)α

(5.4)

on the parameter spaces (in logarithmic form) of the associated integrals. We obtain two different trigonometric degenerations, depending on whether we degenerate along an orbit vector of the form α = α +jk , or of the form γ jk . Specifically, we consider the trigonometric degenerations St (t) respectively Ut (t) + and γ (t ∈ H1 ) of Se (t) (t ∈ H pq ) along the orbit vector α18 18 respectively. The + orbit vector α18 (respectively γ18 ) is the additional simple root turning the basis 1 (respectively 2 ) of R(E 7 ) into the basis 1 (respectively 2 ) of R(E 8 ), see Sect. 2. + of The induced symmetry group of St (t) (t ∈ H1 ) is the isotropy subgroup W (E 7 )α18 W (E 7 ), while the induced symmetry group of Ut (t) (t ∈ H1 ) is W (E 7 )γ18 . It follows + = W (E 7 )γ from the analysis in Sect. 2 that W (E 7 )α18 18 is a maximal, standard parabolic subgroup of W (E 7 ) with respect to both bases 1 and 2 , isomorphic to the − Weyl group W (E 6 ) of type E 6 , with corresponding simple roots 1 = 1 \ {α21 } and −  2 = 2 \ {α87 }, and with corresponding Dynkin diagrams,

66

F. J. van de Bult, E. M. Rains, J. V. Stokman

− α21 ◦

− α32 •

− α43 •

− α54 •

− α65 •

− α76 •

− α87 ◦

− α18 •

β5678 •

− α45 •

− α34 •

− α23 •

− • β1234 • α56 − − Observe that α18 and α76 are the highest roots of the standard parabolic root system R(E 6 ) of type E 6 in R(E 7 ) corresponding to the bases 1 and 2 respectively. From now on we write + = W (E 7 )γ . W (E 6 ) := W (E 7 )α18 18

We first introduce the trigonometric hypergeometric integrals St (t) and Ut (t) (t ∈ H1 ) explicitly. Their integrands are defined by (z ±2 , t1−1 z ±1 , t8−1 z ±1 ; q)∞ , 7 ±1 j=2 (t j z ; q)∞    7  z 2  z/ti ; q ∞  1 θ (t1 t8 /µz, z/µ; q) µ     , 1− Jt (t; z) = 2 θ (t1 /µ, t8 /µ; q) q t t z; q z/q, t i j /z; q ∞ ∞ j=1,8 j i=2 It (t; z) =

 8  where t = (t1 , . . . , t8 ) ∈ C× . For generic t = (t1 , . . . , t8 ) ∈ C8 satisfying 8j=1 t j = 1 and generic µ ∈ C we now define the resulting trigonometric hypergeometric integrals as   dz dz µ µ , Ut (t) = , It (t; z) Jt (t; q) St (t) =  2πi z 2πi z C C where C (respectively C  ) is a deformation of the positively oriented unit circle T including the pole sequences t j q Z≥0 ( j = 2, . . . , 7) of It (t; z) and excluding their reciprocals µ (respectively including the pole sequences t j q Z≥0 ( j = 1, 8) of Jt (t; z) and excluding Z≤1 ( j = 1, 8) and t −1 q Z≤0 (i = 2, . . . , 7)). As in the ellipthe pole sequences t −1 j q i µ tic and hyperbolic cases, one observes that St (t) (respectively Ut (t)) admits a unique  8 meromorphic extension to the parameter domain {t ∈ C8 | j=1 t j = 1} (respectively  {(µ, t) ∈ C× × C8 | 8j=1 t j = 1}). We call St (t) the trigonometric hypergeometric function. µ

Lemma 5.2. The integral Ut (t) is independent of µ ∈ C× . Proof. There are several different, elementary arguments to prove the lemma; we give qµ µ here the argument based on Liouville’s Theorem. Note that Ut (t) = Ut (t), and that µ the possible poles of µ → Ut (t) are at t j q Z ( j = 1, 8). Without loss of generality we  µ assume the generic conditions on the parameters t ∈ C8 ( 8j=1 t j = 1) such that Ut (t) admits the integral representation as above, and such that t1 ∈ t8 q Z . The latter condition µ ensures that the possible poles t1 q Z , t8 q Z of µ → Ut (t) are at most simple. But the µ residue of Ut (t) at µ = t j ( j = 1, 8) is zero, since it is an integral over a deformation C  of T whose integrand is analytic within the integration contour C  and vanishes at the µ origin. Hence C×  µ → Ut (t) is bounded and analytic, hence constant by Liouville’s Theorem.  

Properties of Generalized Univariate Hypergeometric Functions

67 µ

In view of Lemma 5.2, we omit the µ-dependence in the notation for Ut (t). Since µ −µ It (−t; z) = It (t; −z) and Jt (−t; z) = Jt (t; −z), we may and will view St and Ut as meromorphic function on H1 . By choosing a special value of µ, we are able to derive another, “unfolded” integral representation of Ut (t) as follows. Let H+ be the upper half plane in C. Choose τ ∈ H+ such that q = e(τ ), where e(x) is a shorthand notation for exp(2πi x). Recall the surjective map ψ0 : G0 → H1 from Sect. 2. Corollary 5.3. For generic parameters u ∈ G0 we have Ut (ψ0 (2πiu))

  e((u 8 − u 1 )/τ ) − 1 2  =  τ q, q; q ∞ e(u 1 )θ (e(u 8 − u 1 ); q) ⎧     7   ⎨ e(2x)  e(x − u j ); q ∞ qe(x − u 1 ), qe(x − u 8 ); q ∞     1− × −1 e(x + u ), q −1 e(x + u ); q q e(x + u q ); q L⎩ j 1 8 ∞ ∞ j=2  e(x)   d x, ×  1 − e((u 8 − x)/τ ) 1 − e((x − u 1 )/τ )

where the integration contour L is some translate ξ + R (ξ ∈ iR) of the real line with a finite number of indentations, such that C separates the pole sequences −u 1 + Z + Z≤1 τ , −u 8 + Z + Z≤1 τ and −u j + Z + Z≤0 τ ( j = 2, . . . , 7) of the integrand from the pole sequences u 1 + Z≥0 τ and u 8 + Z≥0 τ . Remark 5.4. Note that always ξ = 0 in Corollary 5.3. Due to the balancing condition 8 j=1 u j = 0, there are no parameter choices for which L = R can be taken as an integration contour. This is a reflection of the fact that there are no parameters t ∈ H1 such that the unit circleT can be chosen as an integration cycle in the original integral µ dz representation Ut (t) = C  Jt (t; z) 2πi z of Ut (t). Proof. In the integral expression Ut (ψc (2πiu)) =



µ

C

Jt (ψ0 (2πiu); z)

dz , 2πi z

we change the integration variable to z = e(x), take µ = e(η(u 1 , u 8 )), and we use Lemma 5.1 to rewrite the quotient of theta-functions in the integrand as a bilateral sum. Changing the integration over the indented line segment with the bilateral sum using Fubini’s Theorem, we can rewrite the resulting expression as a single integral over a noncompact integration cycle L. This leads directly to the desired result.   In the following lemma we show that Ut (t) can be expressed as a sum of two nonterminating very-well-poised 10 φ9 series. Lemma 5.5. As meromorphic functions in t ∈ H1 , we have  7   t1 /t j ; q ∞ 2    Ut (t) =  2 q, t1 , t1 t8 /q, t8 /t1 ; q ∞ j=2 t1 t j ; q ∞  2    t1 t1 t8 ; t1 t2 , t1 t3 , . . . , t1 t7 , ; q, q + t1 ↔ t8 . ×10 W9 q q

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Proof. For generic t ∈ H1 we shrink the contour C  in the integral representation  µ µ dz of Ut (t) = C  Jt (t; z) 2πi z to the origin while picking up the residues at the pole µ Z Z ≥0 ≥0 sequences t1 q and t8 q of the integrand Jt (t; z). The resulting sum of residues can be directly rewritten as a sum of two very-well-poised 10 φ9 series, leading to the desired identity (cf. the general residue techniques in [5, §4.10]).   Remark 5.6. Lemma 5.5 yields that Ut (t) is, up to an explicit rescaling factor, an integral form of the particular sum  of two very-well-poised 10 φ9 series as e.g. studied in [7] and [15] (see [7, (1.8)], [15, (9c)]). Note furthermore that the explicit µ-dependent µ quotient of theta-functions in the integrand of Ut (t) has the effect that it balances the µ very-well-poised 10 φ9 series when picking up the residues of Jt (t; z) at the two pole sequences t1 q Z≥0 and t8 q Z≥0 . In the following proposition we show that St (respectively Ut ) is the degeneration of + (respectively γ ). Se along the root vector α18 18 Proposition 5.7. Let t = (t1 , . . . , t8 ) ∈ C8 be generic parameters satisfying the bal ancing condition 8j=1 t j = 1. Then St (t) = lim Se ( pqt1 , t2 , . . . , t7 , pqt8 ), p→0

(5.5)    1 1 1 1 Ut (t) = lim θ t1 t8 / pq; q Se ( pq)− 2 t1 , ( pq) 2 t2 , . . . , ( pq) 2 t7 , ( pq)− 2 t8 ). p→0

Proof. For the degeneration to St (t) we use that 7 Ie ( pqt1 , t2 , . . . , t7 , pqt8 ; z) =

±1 j=2 e (t j z ; p, q) e (z ±2 , t1−1 z ±1 , t8−1 z ±1 ; p, q)

in view of the reflection equation for e , which (pointwise) tends to It (t; z) as p → 0 in view of (5.1). A standard application of Lebesgue’s dominated convergence theorem leads to the limit of the associated integrals. The degeneration to Ut (t) is more involved, since one needs to use a nontrivial symmetry argument to cancel some unwanted sequences of poles of Ie (t; z). To ease the notations we set  1 1 1 1  t p = ( pq)− 2 t1 , ( pq) 2 t2 , . . . , ( pq) 2 t7 , ( pq)− 2 t8 and we denote   1 1 1 1 θ ( pq)− 2 t1 t8 /µz, ( pq)− 2 t1 z, ( pq)− 2 t8 z, ( pq) 2 z/µ; q   Q(z) = . θ z2; q By (3.6), we have the identity   Q(z) + Q(z −1 ) = θ t1 t8 / pq, t1 /µ, t8 µ; q . Since the integrand Ie (τ p ; z) is invariant under z → z −1 , we can consequently write    2 dz  θ t1 t8 / pq; q Se (t p ; z) =  , Q(z)Ie (t p ; z) 2πi z θ t1 /µ, t8 /µ; q C

Properties of Generalized Univariate Hypergeometric Functions

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with C a deformation of the positively oriented unit circle T separating the downward 1 pole sequences of the integrand from the upward pole sequences. Taking ( pq) 2 z as a new integration variable and using the functional equation and reflection equation of e , we obtain the integral representation   7  θ t1 t8 /µz, z/µ; q  2 e (t j z; p, q)   θ z /q; p  (z/t j ; p, q) C θ t1 /µ, t8 /µ; q j=2 e    dz , (5.6) e t j z/q, t j /z; p, q × 2πi z

  θ t1 t8 / pq; q Se (t p ; z) = 2



j=1,8

where C is a deformation of the positively oriented unit circle T which includes the pole sequences t1 p Z≥0 q Z≥0 , t8 p Z≥0 q Z≥0 and t j p Z≥1 q Z≥1 ( j = 2, . . . , 7), and which excludes Z≤0 q Z≤0 ( j = 2, . . . , 7). We the pole sequences t1−1 p Z≤0 q Z≤1 , t8−1 p Z≤0 q Z≤1 and t −1 j p can now take the limit p → 0 in (5.6) with p-independent, fixed integration contour C, leading to the desired limit relation   µ lim θ t1 t8 / pq; q Se (t p ) = Ut (t).

p→0

  Remark 5.8. Observe that Lemma 5.5 and the proof of Proposition 5.7 entail independent proofs of Lemma 5.2. By specializing the parameters t ∈ H1 in Proposition 5.7 further, we arrive at trigonometric integrals which can be evaluated by (3.2). The resulting trigonometric degenerations lead immediately to the trigonometric Nassrallah-Rahman integral evaluation formula [5, (6.4.1)] and Gasper’s integral evaluation formula [5, (4.11.4)]: Corollary 5.9. For generic t = (t1 , . . . , t6 ) ∈ C6 satisfying the balancing condition 6 j=1 t j = 1 we have      ±2 −1 ±1  2 6j=2 1/t1 t j ; q ∞ z , t1 z ; q ∞ dz     , = 6  ±1  2πi z q; q ∞ 2≤ j
( j = 1, . . . , 4), t5−1 q Z≤1 and t6−1 q Z≤1 .

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Proof. For the first integral evaluation, take t ∈ H1 and t7 = t8−1 in the degeneration from Se to St , and use the elliptic Nassrallah-Rahman integral evaluation formula (3.2). For the second integral evaluation, take t ∈ H1 with t6 = t7−1 in the degeneration µ from Se to Ut and again use (3.2) to evaluate the elliptic integral. It leads to the second integral evaluation formula with generic parameters (t1 , t2 , t3 , t4 , t5 , t8 ) ∈ C6 satisfying t1 · · · t5 t8 = 1.   The second integral in Corollary 5.9 can be unfolded using Lemma 5.1 as in Corol lary 5.3. We obtain for generic parameters u ∈ C6 satisfying 6j=1 u j = 0, ⎧     4   ⎨ e(2x)  e(x − u j ); q ∞ qe(x − u 5 ), qe(x − u 6 ); q ∞     1− −1 e(x + u ), q −1 e(x + u ); q q e(x + u q ); q L⎩ j 6 5 ∞ ∞ j=1  e(x)   dx × 1 − e((u 6 − x)/τ ) 1 − e((x − u 5 )/τ )        q; q ∞ 1≤ j
which is Bailey’s summation formula [5, (2.11.7)] of the sum of two very-well-poised 8 φ7 series. We can now compute the (nontrivial) W (E 6 )-symmetries of the trigonometric hypergeometric integrals St and Ut by taking limits of the corresponding symmetries on the elliptic level using Proposition 5.7. We prefer to give a derivation based on the trigonometric evaluation formulas (see Corollary 5.9), in analogy to our approach in the elliptic and hyperbolic cases. Proposition 5.10. The trigonometric integrals St (t) and Ut (t) (t ∈ H1 ) are invariant under permutations of (t1 , t8 ) and of (t2 , . . . , t7 ). Furthermore,   1/t1 t2 , 1/t1 t3 , 1/t1 t4 , 1/t8 t5 , 1/t8 t6 , 1/t8 t7 ; q ∞   St (t) = St (wt) , t2 t3 , t2 t4 , t3 t4 , t5 t6 , t5 t7 , t6 t7 ; q ∞   (5.7) 1/t2 t3 , 1/t2 t4 , 1/t3 t4 , 1/t5 t6 , 1/t5 t7 , 1/t6 t7 ; q ∞   Ut (t) = Ut (wt) t1 t2 , t1 t3 , t1 t4 , t5 t8 , t6 t8 , t7 t8 ; q ∞

Properties of Generalized Univariate Hypergeometric Functions

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as meromorphic functions in t ∈ H1 . Proof. In order to derive the w-symmetry of St (t) we consider the double integral  ±2 ±2 −1 ±1 −1 ±1   z , x , t1 z , st8 x ; q ∞ dz d x   ±1 ±1 ±1 ±1 ±1 −1 ±1 −1 ±1 −1 ±1 2 2πi z 2πi x T t2 z , t3 z , t4 z , sz x , s t5 x , s t6 x , s t7 x ; q ∞  for parameters (t1 , . . . , t8 ) ∈ C8 satisfying 8j=1 t j = 1, where s 2 t1 t2 t3 t4 = 1 = s −2 t5 t6 t7 t8 and where we assume the additional parameter restraints |t2 |, |t3 |, |t4 |, |s|, |t5 /s|, |t6 /s|, |t7 /s| < 1 to ensure that the integration contour T separates the downward sequences of poles from the upward sequences. The desired transformation then follows by either integrating the double integral first to x, or first to z, using in each case the trigonometric NassrallahRahman integral evaluation formula (see Corollary 5.9). The proof of the w-symmetry of Ut (t) follows the same line of arguments. For  > 0 we denote T for the positively oriented circle in the complex plane with radius  and centered at the origin. The w-symmetry   1/t2 t3 , 1/t2 t4 , 1/t3 t4 , 1/t5 t6 , 1/t5 t7 , 1/t6 t7 ; q ∞ µ/s µ   Ut (t) = Ut (wt) t1 t2 , t1 t3 , t1 t4 , t5 t8 , t6 t8 , t7 t8 ; q ∞ for t ∈ H1 , where s 2 t1 t2 t3 t4 = 1 = s −2 t5 t6 t7 t8 , by considering for (t1 , . . . , t8 ) ∈ C8  satisfying 8j=1 t j = 1 the double integral 

     θ st1 t8 /µz, t8 z/µx, x/µ; q x2 z2   1 − 1 −  2 q q θ st8 /µ, st1 /µ, t8 /sµ; q |qs|T    z/t2 , z/t3 , z/t4 , x z/s, sx/t5 , sx/t6 , sx/t7 ; q ∞ dz d x  ×  t1 z/q, t1 /z, t2 z, t3 z, t4 z, sx z/q, sz/x, sx/z, t5 x/s, t6 x/s, t7 x/s, t8 x/qs, t8 /sx; q ∞ 2πi z 2πi x

with s 2 t1 t2 t3 t4 = 1 = s −2 t5 t6 t7 t8 , where we assume the additional parameter restraints 0 < |s|  |q 2 |, |t1 |, |t2−1 |, |t3−1 |, |t4−1 | < |qs|, |t5 |, |t6 |, |t7 | < |q −1 |, |t8 | < |qs 2 | 1

to ensure a proper separation by the integration contours of the upward sequences of poles from the downward sequences. Using the second trigonometric integral evaluation formula of Corollary 5.9 then yields the desired result for the restricted parameter domain. Analytic continuation completes the proof.   Remark 5.11. Rewriting Ut (t) as a sum of two very-well-poised 10 φ9 series (see Lemma 5.5 and Remark 5.6), the w-symmetry of Ut (t) becomes Bailey’s four-term transformation formula [5, (2.12.9)], see also [7]. The identification of the symmetry group of Ut with the Weyl group of type E 6 has been derived by different methods in [15]. Finally we relate the two trigonometric integrals St and Ut . We can obtain the following proposition as a degeneration of a particular W (E 7 )-symmetry of Se , but we prefer here to give a direct proof using double integrals.

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Proposition 5.12. As a meromorphic function in t ∈ H1 , we have  St (t)

2≤ j
(1/qt1 t8 , 1/t1 t6 , 1/t1 t7 , 1/t8 t6 , 1/t8 t7 ; q)∞

= Ut (t6 /s, st2 , st3 , st4 , st5 , t1 /s, t8 /s, t7 /s),

where t2 t3 t4 t5 s 2 = 1 = t1 t6 t7 t8 /s 2 Proof. For (t1 , . . . , t8 ) ∈ C8 satisfying 

8

j=1 t j

= 1 we consider the double integral

  5 z2  θ (µz, t6 t7 µ/s 2 z) 1 1− ±1 q (t x ; q)∞ z∈ηT x∈T θ (t6 µ/s, t7 µ/s) j=2 j 

×

(szx ±1 , t

d x dz (x ±2 , zx ±1 /s, sz/t1 , sz/t8 ; q)∞ , z/sq, t /sz, t z/sq, t /sz, t z/s, t z/s; q) 2πi x 2πi z 7 7 6 6 1 8 ∞

 1  with s 2 t2 t3 t4 t5 = 1 = s −2 t1 t6 t7 t8 and with 0 < η < min |s −1 |, |q 2 | , where we assume the additional parameter restraints |t2 |, |t3 |, |t4 |, |t5 | < 1,

|t6 |, |t7 | < η|s|,

|t1 |, |t8 | < η−1 |s|

to ensure a proper separation by the integration contours of the upward sequences of poles from the downward sequences. Using Corollary 5.9, we can first integrate over x using the trigonometric Nassrallah-Rahman integral evaluation formula, or first integrate over z using the second integral evaluation formula of Corollary 5.9. The resulting identity gives the desired result for restricted parameter values. Analytic continuation completes the proof.   Remark 5.13. (i) Combining Proposition 5.12 with Lemma 5.5 we obtain an expression of St (t) as a sum of two very-well-poised 10 φ9 series, which is originally due to Rahman [5, (6.4.8)]. (ii) For e.g. t1 t6 = q m (m ∈ Z≥0 ), it follows from (i) (see also [20] and [5, (6.4.10)]) that the St (t; p, q) essentially coincides with the biorthogonal rational function of Rahman [20], which is explicitly given as a terminating very-well-poised 10 φ9 series.

5.3. Contiguous relations. The fundamental equation on this level equals 1 1 1 (1 − vx ±1 )(1 − yz ±1 ) + (1 − vy ±1 )(1 − zx ±1 ) + (1 − vz ±1 )(1 − x y ±1 ) = 0, y z x (5.8) where (1 − ax ±1 ) = (1 − ax)(1 − ax −1 ). The fundamental relation (5.8) is the p = 0 − log(q) acts as in the elliptic case by multiplying reduction of (3.6). In this section τi j = τi j ti by q and dividing t j by q. Formula (5.8) leads as in the elliptic case to the difference equation (1 − t5 t6±1 /q) (1 − t4 t6±1 )

St (τ45 t) +

(1 − t5 t4±1 /q) (1 − t6 t4±1 )

St (τ65 t) = St (t),

t ∈ H1 .

(5.9)

Properties of Generalized Univariate Hypergeometric Functions

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To obtain a second difference equation between trigonometric hypergeometric functions where two times the same parameter is multiplied by q, we can mimic the approach in the elliptic case with the role of the longest Weyl group element taken over by the element u = ws35 s46 w ∈ W (E 6 ). Alternatively, one can rewrite the difference equation (3.8) for Se in the form   θ t3 /qt4 , 1/t1 t5 , 1/t8 t5 , t2 t5 /q, t5 t6 /q, t5 t7 /q; p   t) + (t3 ↔ t5 ) Se (τ45 θ t3 /t5 ; p   = θ 1/qt1 t4 , 1/qt8 t4 , t2 t4 , t4 t6 , t4 t7 ; p Se ( t), where t ∈ H1 and  t = ( pqt1 , t2 , . . . , t7 , pqt8 ), and degenerate it using Proposition 5.7. We arrive at (1 − t3 /qt4 )  (1 − 1/t5 t j )  (1 − t5 t j /q) St (τ45 t) (1 − t3 /t5 ) (1 − 1/qt4 t j ) (1 − t4 t j ) j=1,8

j=2,6,7

+(t3 ↔ t5 ) = St (t), t ∈ H1 .

(5.10)

Together these equations imply the following result. Proposition 5.14. We have A(t)St (τ45 t) + (t4 ↔ t5 ) = B(t)St (t)

(5.11)

as meromorphic functions in t ∈ H1 , where A(t) = B(t) =

t5 t j 1 1  j=2,3,6,7 (1 − q ) t1 t5 )(1 − t8 t5 ) − , t4 (1 − t4qt5 )(1 − qtt54 )(1 − tt45 ) (1 − qt11 t6 )(1 − qt18 t6 )(1 − t3 t6 )(1 − t7 t6 )(1 − t2 t6 ) − t6 (1 − qtt46 )(1 − qtt56 )   t t (1 − tt46 )(1 − t6 t4 ) j=1,8 (1 − t j1t5 ) j=2,3,7 (1 − jq 5 ) + t6 (1 − qtt56 )(1 − t4qt5 )(1 − qtt54 )(1 − tt45 )   t t (1 − tt65 )(1 − t6 t5 ) j=1,8 (1 − t j1t4 ) j=2,3,7 (1 − jq 4 ) + . t6 (1 − qtt46 )(1 − t4qt5 )(1 − qtt45 )(1 − tt54 )

(1 −

Despite the apparent asymmetric expression B still satisfies B(s67 t) = B(t). The contiguous relation for the elliptic hypergeometric function Se with step-size p can also be degenerated to the trigonometric level. A direct derivation is as follows. By (3.6) we have θ (t8−1 t7±1 ; q) θ (t6 t7±1 ; q) +

It (t1 , t2 , . . . , t5 , qt8 , t7 , t6 /q; z)

θ (t8−1 t6±1 ; q) θ (t7 t6±1 ; q)

It (t1 , t2 , . . . , t6 , qt8 , t7 /q; z) = It (t; z).

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Integrating this equation we obtain θ (t8−1 t7±1 ; q) θ (t6 t7±1 ; q)

St (t1 , t2 , . . . , t5 , qt8 , t7 , t6 /q) + (t6 ↔ t7 ) = St (t)

(5.12)

as meromorphic functions in t ∈ H1 , a three term transformation for St . The three term transformation [7, (6.5)] is equivalent to the sum of two equations of this type (in which the parameters are chosen such that two terms coincide and two other terms cancel each other). Remark 5.15. In [15] it is shown that there are essentially five different types of three term transformations for  (see Remark 5.6), or equivalently of the integrals Ut and St . The different types arise from a careful analysis of the three term transformations in terms of the W (E 7 )-action on H1 . It is likely that all five different types of three term transformations for  can be re-obtained by degenerating contiguous relations for Se with step-size p (similarly as the derivation of (5.12)): concretely, the five prototypes are in one-to-one correspondence to the orbits of {(α, β, γ ) ∈ O3 | α, β, γ are pair-wise different} under the diagonal action of W (E 7 ), where O is the W (E 7 )-orbit (5.3). 5.4. Degenerations with D5 symmetries. In this section we consider degenerations of St and Ut with symmetries with respect to the Weyl group of type D5 . Compared to the analysis on the hyperbolic level, we introduce a trigonometric analog of the Euler and Barnes’ type integrals, as well as a third, new type of integral arising as degeneration of Ut . We first introduce the degenerate integrals explicitly.  6 For generic t = (t1 , . . . , t6 ) ∈ C× we define the trigonometric Euler integral as   ±2 −1 ±1  z , t1 z ; q ∞ dz , E t (t) = 6  ±1  2πi z C j=2 t j z ; q ∞

(5.13)

where C is a deformation of the positively oriented unit circle T separating the decreasing pole sequences t j q Z≥0 ( j = 2, . . . , 6) of the integrand from their reciprocals. We have  6 E t (−t) = E t (t), and E t has a unique meromorphic extension to C× . The resulting  6 meromorphic function on C× /C2 is denoted also by E t . For generic µ ∈ C× and generic t = (t1 , . . . , t8 ) ∈ C8 satisfying the balancing  condition 8j=1 t j = 1 we define the trigonometric Barnes integral as  Bt (t) = 2

(z/t1 , z/t8 ; q)∞ θ (t2 t7 /µz, z/µ; q) dz , (5.14) 6 θ (t /µ, t /µ; q) 2πi z 2 7 C j=3 (t j z; q)∞ (t2 /z, t7 /z; q)∞

where C is a deformation of T separating the decreasing pole sequences t2 q Z≥0 and t7 q Z≥0 Z≤0 ( j = 3, . . . , 6). Analoof the integrand from the increasing pole sequences t −1 j q gously to the analysis of the integral Ut (t), we have that the trigonometric Barnes integral  Bt (t) uniquely extends to a meromorphic function in {(µ, t) ∈ C× × C8 | 8j=1 t j = 1}

Properties of Generalized Univariate Hypergeometric Functions

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which is independent of µ (cf. Lemma 5.2). Furthermore, by a change of integration variable we have Bt (−t) = Bt (t), hence Bt may (and will) be interpreted as meromorphic function on H1 .  6 Finally, for generic t = (t1 , . . . , t6 ) ∈ C× we consider      6  θ qt2 t3 t4 t5 t6 z; q z 2  z/t j ; q ∞ 1 dz    1−    , Vt (t) = 2 q 2πi z t t θ qt t t t t t ; q z; q z/q, t /z; q C 1 2 3 4 5 6 j 1 1 ∞ ∞ j=2 

(5.15) where C is a deformation of T separating the decreasing pole sequence t1 q Z≥0 of the integrand from the remaining (increasing) pole sequences. As before, Vt uniquely extends  6 to a meromorphic function on C× /C2 . Similarly as for Ut (t), the trigonometric Barnes integral Bt (t) can be unfolded. Recall that q = e(τ ) with τ ∈ H+ , where e(x) is a shorthand notation for exp(2πi x). Lemma 5.16. For generic parameters u ∈ G0 we have   e((u 7 − u 2 )/τ ) − 1 2  Bt (ψ0 (2πiu)) =  τ q, q; q ∞ e(u 2 )θ (e(u 7 − u 2 ); q)    e(x − u 1 ), qe(x − u 2 ), qe(x − u 7 ), e(x − u 8 ); q ∞   × e(x + u 3 ), e(x + u 4 ), e(x + u 5 ), e(x + u 6 ); q ∞ L  e(x)   d x, × 1 − e((u 7 − x)/τ ) 1 − e((x − u 2 )/τ ) where the integration contour L is some translate ξ + R (ξ ∈ iR) of the real line with a finite number of indentations, such that C separates the pole sequences −u j + Z + Z≤0 τ ( j = 3, . . . , 6) of the integrand from the pole sequences u 2 + Z≥0 τ and u 7 + Z≥0 τ . Proof. The proof is similar to the proof of Corollary 5.3.

 

For Bt (t) and Vt (t) we have the following series expansions in balanced 4 φ3 ’s (respectively in a very-well-poised 8 φ7 ). Lemma 5.17. (a) We have     2 t2 /t1 , t2 /t8 ; q ∞ t t ,t t ,t t ,t t  4 φ3 2 3 2 4 2 5 2 6 ; q, q + (t2 ↔ t7 ) Bt (t) =  qt2 /t7 , t2 /t1 , t2 /t8 q, t7 /t2 , t2 t3 , t2 t4 , t2 t5 , t2 t6 ; q ∞ as meromorphic functions in t ∈ H1 . (b) We have    6   t1 /t j ; q ∞ t12 2 1   8 W7 Vt (t) =  2  ; t1 t2 , t1 t3 , . . . , t1 t6 ; q, q t1 t2 t3 t4 t5 t6 q, t1 ; q ∞ j=2 t1 t j ; q ∞  6 as meromorphic functions in t ∈ C× /C2 : |t1 t2 t3 t4 t5 t6 | > 1.

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Proof. This follows by a straightforward residue computation as in the proof of Lemma 5.5 (cf. also [5, §4.10]). For (a) one picks up the residues at the increasing pole sequences t2 q Z≥0 and t7 q Z≥0 of the integrand of Bt (t); for (b) one picks up the residues at the single increasing pole sequence t1 q Z≥0 of the integrand of Vt (t).   Proposition 5.18. For generic t ∈ H1 we have lim St (t1 , . . . , t6 , t7 u, t8 /u) = E t (t1 , . . . , t6 ),

u→0

  1 1 1 1 1 1 1 1 lim t2 t7 /u; q ∞ St (t1 u − 2 , t2 u − 2 , t3 u 2 , t4 u 2 , t5 u 2 , t6 u 2 , t7 u − 2 , t8 u − 2 ) = Bt (t), u→0   lim t1 t7 /u; q ∞ Ut (t1 , . . . , t6 , t7 /u, t8 u) = Vt (t1 , . . . , t6 ), u→0

1

1

1

1

1

1

1

1

lim Ut (t2 u 2 , t1 u 2 , t3 u − 2 , t4 u − 2 , t5 u − 2 , t6 u − 2 , t8 u 2 , t7 u 2 ) = Bt (t).

u→0

Proof. The first limit is direct. For the second limit, we follow the same approach as in the proof of Proposition 5.7. Define Q(z) as 1

Q(z) =

1

1

1

θ (t2 zs − 2 , t7 zs − 2 , µzs 2 , t2 t7 µs − 2 z −1 ; q) . θ (z 2 ; q)

Using (3.6) we obtain the equation Q(z) + Q(z −1 ) = θ (t2 t7 /s, t2 µ, t7 µ; q), and hence, as in the proof of Proposition 5.7,  (t2 t7 /u; q)∞ dz (t2 t7 /u; q)∞ St (tu ) = 2 It (tu ; z)Q(z) θ (t2 t7 /u, t2 µ, t7 µ; q) C 2πi z for an appropriate contour C, where we use the abbreviated notation 1

1

1

1

1

1

1

1

tu = (t1 u − 2 , t2 u − 2 , t3 u 2 , t4 u 2 , t5 u 2 , t6 u 2 , t7 u − 2 , t8 u − 2 ). 1

Taking u − 2 z as a new integration variable we obtain  (ut2 t7 ; q)∞ St (tu ) = 2

(z/t1 , z/t8 ; q)∞ θ (µz, t2 t7 µ/z; q) 6 C θ (t2 µ, t7 µ; q) j=3 (t j z; q)∞ (t2 /z, t7 /z; q)∞

×(1 −

u (ut1−1 /z, ut8−1 /z, qut2−1 /z, qut7−1 /z; q)∞ dz , )  z2 2πi z (qu/t2 t7 ; q)∞ 6j=3 (t j u/z; q)∞

where, for u small enough, we take C to be a u-independent deformation of T separating the decreasing pole sequences t2 q Z≥0 , t7 q Z≥0 and t j uq Z≥0 ( j = 3, . . . , 6) of the inteZ≤0 ( j = 3, . . . , 6). The limit u → 0 grand from the decreasing pole sequences t −1 j q can be taken in the resulting integral, leading to the desired result.

Properties of Generalized Univariate Hypergeometric Functions

77

third limit, we set µ = q/t1 t7 t8 in the integral expression of Ut (t) =  Toµ prove the dz J (t; z)  C t 2πi z to remove the contribution t7 z; q ∞ in the denominator of the integrand:    z2 θ (t1 t7 t8 /z; q) 1− Ut (t) = 2 q C  θ (t1 t7 , t7 t8 ; q)     6  z/t j ; q z/t7 , q/t7 z; q ∞ dz ∞     . × t j z; q ∞ t1 z/q, t1 /z, t8 z/q, t8 /z; q ∞ 2πi z j=2

In the resulting integral the desired limit can be taken directly, leading to the desired result. For the fourth limit, one easily verifies that 1

µu 2

Bt (t) = lim Ut u→0

1

1

1

1

1

1

1

1

(t2 u 2 , t1 u 2 , t3 u − 2 , t4 u − 2 , t5 u − 2 , t6 u − 2 , t8 u 2 , t7 u 2 ) 1

for generic t ∈ H1 after changing the integration variable z to zu 2 on the right-hand side.   Proposition 5.18 and Corollary 5.9 immediately lead to the following three trigonometric integral evaluations (of which the first is the well known Askey-Wilson integral evaluation [5, (6.1.4)]). Corollary 5.19. For generic parameters t = (t1 , t2 , t3 , t4 ) ∈ C4 we have   ±2    2 t1 t2 t3 t4 ; q ∞ z ;q ∞ dz     , = 4  ±1  2πi z q; q ∞ 1≤ j
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Proof. Specializing the degeneration from St to E t in Proposition 5.18 to generic parameters t ∈ H1 under the additional condition t1 t2 = 1 and using the trigonometric Nassrallah-Rahman integral evaluation (Corollary 5.9) leads to the Askey-Wilson integral evaluation with corresponding parameters (t3 , t4 , t5 , t6 ). Similarly, specializing the degeneration from Ut to Vt (respectively St to Bt ) to generic parameters t ∈ H1 under the additional condition t2 t3 = 1 (respectively t1 t3 = 1) and using the Nassrallah-Rahman integral evaluation we obtain the second (respectively third) integral evaluation with parameters (t1 , t4 , t5 , t6 ) (respectively (t2 , t4 , t5 , t6 , t7 , t8 )).   Various well-known identities are direct consequences of Corollary 5.19. Firstly, analogous to the unfolding of the integrals Ut and Vt (see Corollary 5.3 and Lemma 5.16), the left-hand side of the third integral evaluation can be unfolded. We obtain for  generic u ∈ C6 with 6j=1 u j = 0,   qe(x − u 1 ), qe(x − u 5 ), e(x − u 6 ); q ∞ e(x)      dx e(x + u 1 − e((u ), e(x + u ), e(x + u ); q − x)/τ ) 1 − e((x − u 1 )/τ ) L 2 3 4 5 ∞     q, 1/t2 t6 , 1/t3 t6 , 1/t4 t6 ; q ∞ τ t1 θ t5 /t1 ; q   , =  e((u 5 − u 1 )/τ ) − 1 t1 t2 , t1 t3 , t1 t4 , t2 t5 , t3 t5 , t4 t5 ; q ∞



where τ ∈ H+ such that q = e(τ ), where t j = e(u j ) ( j = 1, . . . , 6) and where the integration contour L is some translate ξ +R (ξ ∈ iR) of the real line with a finite number of indentations such that C separates the pole sequences −u j + Z + Z≤0 τ ( j = 2, 3, 4) of the integrand from the pole sequences u 1 + Z≥0 τ and u 5 + Z≥0 τ . This integral identity is Agarwal’s [5, (4.4.6)] trigonometric analogue of Barnes’ second lemma. The left-hand side of the second integral evaluation in Corollary 5.19 can be rewritten as a unilateral sum by picking up the residues at t1 q Z≥0 , cf. Lemma 5.5. The resulting identity is 6 φ5

 1 1 q −1 t12 , q 2 t1 , −q 2 t1 , t1 t2 , t1 t3 , t1 t4 1

1

q − 2 t1 , −q − 2 t1 , t1 /t2 , t1 /t3 , t1 /t4

; q,

1 t1 t2 t3 t4



 2 t , 1/t2 t3 , 1/t2 t4 , 1/t3 t4 ; q ∞  , =  1 1/t1 t2 t3 t4 , t1 /t2 , t1 /t3 , t1 /t4 ; q ∞

for generic t ∈ C4 satisfying |t1 t2 t3 t4 | > 1, which is the 6 φ5 summation formula [5, (2.7.1)].  For generic t ∈ C6 satisfying 6j=1 t j = 1 the left-hand side of the third integral evaluation in Corollary 5.19 can be written as a sum of two unilateral series by picking up the poles of the integrand at the decreasing sequences t1 q Z≥0 and t5 q Z≥0 of poles of the integrand. The resulting identity is       4    t1 /t6 ; q ∞ 1/t j t6 ; q ∞ t 1 t2 , t1 t3 , t1 t4    3 φ2  ; q, q + t1 ↔ t5 = qt1 /t5 , t1 /t6 t5 /t1 , t1 t2 , t1 t3 , t1 t4 ; q ∞ t t ,t t ;q ∞ j=2 1 j j 5  for generic t ∈ C6 satisfying 6j=1 t j = 1, which is the nonterminating version [5, (2.10.12)] of Saalschütz formula. We now return to the three trigonometric hypergeometric integrals E t , Bt and Vt . + = Recall that the symmetry group of St and Ut is the subgroup W (E 6 ) = W (E 7 )α18 W (E 7 )γ18 , which is a maximal standard parabolic subgroup of W (E 7 ) with respect to both bases 1 and 2 of R(E 7 ) (see Sect. 2), with corresponding sub-bases 1 = − − 1 \ {α12 } and 2 = 2 \ {α87 } respectively. The four limits of Proposition 5.18 now

Properties of Generalized Univariate Hypergeometric Functions

79

imply that the trigonometric integrals E t , Bt and Vt have symmetry groups W (E 6 )α − 78 or W (E 6 )β1278 . The stabilizer subgroup W (E 6 )α − is a standard maximal parabolic sub78 group of W (E 6 ) with respect to both bases 1 or 2 , with corresponding sub-basis − − (D5 ) = 1 \ {α76 } = 2 \ {α18 }

and with corresponding Dynkin sub-diagrams − α32 •

− α43 •

− α54 •

− α65 •

− α76 ◦

− α18 ◦

β5678 •

− α45 •

− α34 •

− α23 •

− • α56

• β1234

respectively. Similarly, W (E 6 )β1278 is a standard maximal parabolic subgroup of W (E 6 ) with respect to the basis 2 , with corresponding sub-basis − }  (D5 ) = 2 \ {α23

and with corresponding Dynkin sub-diagram − α18 •

β5678 •

− α45 •

− α34 •

− α23 ◦

− • α56

We write W (D5 ) = W (E 6 )α − , 78

W  (D5 ) = W (E 6 )β1278

for the corresponding isotropy group, which are both isomorphic to the Weyl group of type D5 .  6 The isotropy group W (D5 ) acts on C× /C2 : the simple reflections corresponding to roots of the form αi−j ∈ (D5 ) act by permuting the i th and j th coordinate, while w acts by   w(±t) = ± st1 , st2 , st3 , st4 , t5 /s, t6 /s , s 2 = 1/t1 t2 t3 t4 . With this action, the degenerations to E t and Vt in Proposition 5.18 are W (D5 )-equivariant in an obvious sense. We can now directly compute the W (D5 )-symmetries of the trigonometric integrals E t and Vt , as well as W  (D5 )-symmetries of Bt , by taking limits of the corresponding symmetries for St and Ut using Proposition 5.18. This yields the following result. Proposition 5.20. a) The trigonometric hypergeometric integrals E t (t) and Vt (t)  6 (t ∈ C× /C2 ) are invariant under permutations of (t2 , . . . , t6 ). Furthermore, (1/t1 t2 , 1/t1 t3 , 1/t1 t4 , t1 t2 t3 t4 t5 t6 ; q)∞ , (t2 t3 , t2 t4 , t3 t4 , t5 t6 ; q)∞ (1/t2 t3 , 1/t2 t4 , 1/t3 t4 , 1/t5 t6 ; q)∞ Vt (t) = Vt (wt) . (t1 t2 , t1 t3 , t1 t4 , 1/t1 t2 t3 t4 t5 t6 ; q)∞  6 as meromorphic functions in t ∈ C× /C2 . E t (t) = E t (wt)

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b) The trigonometric Barnes integral Bt (t) (t ∈ H1 ) is invariant under permutations of the pairs (t1 , t8 ), (t2 , t7 ) and of (t3 , t4 , t5 , t6 ). Furthermore,   1/t1 t3 , 1/t1 t4 , 1/t5 t8 , 1/t6 t8 ; q ∞   Bt (t) = Bt (wt) t2 t4 , t2 t3 , t5 t7 , t6 t7 ; q ∞ as meromorphic functions in t ∈ H1 . Remark 5.21. The w-symmetry of Vt , rewritten in series form using Lemma 5.17, gives the transformation formula [5, (2.10.1)] for very-well-poised 8 φ7 basic hypergeometric series. Similarly as in the hyperbolic theory, the w-symmetry of E t generalizes to the following integral transformation formula for the trigonometric Euler integral E t .  6 Proposition 5.22. For t ∈ C× and s ∈ C× satisfying |t2 |, |t3 |, |t4 |, |s|, |t5 /s|, |t6 /s| < 1 we have



 T

E t (t1 , t2 , t3 , t4 , sx, sx   2 t5 t 6 ; q ∞

−1

x ±2 ; q



dx ∞  )  ±1 t5 x /s, t6 x ±1 /s; q ∞ 2πi x

 E t (t1 , . . . , t6 ). = q, s 2 , t5 t6 /s 2 ; q ∞ Proof. The proof is similar to the hyperbolic case (see Proposition 4.13), now using the double integral  ±2 ±1   z , z /t1 , x ±2 ; q ∞ dz d x .   4   6   ±1 ±1 ±1 ±1 2πi z 2πi x T2 sz x ; q j=2 t j z ; q k=5 tk x /s; q ∞

∞

∞

  Specializing s 2 = 1/t1 t2 t3 t4 in Proposition 5.22 and using the trigonometric Nassrallah-Rahman integral (see Corollary 5.19), we re-obtain the w-symmetry of E t (see Proposition 5.20). The three trigonometric integrals E t , Bt and Vt are interconnected as follows. Proposition 5.23. We have     1/t1 t6 , 1/t8 t6 ; q ∞  Vt t7 /s, t3 s, t4 s, t5 s, t1 /s, t8 /s , Bt (t) =  t2 t3 , t2 t4 , t2 t5 , t6 t7 ; q ∞    1/t1 t3 , 1/t1 t4 , 1/t1 t5 , 1/t6 t8 ; q ∞    = E t t8 /v, t7 /v, t3 v, t4 v, t5 v, t2 /v , t2 t6 , t7 t6 ; q ∞ as meromorphic functions in t ∈ H1 , where s 2 = t1 t6 t7 t8 = 1/t2 t3 t4 t5 and v 2 = t2 t6 t7 t8 = 1/t1 t3 t4 t5 .

Properties of Generalized Univariate Hypergeometric Functions

81

Proof. This follows by combining Proposition 5.12 and Proposition 5.18. Concretely, to relate Bt and Vt one computes for generic t ∈ H1 and with s 2 = 1/t2 t3 t4 t5 ,    1 1 1 1 1 1 1 1 Bt (t) = lim t2 t7 /u; q ∞ St t1 u − 2 , t2 u − 2 , t3 u 2 , t4 u 2 , t5 u 2 , t6 u 2 , t7 u − 2 , t8 u − 2 u→0   1/t1 t6 , 1/t8 t6 ; q ∞  =  t2 t3 , t2 t4 , t2 t5 , t6 t7 ; q ∞     × lim t2 t7 /u; q ∞ Ut t7 /s, t3 s, t4 s, t5 s, t1 /s, t8 /s, t2 s/u, t6 u/s u→0     1/t1 t6 , 1/t8 t6 ; q ∞  Vt t7 /s, t3 s, t4 s, t5 s, t1 /s, t8 /s , =  t2 t3 , t2 t4 , t2 t5 , t6 t7 ; q ∞ where the first and third equality follows from Proposition 5.18 and the second equality follows from Proposition 5.12. To relate Bt and E t , we first note that Proposition 5.12 is equivalent to the identity Ut (t) =

     1/t6 t7 ; q ∞ 2≤ j
where t ∈ H1 and s 2 = 1/t2 t3 t4 t5 . For generic t ∈ H1 and with v 2 = 1/t1 t3 t4 t5 we then compute  1 1 1 1 1 1 1 1 Bt (t) = lim Ut t2 u 2 , t1 u 2 , t3 u − 2 , t4 u − 2 , t5 u − 2 , t6 u − 2 , t8 u 2 , t7 u 2 u→0   1/t1 t3 , 1/t1 t4 , 1/t1 t5 , 1/t6 t8 ; q ∞   = t2 t6 , t7 t6 ; q ∞   × lim St t8 /v, t7 /v, t3 v, t4 v, t5 v, t2 /v, t1 vu, t6 /vu u→0    1/t1 t3 , 1/t1 t4 , 1/t1 t5 , 1/t6 t8 ; q ∞    E t t8 /v, t7 /v, t3 v, t4 v, t5 v, t2 /v , = t2 t6 , t7 t6 ; q ∞ where the first and third equality follows from Proposition 5.18 and the second equality follows from (5.16).   Remark 5.24. a) Combining the interconnection between E t and Vt from Proposition 5.23 with the expression of Vt as a very-well-poised 8 φ7 series from Lemma 5.17 yields the Nassrallah-Rahman integral representation [5, (6.3.7)]. b) Similarly, combining the interconnection between Bt and Vt from Proposition 5.23 with their series expressions from Lemma 5.17 yields the expression [5, (2.10.10)] of a very-well-poised 8 φ7 series as a sum of two balanced 4 φ3 series. Degenerating the contiguous relations of St using Proposition 5.18 leads directly to contiguous relations for E t , Bt and Vt . For instance, we obtain Proposition 5.25. We have A(t)E t (t1 , t2 , t3 , qt4 , t5 /q, t6 ) + (t4 ↔ t5 ) = B(t)E t (t)

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F. J. van de Bult, E. M. Rains, J. V. Stokman

 6 as meromorphic functions in t ∈ C× /C2 , where A(t) = −

(1 −

t2 t5 t3 t5 t6 t5 1 t1 t5 )(1 − q )(1 − q )(1 − q ) , t4 (1 − t4qt5 )(1 − qtt54 )(1 − tt45 )

qt4 qt5 q − t2 t3 t6 + A(t) + A(s45 t). t1 t4 t5 t5 t4  Proof. Substitute t = (t1 , t2 , t3 , t4 , t5 , t7 u, t6 , t8 /u) with 8j=1 t j = 1 in (5.11) and take the limit u → 0.   B(t) =

For later purposes, we also formulate the corresponding result for Bt (t). We substitute  1 1 1 1 1 1 1 1 t = t1 u − 2 , t7 u − 2 , t3 u 2 , t4 u 2 , t5 u 2 , t2 u − 2 , t6 u 2 , t8 u − 2 for generic t ∈ C8 satisfying  8 1 2 j=1 t j = 1 in (5.11), multiply the resulting equation by u t2 t7 /u; q ∞ , and take the limit u → 0. We arrive at α(t)Bt (t1 , t2 , t3 , qt4 , t5 /q, t6 , t7 , t8 ) + (t4 ↔ t5 ) = β(t)Bt (t), where



t ∈ H1 , (5.17)

    1 − t2qt5 1 − t5qt7 1 − t51t8    α(t) = − , t4 1 − tt45 1 − qtt54     1 − t2qt4    1 − t2 t4  α(s45 t).  α(t) +  β(t) = t7 1 − t2 t3 1 − t2 t6 +  1 − t2 t5 1 − t2qt5 1−

1 t1 t5

The degeneration of (5.12) yields Bailey’s [5, (2.11.1)] three term transformation formula for very-well-poised 8 φ7 ’s: Proposition 5.26. We have θ (t1−1 t2±1 ; q) θ (t3 t2±1 ; q)

E t (t3 /q, qt1 , t2 , t4 , t5 , t6 ) + (t2 ↔ t3 ) = E t (t).

Proof. Consider (5.12) with t1 and t8 , t2 and t7 , and t3 and t6 interchanged. Subsequently  substitute the parameters (t1 , t2 , t3 , t4 , t5 , t6 , t7 u, t8 /u) with 8j=1 t j = 1 and take the limit u → 0.   5.5. The Askey-Wilson function. In this subsection we relate the trigonometric hypergeometric integrals with D5 symmetry to the nonpolynomial eigenfunction of the Askey-Wilson second order difference operator, known as the Askey-Wilson function. The Askey-Wilson function is the trigonometric analog of Ruijsenaars’ R-function, and is closely related to harmonic analysis on the quantum SU(1, 1) group. As for the R-function, we introduce the Askey-Wilson function in terms of the trigonometric Barnes integral Bt . Besides the usual Askey-Wilson parameters we also use logarithmic variables in order to make the connection to the R-function more transparent. We write the base q ∈ C× with |q| < 1 as q = e(ω1 /ω2 ) with τ = ω1 /ω2 ∈ H+ and e(x) = exp(2πi x) as before. From the previous subsection it follows that the parameter space of Bt (t) is H1 /C× e(β1278 ). In logarithmic coordinates, this relates to

Properties of Generalized Univariate Hypergeometric Functions

83

G0 /Cβ1278 . We identify G0 /Cβ1278 with C6 by assigning to the six-tuple (γ , x, λ) = (γ0 , γ1 , γ2 , γ3 , λ, x) the class in G0 /Cβ1278 represented by u = (u 1 , . . . , u 8 ) ∈ G0 with   u 1 = −γ0 − γ1 − 2ω /ω2 , u 2 = 0,     u 4 = γ0 + ω − i x /ω2 , u 3 = γˆ0 + ω − iλ /ω2 ,     (5.18) u 6 = γˆ0 + ω + iλ /ω2 , u 5 = γ0 + ω + i x /ω2 ,     u 8 = −γ0 − γ2 − 2ω /ω2 , u 7 = −γ0 − γ3 /ω2 , where ω = 21 (ω1 +ω2 ) as before. We define the corresponding six-tuple of Askey-Wilson parameters (a, b, c, d, µ, z) by   (a, b, c, d) = e((γ0 + ω)/ω2 ), e((γ1 + ω)/ω2 ), e((γ2 + ω)/ω2 ), e(γ3 + ω)/ω2 ) ,   (5.19) (µ, z) = e(−iλ/ω2 ), e(−i x/ω2 ) . The four-tuple (a, b, c, d) represents the four parameter freedom in the AskeyWilson theory, while z (respectively µ) plays the role of geometric (respectively spectral) parameter. Furthermore, we define the dual Askey-Wilson parameters by     = e((γˆ0 + ω)/ω2 ), e((γˆ1 + ω)/ω2 ), e((γˆ2 + ω)/ω2 ), e((γˆ3 + ω)/ω2 ) ,  a,  b, c, d) with γˆ the dual parameters defined by (4.29). We furthermore associate to the logarithmic parameters u ∈ G0 (see (5.18)) the parameters t = ψ0 (2πiu) ∈ H1 , so that   t = ψ0 (2πiu) = 1/ab, 1, a µ, az, a/z, a /µ, q/ad, 1/ac . (5.20)  We define the Askey-Wilson function φ(γ ; x, λ) = φ γ ; x, λ; ω1 , ω2 ) by   q, t2 t3 , t2 t4 , t2 t5 , t2 t6 ; q ∞ Bt (t) φ(γ ; x, λ) = (5.21) 2 with t = ψ0 (2πiu) ∈ H1 and u given by (5.18). Note the similarity to the definition of Ruijsenaars’ R-function, see (4.27). From the series expansion of Bt as sum of two balanced 4 φ3 , we have in terms of Askey-Wilson parameters (5.19),    ±1  ab, ac; q ∞ a µ±1 az ,  4 φ3 ; q, q φ(γ ; x, λ) =  ab, ac, ad q/ad; q ∞ (5.22)    ±1  qb/d, qc/d, a µ±1 , az ±1 ; q ∞ qz /d, q a µ±1 /ad  4 φ3 ; q, q , + q 2 /ad, qb/d, qc/d ad/q, q a µ±1 /ad, qz ±1 /d; q ∞ which shows that φ(γ ; x, λ) is, up to a (z, γ )-independent rescaling factor, the AskeyWilson function as defined in e.g. [11]. We now re-derive several fundamental properties of the Askey-Wilson function using the results of the previous subsection. Comparing the symmetries of the Askey-Wilson function φ(γ ; x, λ) to the symmetries of the R-function (Proposition 4.18), the symmetry in the parameters γ is broken (from the Weyl group of type D4 to the Weyl group of type D3 ). The most important symmetry (self-duality) is also valid for the Askey-Wilson function and has played a fundamental role in the study of the associated generalized Fourier transform (see [11]). Self-duality of the Askey-Wilson function has a natural interpretation in terms of Cherednik’s theory on double affine Hecke algebras, see [36]. Concretely, the symmetries of the Askey-Wilson function are as follows.

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Proposition 5.27. The Askey-Wilson function φ(γ ; x, λ) is even in x and λ and is selfdual, φ(γ ; x, λ) = φ(γ ; −x, λ) = φ(γ ; x, −λ) = φ(γˆ ; λ, x). Furthermore, φ(γ ; x, λ) has a W (D3 )-symmetry in the parameters γ , given by   e((−γˆ3 + ω ± iλ)/ω2 ); q ∞  φ(γ ; x, λ), φ(γ1 , γ0 , γ2 , γ3 ; x, λ) =  e((γˆ2 + ω ± iλ)/ω2 ); q ∞ φ(γ0 , γ2 , γ1 , γ3 ; x, λ) = φ(γ ; x, λ),   e((−γ3 + ω ± i x)/ω2 ); q ∞  φ(γ0 , γ1 , −γ3 , −γ2 ; x, λ) =  φ(γ ; x, λ). e((γ2 + ω ± i x)/ω2 ); q ∞ Proof. Similarly as for the R-function (see Proposition 4.18), the symmetries of the Askey-Wilson function correspond to the W (D5 )-symmetries of Bt . Alternatively, all symmetries follow trivially from the series expansion (5.22) of the Askey-Wilson function, besides its symmetry with respect to γ0 ↔ γ1 and γ2 ↔ −γ3 . These two symmetry relations are equivalent under duality, since (γ1 , γ0 , γ2 , γ3 )ˆ = (γˆ0 , γˆ1 , −γˆ3 , −γˆ2 ), so we only discuss the symmetry with respect to γ0 ↔ γ1 . By Proposition 5.20 we have, with parameters t given by (5.20) and (5.18),   q, e((γ1 + ω ± i x)/ω2 ), e((γˆ0 + ω ± iλ)/ω2 ); q ∞ φ(γ1 , γ0 , γ2 , γ3 ; x, λ) = Bt (ws35 t) 2  e((−γˆ3 + ω ± iλ)/ω2 ); q ∞  =  φ(γ ; x, λ), e((γˆ2 + ω ± iλ)/ω2 ); q ∞ as desired.

 

Next we show that the Askey-Wilson function satisfies the same Askey-Wilson second order difference equation (with step-size iω1 ) as Ruijsenaars’ R-function, a result which has previously been derived from detailed studies of the associated Askey-Wilson polynomials in [9], cf. also [11]. Lemma 5.28. The Askey-Wilson function φ(γ ; x, λ) satisfies the second order difference equation A(γ ; x; ω1 , ω2 )(φ(γ ; x + iω1 , λ; ω1 , ω2 ) − φ(γ ; x, λ; ω1 , ω2 )) + (x ↔ −x) = B(γ ; λ; ω1 , ω2 )φ(γ ; x, λ; ω1 , ω2 ), where A and B are given by 3

j=0 sinh(π(iω

+ x + iγ j )/ω2 )

, sinh(2π x/ω2 ) sinh(2π(iω + x)/ω2 ) B(γ ; λ; ω1 , ω2 ) = sinh(π(λ − iω − i γˆ0 )/ω2 ) sinh(π(λ + iω + i γˆ0 )/ω2 ). A(γ ; x; ω1 , ω2 ) =

Proof. Specialize the parameters according to (5.20) and (5.18) in (5.17). Subsequently express Bt (t), Bt (τ45 t) and Bt (τ54 t) in terms of φ(γ ; x, λ), φ(γ ; x +iω1 , λ) and φ(γ ; x − iω1 , λ) respectively. The resulting equation is the desired difference equation.  

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Remark 5.29. Denoting (z; µ) = (a, b, c, d; z, µ) for the Askey-Wilson function in the usual Askey-Wilson parameters, Lemma 5.28 becomes the Askey-Wilson second order difference equation A(z)((qz, µ) − (z, µ)) + A(z −1 )((z/q, µ) − (z, µ))   =  a (γ + γ −1 ) − 1 −  a 2 (z, µ), where A(z) =

(1 − az)(1 − bz)(1 − cz)(1 − dz) (1 − qz 2 )(1 − z 2 )

and  a = e((γˆ0 + ω)/ω2 ). We have now seen that the R-function R(γ ; x, λ; ω1 , ω2 ) as well as the Askey-Wilson function φ(γ ; x, λ; ω1 , ω2 ) are solutions to the eigenvalue problem Lωγ 1 ,ω2 f = B(γ ; λ; ω1 , ω2 ) f

(5.23)

for the Askey-Wilson second order difference operator Lωγ 1 ,ω2 (4.31) with step-size iω1 . These two solutions have essentially different behaviour in the iω2 -step direction: the Askey-Wilson function φ(γ ; x, λ) is iω2 -periodic, while the R-function R(γ ; x, λ) is ω1 ↔ ω2 invariant (hence is also an eigenfunction of the Askey-Wilson second order difference operator Lωγ 2 ,ω1 with step-size iω2 , with eigenvalue B(γ ; λ; ω2 , ω1 )). On the other hand, note that  τ = −ω2 /ω1 ∈ H+ and that A(γ ; x; ω2 , ω1 ) = A(−γ ; −x; −ω2 , ω1 ),

B(γ ; λ; ω2 , ω1 ) = B(−γ ; λ; −ω2 , ω1 )

with −γ = (−γ0 , −γ1 , −γ2 , −γ3 ), so that the Askey-Wilson function φ(−γ ; x, λ; −ω2 , ω1 ) (with associated modular inverted base  q = e(−ω2 /ω1 )) does satisfy the AskeyWilson second order difference equation   ω ,ω Lγ 2 1 φ(−γ ; · , λ; −ω2 , ω1 ) (x) = B(γ ; λ; ω2 , ω1 )φ(−γ ; x, λ; −ω2 , ω1 ), (5.24) cf. [27, §6.6]. In the next section we match the hyperbolic theory to the trigonometric theory, which in particular entails an explicit expression of the R-function in terms of products of Askey-Wilson functions in base q and base  q. Note furthermore that Proposition 5.27 hints at the fact that the solution space to the Askey-Wilson eigenvalue problem (5.23) admits a natural twisted W (D4 )-action on the parameters γ . In fact, the solution space to (5.23) is invariant under permutations of (γ0 , γ1 , γ2 , γ3 ). Furthermore, a straightforward computation shows that 2 g(γ ; ·)−1 ◦ Lωγ 1 ,ω2 ◦ g(γ ; ·) = Lω(γ10,ω ,γ1 ,−γ3 ,−γ2 ) + B(γ ; λ; ω1 , ω2 )

−B(γ0 , γ1 , −γ3 , −γ2 ; λ; ω1 , ω2 ) for the gauge factor

  e((γ2 + ω ± i x)/ω2 ); q ∞  , g(γ ; x) =  e((−γ3 + ω ± i x)/ω2 ); q ∞

which implies that for a given solution Fλ (γ0 , γ1 , −γ3 , −γ2 ; · ) to the eigenvalue problem 2 Lω(γ10,ω ,γ1 ,−γ3 ,−γ2 ) f = B(γ0 , γ1 , −γ3 , −γ2 ; λ; ω1 , ω2 ) f,

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we obtain a solution λ (γ ; x) := g(γ ; x)Fλ (γ0 , γ1 , −γ3 , −γ2 ; x) F to the eigenvalue problem (5.23). A similar observation forms the starting point of Ruijsenaars’ [28] analysis of the W (D4 )-symmetries of the R-function (see also Sect. 4.6). Remark 5.30. A convenient way to formalize the W (D4 )-symmetries of the eigenvalue problem (5.23) (in the present trigonometric setting) is by interpreting iω2 -periodic solutions to (5.23), depending meromorphically on (γ , x, λ), as defining a sub-vector bundle   0 (ω1 , ω2 ) of the meromorphic vector bundle  0 (ω1 , ω2 ) over 4  X = C/Zω2 × C/(Ziω1 + Ziω2 ) × C/Ziω2 4  consisting of meromorphic functions in (γ , x, λ) ∈ C/Zω2 × C/Ziω2 × C/Ziω2 . The above analysis can now equivalently be reformulated as the following property of   0 (ω1 , ω2 ): the sub-vector bundle   0 (ω1 , ω2 ) is W (D4 )-invariant with respect to the twisted W (D4 )-action (σ · f )(γ ; x, λ) := Vσ (γ ; x, λ)−1 f (σ −1 γ ; x, λ),  0 (ω1 , ω2 ),

σ ∈ W (D4 )

(5.25)

h(σ −1 γ ; x, λ)/ h(γ ; x, λ)

where Vσ (γ ; x, λ) = (σ ∈ W (D4 )) is the on 1-coboundary with h(γ ; x, λ) = h(γ ; x, λ; ω1 , ω2 ) e.g. given by h(γ ; x, λ; ω1 , ω2 )

  θ e((γ3 − γˆ0 + i x)/ω2 ); q =   3   ∈ 0 (ω1 , ω2 )× , e((ω − γ θ e((ω − γ3 − i x)/ω2 ); q + i x)/ω ); q j 2 j=0 ∞ (5.26)

and where W (D4 ) acts on the γ parameters by permutations and even sign changes. By a straightforward analysis using Casorati-determinants and the asymptotically free solutions to the eigenvalue problem (5.23), one can furthermore show that   0 (ω1 , ω2 ) is a (trivial) meromorphic vector bundle over X of rank two (compare with the general theory on difference equations in [19]). We end this subsection by expressing the Askey-Wilson function φ(γ ; x, λ) in terms of the trigonometric integrals E t and Vt using Proposition 5.23. Note its close resemblance with the hyperbolic case, cf. Theorem 4.21. Lemma 5.31. a) We have φ(γ ; x, λ)     q; q ∞ e((γˆ0 + ω − iλ)/ω2 ), e((γˆ1 + ω + iλ)/ω2 ), e((γˆ2 + ω + iλ)/ω2 ); q ∞   = 2 e((−γˆ3 + ω − iλ)/ω2 ); q ∞  2  j=0 e((γ j + ω ± i x)/ω2 ); q ∞  ×  E t (t) e((−γ3 + ω ± i x)/ω2 ); q ∞ with 3ω γˆ0 iλ + γ3 − + )/ω2 ), 2 2 2 ω γˆ0 iλ + )/ω2 ), t3 = e(( + γ1 − 2 2 2 ω γˆ0 iλ t5 = e(( + + i x − )/ω2 ), 2 2 2

t1 = e((−

ω γˆ0 iλ + γ2 − + )/ω2 ), 2 2 2 ω γˆ0 iλ + )/ω2 ), t4 = e(( + γ0 − 2 2 2 ω γˆ0 iλ t6 = e(( + − i x − )/ω2 ). 2 2 2

t2 = e((

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b) We have φ(γ ; x, λ)     q; q ∞ e((γˆ0 + ω + iλ)/ω2 ), e((γˆ1 + ω − iλ)/ω2 ), e((γˆ2 + ω − iλ)/ω2 ); q ∞   Vt (t) = 2 e((−γˆ3 + ω + iλ)/ω2 ); q ∞

with iλ 3ω γˆ0 − γ3 + − )/ω2 ), 2 2 2 iλ ω γˆ0 − )/ω2 ), t3 = e((− − γ1 + 2 2 2 iλ ω γˆ0 + i x + )/ω2 ), t5 = e((− − 2 2 2

t1 = e((

iλ ω γˆ0 − γ2 + − )/ω2 ), 2 2 2 iλ ω γˆ0 − )/ω2 ), t4 = e((− − γ0 + 2 2 2 iλ ω γˆ0 − i x + )/ω2 ). t6 = e((− − 2 2 2

t2 = e((−

Proof. a) We use Proposition 5.27 to rewrite φ(γ ; x, λ) in terms of φ(γ0 , γ1 , −γ3 , −γ2 ; x, −λ). Subsequently we use the defining expression of φ(γ0 , γ1 , −γ3 , −γ2 ; x, −λ) to obtain φ(γ ; x, λ)   q, e((γˆ1 + ω ± iλ)/ω2 ), e((γ0 + ω ± i x)/ω2 ), e((γ2 + ω ± i x)/ω2 ); q ∞   Bt (ξ ) = 2 e((−γ3 + ω ± i x)/ω2 ); q ∞

with  ξ = e((−γ0 − γ1 − 2ω)/ω2 ), 1, e((γˆ1 + ω + iλ)/ω2 ), e((γ0 + ω − i x)/ω2 ), e((γ0 + ω + i x)/ω2 ), e((γˆ1 + ω − iλ)/ω2 ), e((γ2 − γ0 )/ω2 ),  e((γ3 − γ0 − 2ω)/ω2 ) . With this specific ordered set ξ of parameters we apply Proposition 5.23 to rewrite Bt (ξ ) in terms of E t , which results in the desired identity. b) This follows from applying Proposition 5.23 directly to the definition (5.21) of φ(γ ; x, λ).   Using the expression of the Askey-Wilson function in terms of Vt and using Lemma 5.17, we thus obtain an expression of the Askey-Wilson function as very-wellpoised 8 φ7 series. 6. Hyperbolic Versus Trigonometric Theory 6.1. Hyperbolic versus trigonometric gamma functions. We fix throughout this section periods ω1 , ω2 ∈ C with (ω1 ) > 0, (ω2 ) > 0 and τ = ω1 /ω2 ∈ H+ . We set q = qω1 ,ω2 = e(ω1 /ω2 ),

 q = qω1 ,ω2 = e(−ω2 /ω1 ),

where e(x) = exp(2πi x) as before, so that |q|, | q | < 1. Shintani’s [30] product expansion is      e((i x + ω)/ω2 ); q ∞ x2 1  ω1 ω2    , (6.1) G(ω1 , ω2 ; x) = e − e − + 48 ω2 ω1 4ω1 ω2 e((i x − ω)/ω1 );  q ∞

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where ω = 21 (ω1 +ω2 ) as before. For a proof of (6.1), see [37, Prop. A.1]. In other words, the product expansion (6.1) expresses the hyperbolic gamma function as a quotient of two trigonometric gamma functions (one in base q, the other in the modular inverted base  q ). In this section we explicitly write the base-dependence; e.g. we write St (t; q) (t ∈ H1 ) to denote the trigonometric hypergeometric function St (t) in base q. 6.2. Hyperbolic versus trigonometric hypergeometric integrals. We explore (6.1) to relate the hyperbolic integrals to their trigonometric analogs. We start with the hyperbolic hypergeometric function Sh (u) (u ∈ G2iω ). For u ∈ G2iω we write      t j = e (iu j + ω)/ω2 , t j = e (iu j − ω)/ω1 , j = 1, . . . , 8. (6.2)   q 6. Observe that 8j=1 t j = q 2 and 8j=1  tj =  Theorem 6.1. As meromorphic functions of u ∈ G2iω we have     5 5
q;  q t7 / t j    q,  q;  q ∞ j=1 θ  2 2 2 2 2   Sh (u) = ω2 e (2ω + u j − u 6 + u 7 − u 8 )/2ω1 ω2   2 θ / t ;  q t 7 6 j=1  3    (6.3) 1 1 3 × Ut  q 2 / t8 ,  q 2 / t1 , . . . ,  q 2 / t6 ,  q 2 / t7 ;  q St t6 /q, t1 , . . . , t5 , t7 , t8 /q; q + (u 6 ↔ u 7 ), with the parameters t j and  t j given by (6.2). Proof. We put several additional conditions on the parameters, which can later be removed by analytic continuity. We assume that ω1 , −ω2 ∈ H+ and that (iω) < 0. We furthermore choose parameters u ∈ G2iω satisfying (u j −iω) > 0 and (u j −iω) < 0 for j = 1, . . . , 8. Then  G(iω ± 2x) dx Sh (u) = 8 R j=1 G(u j ± x)    8  7  ω1 ω2   2
2  (x)d x, =e e (ω + + u j )/2ω1 ω2 W (x)W 24 ω2 ω1 R j=1

where

  e(±2i x/ω2 ); q ∞ W (x) = 8   , j=1 t j e(±i x/ω2 ); q ∞  8  q ∞ t j e(±i x/ω1 );   j=1   2    W (x) = e 2x /ω1 ω2  q e(±2i x/ω1 );  q ∞

by (6.1). Using Cauchy’s Theorem and elementary asymptotic estimates of the integrand, we may rotate the integration contour R to iω2 R. Since the factor W (x) is iω2 -periodic, we can fold the resulting integral, interchange summation and integration by Fubini’s Theorem, to obtain the expression    8  iω2 7  ω1 ω2   2
2 Sh (u) = e e (ω + + u j )/2ω1 ω2 W (x)F(x)d x, 24 ω2 ω1 0 j=1

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where F(x) =

∞


 (x + miω2 ) W

n=−∞

  q e(i x/ω1 ), { q t −1 e(i x/ω1 )}8j=1  q e(i x/ω1 ), − j  = W (x) 10 ψ10 ; q,  q . e(i x/ω1 ), −e(i x/ω1 ), { t j e(i x/ω1 )}8j=1 At this stage we have to resort to [5, (5.6.3)], which expresses a very-well-poised 10 ψ10 bilateral series as a sum of three very-well-poised 10 φ9 unilateral series. This results in the formula     t6 e(±i x/ω1 ),  F(x) = e 2x 2 /ω1 ω2 θ  t7 e(±i x/ω1 );  q       q;  q ∞ 5j=1  t8 / t j q,  q t j / t8 ;  q ∞  2 2  2  7   10 W9  × 2 q / t ; { q / t } ;  q ,  q t j 8 8 j=1 2 t8 ,  t8 ,  t6 q 2 / t7 t8 / t6 ,  t8 / t7 ,  q 3 / t8 ;  q ∞  q / +(u 8 ; u 6 , u 7 ),

where (u 8 ; u 6 , u 7 ) means cyclic permutation of the parameters (u 8 , u 6 , u 7 ). Note that the 10 φ9 series in the expression of F(x) are independent of x. Combining Jacobi’s inversion formula, the Jacobi triple product identity and the modularity      q; q ∞ ω2 1  ω1 ω2    = e − + (6.4) −iω1 24 ω2 ω1  q;  q ∞ of Dedekind’s eta function, we obtain        1  ω1 ω2   e (u + ω)2 /2ω1 ω2 θ e(−u/ω2 ); q (6.5) q =e − + θ e(u/ω1 );  24 ω2 ω1 for the rescaled Jacobi theta function θ (·), see e.g. [5] or [37]. As a result, we can rewrite the theta functions in the expression of F(x) as theta functions in base q,     t6 e(±i x/ω1 ),  e 2x 2 /ω1 ω2 θ  t7 e(±i x/ω1 );  q      1  ω1 ω2   =e − + e −(u 26 + u 27 )/ω1 ω2 θ qe(±i x/ω2 )/t6 , qe(±i x/ω2 )/t7 ; q . 6 ω2 ω1 We thus obtain the expression  iω2   θ qe(±i x/ω2 )/t6 , qe(±i x/ω2 )/t7 ; q W (x)d x Sh (u) = C(u 8 ; u 6 , u 7 ) 0

+ (u 8 ; u 6 , u 7 )

(6.6)

 = iω2 C(u 8 ; u 6 , u 7 )St t6 /q, t1 , . . . , t5 , t8 , t7 /q; q) + (u 8 ; u 6 , u 7 ), where we have used that |t j | < 1 for j = 1, . . . , 6, with ⎛ ⎞   5 1  ω1 ω2  ⎝ 2
2 e (ω + + u j − u 26 − u 27 + u 28 )/2ω1 ω2 ⎠ C(u 8 ; u 6 , u 7 ) = e 8 ω2 ω1 j=1   5    q;  q ∞ j=1  q,  q t j / t8 ;  q ∞ t8 / t j  × 2  q / t6 q 2 / t7 t8 / t6 ,  t8 / t7 ,  q 3 / t82 ;  q ∞ t8 ,  t8 ,   2 2  t8 }7j=1 ;  q / ×10 W9  t8 ; { q 2 / t j q,  q .

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We thus have obtained an expression of Sh (u) as a sum of three trigonometric hypergeometric functions St in base q, with coefficients expressed as very-well-poised 10 φ9 series in base  q . The next step is to use a three term transformation for St to write Sh (u) as a sum of two trigonometric hypergeometric functions St in base q, with coefficients now being a sum of two very-well-poised 10 φ9 series. Concretely, we consider the contiguous relation (3.8) for Se with p ↔ q and with parameters specialized to (t1 , . . . , t5 , pt6 , pt7 , t8 ). Taking the limit p → 0 leads to the three term transformation  5      θ t7 /t8 ; q  θ t j t6 ; q    St t6 /q, t1 , . . . , t5 , t8 , t7 /q; q =  θ t7 /t6 ; q j=1 θ t j t8 ; q   ×St t7 /q, t1 , . . . , t6 , t8 /q; q + (u 6 ↔ u 7 )(6.7) for St . Rewriting the coefficients in (6.7) in base  q using the Jacobi inversion formula (6.5),   5    5  2   θ   2 t6 ;  θ t7 /t8 ; q  θ t j t6 ; q t7 ;  q θ  q / t j q t8 / 2     = e (u 6 − u 8 )/ω1 ω2    , t8 ;  θ t7 /t6 ; q j=1 θ t j t8 ; q θ t6 / t7 ;  q j=1 θ  q 2 / t j q and using the resulting three term transformation in (6.6), we obtain Sh (u) = D(u 6 , u 7 )St (t7 /q, t1 , . . . , t6 , t8 /q; q) + (u 6 ↔ u 7 ) with    θ  t7 ;  q t8 / 2 2  D(u 6 , u 7 ) = iω2 e (u 6 − u 8 )/ω1 ω2  θ t7 ;  q t6 /  2  5

 t6 ;  θ  q / t j q   C(u 8 ; u 6 , u 7 ) + C(u 6 ; u 7 , u 8 ) . × t8 ;  θ  q 2 / t j q j=1

q , which The coefficient D(u 6 , u 7 ) is a sum of two very-well-poised 10 φ9 series in base  can be expressed in terms of the trigonometric integral Ut (in base  q ) by direct computations using Lemma 5.5. This yields the desired result.   Remark 6.2. i) Note that the W (E 6 )-symmetry of the trigonometric integrals St and Ut is upgraded to a W (E 7 )-symmetry in Theorem 6.1 since the second term in the right-hand side of (6.3) is the first term with the role of u 6 and u 7 interchanged. ii) Specializing the parameters in Theorem 6.1 to u ∈ G2iω with u 1 = −u 6 (so that t1 t6 = q and  t1 q ), the left-hand side of the identity can be evaluated by the t6 =  hyperbolic Nassrallah-Rahman integral evaluation  (4.6). For the right-hand side of the identity, the second term vanishes because θ  q;  q ) = 0 under the particular t6 / t1 parameter specialization. The remaining product of two trigonometric integrals can be evaluated by Corollary 5.9. The equality of both sides of the resulting identity can be reconfirmed using (6.1) and (6.4). It follows from this argument that the evaluation of the hyperbolic Nassrallah-Rahman integral is in fact a consequence of fusing trigonometric identities, an approach to hyperbolic beta integrals which was analyzed in detail in [37].

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iii) More generally, specializing (6.3) at generic u ∈ G2iω satisfying u 1 + u 6 = niω1 + miω2 (n, m ∈ Z≥0 ), the second term on the right-hand side of (6.3) still vanishes while the first term reduces to the product of two terminating very-well-poised q . The terminating 10 φ9 series is 10 φ9 series, one in base q and the other in base  Rahman’s [20] biorthogonal rational 10 φ9 function (cf. Remark 5.13(ii)), while the resulting expression for Sh is the corresponding two-index hyperbolic analogue of Rahman’s biorthogonal rational function, considered by Spiridonov [33, §8.3] (cf. Remark 3.1 on the elliptic level). Corollary 6.3. We have  2 4     4 2ω + j=1 u 2j − u 25 + u 26 q;  q t6 / t j  q,  q;  q ∞ j=1 θ    E h (u) = ω2 e 2ω1 ω2 2 θ t6 / t5 ;  q  3    1 1  q2  t5 q2  q2 , t1 , . . . , t4 , t6 ; q + (u 5 ↔ u 6 ) ×Vt , ,..., ; q Et   q t6  t1 t5    6 , where t = e (iu + ω)/ω and  t = e (iu j − as meromorphic functions in u ∈ C j j 2 j  ω)/ω1 ( j = 1, . . . , 6) as before. Proof. For generic u ∈ G2iω we have



(u 8 − u 1 − 2s)(u 1 + u 8 ) E h (u 2 , . . . , u 7 ) = lim Sh (u 1 + s, u 2 , . . . , u 7 , u 8 − s)e s→∞ 2ω1 ω2



under suitable parameter restraints by Proposition 4.8. By Theorem 6.1, we alternatively have   (u 8 − u 1 − 2s)(u 1 + u 8 ) lim Sh (u 1 + s, u 2 , . . . , u 7 , u 8 − s)e s→∞ 2ω1 ω2 ⎛ ⎞ 5  5     2 2 2 2 2ω + j=2 u j − u 6 + u 7 u / q;  q θ q;  q t7 t7 / t j t1  q,  q;  q ∞θ j=2 ⎠   = lim ω2 e ⎝ s→∞ 2ω1 ω2 2 θ t6 ;  q t7 /

    3 1 1 1 3 t7 ,  q 2 / t2 , . . . ,  q 2 / t6 ,  q 2 / t1 u,  q 2 u / t8 ;  q St t6 /q, t2 , . . . , t5 , t7 , t1 u, t8 /qu; q q 2 / ×Ut  +(u 6 ↔ u 7 ),

    where u = e is/ω2 and  u = e is/ω1 . We have u,  u → 0 as s → ∞ since (ω1 ), (ω2 ) > 0, hence application of Proposition 5.18 gives the right-hand side of the desired identity with respect to the parameters (u 2 , . . . , u 7 ).   Remark 6.4. Alternatively Corollary 6.3 can be proved by repeating the arguments of Theorem 6.1. The argument simplifies, since one now only needs the expression [5, (5.6.1)] of a very-well-poised 8 ψ8 as a sum of two very-well-poised 8 φ7 series, and one does not need to use three term transformations for the trigonometric integrals. We conclude this section by relating the R-function to the Askey-Wilson function using Corollary 6.3. The answer deviates from Ruijsenaars’ [27, §6.6] hunch that R is (up to an elliptic prefactor) the product of an Askey-Wilson function in base q and an Askey-Wilson function in base  q : it is the appearance below of two such terms which upgrades the W (D3 )-symmetry of the Askey-Wilson functions to the W (D4 )-symmetry of R (cf. Remark 6.2i)). For notational convenience, we write w0 = −1 ∈ W (D4 ) for

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the longest Weyl group element, acting as w0 γ = −γ on the Askey-Wilson parameters γ . We define ψ(γ ; x, λ) = ψ(γ ; x, λ; ω1 , ω2 ) by  h(γˆ ; λ, x; ω1 , ω2 )  w0 · φ (γ ; x, λ; ω1 , ω2 ) h(−γˆ ; λ, x; ω1 , ω2 ) h(γ ; x, λ; ω1 , ω2 )h(γˆ ; λ, x; ω1 , ω2 ) φ(−γ ; x, λ; ω1 , ω2 ), = h(−γ ; x, λ; ω1 , ω2 )h(−γˆ ; λ, x; ω1 , ω2 )

ψ(γ ; x, λ; ω1 , ω2 ) =

where the gauge factor h is given by (5.26). Note that ψ(γ ; x, λ; ω1 , ω2 ) is a self-dual solution of the Askey-Wilson difference equation (5.23), see Remark 5.30. We furthermore define the multiplier   θ e((γˆ0 − γ3 − i x)/ω2 ), e((ω + γ2 + i x)/ω2 ), e((ω + γ3 − i x)/ω2 ); q , M(γ ; x) =  θ e((γ3 − γˆ0 − i x)/ω2 ), e((ω + γ0 − i x)/ω2 ), e((ω + γ1 − i x)/ω2 ); q which is elliptic in x with respect to the period lattice Ziω1 + Ziω2 . Theorem 6.5. We have R(γ ; x, λ; ω1 , ω2 ) = K (γ )M(γ ; x)M(γˆ ; λ)φ(s23 γ ; x, λ; ω1 , ω2 ) ×ψ(−γ ; x, λ; −ω2 , ω1 ) + (γ2 ↔ γ3 ), where s23 γ = (γ0 , γ1 , γ3 , γ2 ) and with K (γ ; ω1 , ω2 )     √ 3ω(γ0 + γ1 + γ2 − γ3 ) 1 ω1 ω2 = −i e − ( + ) e − 24 ω2 ω1 2ω1 ω2 3   2 2 −γ0 − γ1 − γ22 + γ32 − 2γ0 γ1 − 2γ0 γ2 + 2γ0 γ3 j=1 G(iω + iγ0 + iγ j )

e . × 4ω1 ω2 θ e((γ2 − γ3 )/ω2 ); q Proof. Using the second hyperbolic Euler integral representation of R from Theorem 4.21 and subsequently applying Corollary 6.3, we obtain an expression of R(γ ; x, λ; ω1 , ω2 ) in terms of trigonometric integrals E t and Vt with parameter specializations which allows us to rewrite them as Askey-Wilson functions by Lemma 5.31. This leads to the expression R(γ ; x, λ; ω1 , ω2 ) = C(γ ; x, λ; ω1 , ω2 )φ(s23 γ ; x, λ; ω1 , ω2 )ψ(−γ ; x, λ; −ω2 , ω1 ) +(γ2 ↔ γ3 ), (6.8)

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with the explicit prefactor C(γ ; λ, x; ω1 , ω2 )  2  ω2 8ω + 8x 2 + 2(iω − i γˆ0 − λ)2 + (iω − 2iγ0 + i γˆ0 + λ)2 = e ω1 8ω1 ω2 (iω − 2iγ1 + i γˆ0 + λ)2 − (iω − 2iγ2 + i γˆ0 + λ)2 + (iω − 2iγ3 + i γˆ0 + λ)2 + 8ω1 ω2   3  2   q;  q ∞ j=1 G(iω + iγ0 + iγ j )  e((−ω + γ j + i x)/ω1 );  q ∞      × q; q ∞ θ e((γ3 − γ2 )/ω1 );  q j=0 e((−ω − γ j + i x)/ω1 );  q ∞



    θ e((−ω + γ3 ± i x)/ω1 );  q e((ω − γ2 ± i x)/ω2 ); q ∞ 3j=0 G(−iγ j ± x)  ×  e((ω + γ0 ± i x)/ω2 ), e((ω + γ1 ± i x)/ω2 ), e((ω + γ3 ± i x)/ω2 ); q ∞    q ∞ q e((−ω + γ3 − i x)/ω1 );  θ e((γ3 − γˆ0 + i x)/ω1 );     ×  e((−ω − γ3 − i x)/ω1 ); q ∞ θ (γˆ0 − γ3 + i x)/ω1 );  q   q θ e((γˆ3 − γ0 + iλ)/ω1 ), e((−ω − γˆ2 + iλ)/ω1 ), e((−ω + γˆ3 + iλ)/ω1 );    × θ e((γ0 − γˆ3 + iλ)/ω1 );  q G(λ − i γˆ1 , λ − i γˆ2 , λ − i γˆ3 ) G(λ + i γˆ0 )    2  e((−ω + γˆ3 − iλ)/ω1 );  e((−ω + γˆ j + iλ)/ω1 );  q ∞  q ∞    × q ∞ j=0 e((−ω − γˆ j + iλ)/ω1 );  q ∞ e((−ω − γˆ3 − iλ)/ω1 );    e((−ω − γˆ3 + iλ)/ω1 );  q ∞   × e((−ω + γˆ0 + iλ)/ω1 ), e((−ω + γˆ1 − iλ)/ω1 ), e((−ω + γˆ2 − iλ)/ω1 );  q ∞   e((ω − γˆ2 − iλ)/ω2 ); q ∞  . × e((ω + γˆ0 − iλ)/ω2 ), e((ω + γˆ1 + iλ)/ω2 ), e((ω + γˆ3 + iλ)/ω2 ); q ∞ ×

Elaborate but straightforward computations using (6.1), (6.4) and (6.5) now yields the desired result.   Acknowledgements. Rains was supported in part by NSF Grant No. DMS-0401387. Stokman was supported by the Netherlands Organization for Scientific Research (NWO) in the VIDI-project “Symmetry and modularity in exactly solvable models”.

References 1. van de Bult, F.J.: Ruijsenaars’ hypergeometric function and the modular double of Uq (sl2 (C)). Adv. Math. 204(2), 539–571 (2006) 2. van Diejen, J.F.: Integrability of difference Calogero-Moser systems. J. Math. Phys. 35(6), 2983– 3004 (1994) 3. Faddeev, L.: Modular double of a Quantum Group. In: “Conférence Moshé Flato” 1999, Vol I (Dijon), Math. Phys. Stud. 21, Dordrecht: Kluwer Acad. Publ., 2000, pp. 149–156

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4. Frenkel, I.N., Turaev, V.G.: Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions. In: Arnold, V.I., Gelfand, I.M., Retakh, V.S., Smirnov, M. (eds.) The Arnold-Gelfand mathematical seminars., pp. 171–204. Birkhäuser, Boston, 1997 5. Gasper, G., Rahman, M.: Basic Hypergeometric Series, Encyclopedia of Mathematics and its Applications, Vol. 96, 2nd ed., Cambridge Univ. Press, Cambridge, 2004 6. Groenevelt, W.: The Wilson function transform. Int. Math. Res. Not. 2003(52), 2779–2817 (2003) 7. Gupta, D.P., Masson, D.R.: Contiguous relations, continued fractions and orthogonality. Trans. Amer. Math. Soc. 350(2), 769–808 (1998) 8. Heckman, G., Schlichtkrull, H.: Harmonic Analysis and Special Functions on Symmetric Spaces. Perspectives in Mathematics, Vol. 16, 1994 9. Ismail, M.E.H., Rahman, M.: Associated Askey-Wilson polynomials. Trans. Amer. Math. Soc. 328, 201–239 (1991) 10. Koelink, E., Stokman, J.V.: Fourier transforms on the quantum SU(1, 1) quantum group (with an appendix of M. Rahman). Publ. Res. Math. Sci. 37(4), 621–715 (2001) 11. Koelink, E., Stokman, J.V.: The Askey-Wilson function transform. Int. Math. Res. Not. 2001(22), 1203–1227 (2001) 12. Koelink, E., van Norden, Y., Rosengren, H.: Elliptic U(2) quantum group and elliptic hypergeometric series. Commun Math. Phys 245(3), 519–537 (2004) 13. Komori, Y., Hikami, K.: Quantum integrability of the generalized elliptic Ruijsenaars models. J. Phys. A: Math. Gen. 30, 4341–4364 (1997) 14. Letzter, G.: Quantum zonal spherical functions and Macdonald polynomials. Adv. Math. 189(1), 88– 147 (2004) 15. Lievens, S., Van der Jeugt, J.: Symmetry groups of Bailey’s transformations for 10 φ9 -series. J. Comput. Appl. Math. 206(1), 498–519 (2007) 16. Macdonald, I.G.: Orthogonal polynomials associated to root systems. Sém. Lothar. Combin. 45, Art. B45a (2000/01) 17. Nassrallah, B., Rahman, M.: Projection formulas, a reproducing kernel and a generating function for q-Wilson polynomials. SIAM J. Math. Anal. 16, 186–197 (1985) 18. Noumi, M.: Macdonald’s symmetric polynomials and as zonal spherical functions on some quantum homogeneous spaces. Adv. Math. 123(1), 16–77 (1996) 19. van der Put, M., Singer, M.F.: Galois theory of difference equations. Lecture Notes in Mathematics 1666. Springer-Verlag, Berlin, 1997 20. Rahman, M.: An integral representation of a 10 φ9 and continuous biorthogonal 10 φ9 rational functions. Canad. J. Math. 38, 605–618 (1986) 21. Rains, E.M.: Tranformations of elliptic hypergeometric integrals. Ann. Math., to appear 2008 22. Rains, E.M.: Recurrences for elliptic hypergeometric integrals in elliptic integrable systems. In: Noumi, M., Takasaki, K. (eds.) Rokko Lectures in Mathematics 18, pp. 183–199, Kobe, Japan (2005) 23. Rains, E.M.: Limits of elliptic hypergeometric integrals. http://arxiv.org/list/math.CA/0607093, 2006 24. Ruijsenaars, S.N.M.: First order analytic difference equations and integrable quantum systems. J. Math. Phys. 38, 1069–1146 (1997) 25. Ruijsenaars, S.N.M.: Systems of Calogero-Moser type. In: Particles and Fields (Banff, AB, 1994), CRM Ser. Math. Phys., New York: Springer, 1999, pp. 251–352 26. Ruijsenaars, S.N.M.: A generalized hypergeometric function satisfying four analytic difference equations of Askey-Wilson type. Commun Math. Phys. 206(3), 639–690 (1999) 27. Ruijsenaars, S.N.M.: Special functions defined by analytic difference equations. In: Special functions 2000: Current perspective and future directions (Tempe, AZ), NATO Sci. Ser. II Math. Phys. Chem. 30, Dordrecht: Kluwer Acad. Publ., 2001, pp. 281–333 28. Ruijsenaars, S.N.M.: A generalized hypergeometric function II. Asymptotics and D4 symmetry. Comm. Math. Phys. 244(3), 389–412 (2003) 29. Ruijsenaars, S.N.M.: A generalized hypergeometric function III. Associated Hilbert space transform. Commun. Math. Phys. 243(3), 413–448 (2003) 30. Shintani, T.: On a Kronecker limit formula for real quadratic fields. J. Fac. Sci. Univ. Tokyo, Sect. 1A 24, 167–199 (1977) 31. Spiridonov, V.P.: On the elliptic beta function. Russ. Math. Surv. 56(1), 185–186 (2001) 32. Spiridonov, V.P.: Short proofs of the elliptic beta integrals. Ramanujan J. 13, 265–283 (2007) 33. Spiridonov, V.P.: Classical elliptic hypergeometric functions and their applications. Rokko Lect. in Math. Vol. 18, Dept. of Mth., Kobe Univ., 2005, pp. 253–287 34. Spiridonov, V.P.: Theta hypergeometric integrals. Algebra i Analiz 15, 161–215 (2003) (St. Petersburg Math J. 15, 929–967) (2004) 35. Spiridonov, V.P., Zhedanov, A.S.: Spectral transformation chains and some new biorthogonal rational functions. Commun Math. Phys. 210, 49–83 (2000)

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36. Stokman, J.V.: Difference Fourier transforms for nonreduced root systems. Selecta Math. (N.S.) 9(3), 409– 494 (2003) 37. Stokman, J.V.: Hyperbolic beta integrals. Adv. Math. 190, 119–160 (2005) Communicated by L. Takhtajan

Commun. Math. Phys. 275, 97–137 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0287-2

Communications in

Mathematical Physics

A KAM Theorem with Applications to Partial Differential Equations of Higher Dimensions Xiaoping Yuan
School of Mathematical Sciences and Key Lab of Mathematics for Nonlinear Science, Fudan University, Shanghai 200433, China. E-mail: [email protected]; [email protected] Received: 24 August 2006 / Accepted: 24 February 2007 Published online: 24 July 2007 – © Springer-Verlag 2007

Abstract: The existence of lower dimensional KAM tori is shown for a class of nearly integrable Hamiltonian systems of infinite dimensions where the second Melnikov’s conditions are completely eliminated and the algebraic structure of the normal frequencies are not needed. As a consequence, it is proved that there exist many invariant tori and thus quasi-periodic solutions for nonlinear wave equations, Schrödinger equations and other equations of any spatial dimensions. 1. Introduction and Main Results Let us begin with the nonlinear wave (NLW) equation u tt − u θθ + V (θ )u + h(θ, u) = 0

(1.1)

subject to Dirichlet boundary conditions. The existence of solutions, periodic in time, for NLW equations has been studied by many authors. See [B, B-B, B-Bo, B-G, B-P, Br, G-M-P, L-S] and the references therein, for example. Looking for quasi-periodic solutions, one inevitably encounters the so-called small divisor problems. The KAM (Kolmogorov-Arnold-Moser) theory is a very powerful tool to deal with the problems. The classical KAM theory is concerned with the existence of invariant tori (thus quasiperiodic solutions ) for nearly integrable Hamiltonian systems. In order to obtain the quasi-periodic solutions of partial differential equations (PDEs), one may show the existence of the lower (finite) dimensional invariant tori for the infinite dimensional Hamiltonian system defined by PDEs. Now consider a Hamiltonian of the form: H = (ω0 , y) +

ι


0j z j z¯ j + R 0 (x, y, z, z¯ ), ι ≤ ∞,

j=1
Supported by NNSFC and NCET-04-0365 and STCSM-06ZR14014.

(1.2)

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with tangential frequency vector ω0 = (ω10 , . . . , ωn0 ) and normal frequency vector  = (01 , . . . , 0ι ). When R 0 ≡ 0, there is a trivial invariant torus x = ω0 t, y = 0, z = z¯ = 0. The KAM theory guarantees the persistence of the trivial invariant torus with a small deformation under a sufficiently small perturbation R 0 , provided that the well-known Melnikov conditions are fulfilled: (k, ω0 ) − 0j = 0

(the first Melnikov)

(1.3)

for all k ∈ Zn and 1 ≤ j ≤ ι, and (k, ω0 ) + 0j1 − 0j2 = 0

(the second Melnikov)

(1.4)

for all k ∈ Zn and 1 ≤ j1 , j2 ≤ ι, j1 = j2 . See [E, K1, P1, W] for the details. This KAM theorem can be applied to a wide array of Hamiltonian partial differential equations of 1-dimensional spatial variables, including (1.1). It is first proved that, among certain classes of potentials V (x), there exist good ones such that (1.1) possesses quasi-periodic solutions. In this spirit, Kuksin [K1, K2, K3] considers potentials V (x) with n nondegenerate parameters and proves that for some parameters there exist quasi-periodic solutions of (1.1). Wayne [W] obtains also the existence of the quasi-periodic solutions of (1.1) when the potential V does not belong to some set of “bad” potentials. In [W], the set of all potentials is given a Gaussian measure and the set of “bad” potentials is proved to be of small measure. In the situation when a fixed potential is prescribed, Bobenko & Kuksin [Bo-K] and Pöschel [P2] derived the existence of invariant tori and quasi-periodic solutions for the potential V (x) ≡ m ∈ (0, ∞). By the remark in [P2], the same result holds true also for the parameter values −1 < m < 0. When m ∈ (−∞, −1) \ Z, it is shown in [Y1] that there are many hyperbolic-elliptic invariant tori. When m ∈ (−∞, 0) ∩ Z, there is a normal frequency vanishing, that is, there is a j0 such that 0j0 = 0 in (1.2). Therefore (1.2) has some singularity which partly violates the first Melnikov condition. Generally speaking, as stated in [K1], no preservation theorem for invariant tori is known for this case yet. Using the singular normal form technique developed by [Y3], the singularity coming from 0j0 = 0 is overcome and the existence of invariant tori is derived in [Ch-Y]. More recently, the existence of invariant tori (thus quasi-periodic solutions) of (1.1) is shown for any prescribed non-vanishing and smooth potential V (x) in [Y2]. When V (x) ≡ 0, Eq. (1.1) is completely resonant. For this case, the invariant tori of any dimension are constructed in [Y3], and the quasi-periodic travelling-wave-type solutions of 2-dimensional frequency are constructed in [Pr, Ba]. Equation (1.1) subject to periodic boundary conditions is investigated in [C-W, Bo1, C-Yo, Br-K-S] and others. For the NLW equation (1.1) of spatial dimension 1, the multiplicity of normal frequency 0j is 1 under the Dirichlet boundary condition and 2 under the periodic boundary condition. When Hamiltonian partial differential equations with spatial dimension greater than 1 are considered, a significant new problem arises due to the presence of clusters of normal frequencies of the Hamiltonian systems defined by these PDEs. In this case, the multiplicity of 0j may tend to ∞ as | j| → ∞; consequently, the second Melnikov condition is seriously violated, thus preventing the KAM theorems mentioned above from being applied to Hamiltonian partial differential equations of higher spatial dimensions. In general, the study of quasi-periodic solutions and invariant tori for PDEs in higher spatial dimension is at its early stage. In a series of papers [Bo1, Bo2, Bo3, Bo4, Bo5], Bourgain developed another profound approach which was originally proposed by Craig-Wayne in [C-W], in order to overcome the difficulty that the second

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Melnikov condition can not be imposed. In 1998 [Bo3] Bourgain successfully obtained the existence of quasi-periodic solutions of the nonlinear Schrödinger (NLS) equations of dimension d = 2 in space. Instead of KAM theory, this approach is based on a generalization of the Lyapunov-Schmidt procedure and a technique by Fröhlich and Spencer [F-S]. Bambusi initially calls this approach the “C-W-B method”. In 2003, by observing some symmetries in NLS and NLW equations, the present author [Y4] showed that there are many quasi-periodic solutions of travelling wave type for any spatial dimension d. Here the difficulty of small divisors was avoided owing to the symmetries. However, one can not avoid this difficulty in searching for more general solutions. In 2005, the new technique developed by Bourgain [Bo4] allowed him to prove the existence of more general quasi- periodic solutions for NLW and NLS equations of any spatial dimension d. Two advantages of the KAM approach are that, on one hand, it possibly simplifies the proof and, on the other hand, it allows the construction of local normal forms close to the considered torus, which could provide better understanding of the local dynamics. For example, in general, one can easily check the linear stability and the vanishing Lyapunov exponents using the normal form. Opposed to the KAM theory, the C-W-B method is more flexible in dealing with resonant cases where the second Melnikov condition is violated seriously, although a (local) normal form can not be obtained. Therefore, it is conceivable that a method might exist, using which the resonant cases can be more easily dealt with and at the same time the normal form can be obtained. To the best of our knowledge, the first result in that direction is due to Bourgain [Bo5]. In order to introduce Bourgain’s idea in [Bo5], let us start with the basic idea of constructing lower dimensional KAM tori and readers can find more details in [E, K1, P1]. Consider a finite dimensional Hamiltonian (1.2) with ι < ∞. Decompose the perturbation R 0 in (1.2) into R 0 = R x (x) + (R y (x), y) +R z (x), z + R z¯ (x), z¯ +R zz (x)z, z + R z z¯ (x)z, z¯ + R zz (x)¯z , z¯ +O(|y|2 + |y||z| + |z|3 ),

(1.5) (1.6) (1.7) (1.8)

where (x, y, z, z¯ ) is in some subset of Tn × Rn × Cι × Cι , R x , R y : Tn → Cn , R z , R z¯ : Tn → Cι , and R zz , R z z¯ , R z¯ z¯ are complex ι × ι matrices. (The space Cι should be replaced by some Hilbert space, say the usual Sobolev space H p , if ι = ∞.) In the classic KAM theory, one finds a series of symplectic transformations to eliminate the perturbation terms in (1.5), (1.6) and (1.7) such that the transformed Hamiltonian is of the form H = (ω, y) +

ι


 j z j z¯ j + O(|y|2 + |y||z| + |z|3 ),

(1.9)

j=1

where ω,  are new frequencies with |ω0j − ω| 1 and |0j −  j | 1. One sees easily that Tn × {y = 0}| × {z = z¯ = 0} is an invariant torus of (1.9) which corresponds to an invariant torus of (1.2). The second Melnikov condition (1.4) appears in the procedure of eliminating the perturbation terms in (1.7). Bourgain [Bo5] modifies the above idea in an essential way. He eliminates the perturbation terms in (1.5) and (1.6) and puts the perturbation terms in (1.7) into the “integrable” part of the Hamiltonian H . Consequently,

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he obtains a Hamiltonian of the form H˜ = (ω, y) +

ι


 j z j z¯ j + R zz (x)z, z + R z z¯ (x)z, z¯ + R zz (x)¯z , z¯ (1.10)

j=1

+O(|y| + |y||z| + |z|3 ). 2

(1.11)

This Hamiltonian H˜ can be regarded as a counterpart of the local normal form of the classic KAM theory. It is clear that Tn × {y = 0} × {z = z¯ = 0} is still an invariant torus of H˜ . Without the need to eliminate the terms in (1.7), the second Melnikov condition (1.4) are not required. Therefore, using Bourgain’s idea, one can deal with resonant cases where the second Melnikov condition (1.4) is violated and, at the same time, one obtains a normal form (1.10+11). However, the “integrable” terms in (1.10) are not really integrable, since it contains the angle-variable x in R zz (x)’s. Due to this fact, while eliminating (1.6), one has to solve a homological equation with variable coefficients √ −1(ω0 , ∂x )F z + (0 + R zz (x) + · · · )F z = R z , (1.12) where F z is the unknown function. By contrast, in the proof of the classical KAM theorem, the homological equation is of constant coefficients. To solve (1.12), one needs to investigate the inverse of a “big” matrix A of the form: R zz (k − l) : k, l ∈ Zn ). (1.13) A = diag ((k, ω0 ) + 0j : k ∈ Zn , j = 1, . . . , ι) + (  As in the C-W-B method, Bourgain [Bo5] uses and develops the technique of Fröhlich and Spencer [F-S] to investigate the inverse of A. Even if the inverse A−1 is obtained, it remains to show that A−1 has off-diagonal decay in order to control the exponentially weighted norm of A−1 . This technique depends heavily on the algebraic structure of (k, ω) +  j . For nonlinear wave equations of dimension d,   0j ≈ | j|2 = j12 + · · · + jd2 , j = ( j1 , . . . , jd ). This makes the algebraic structure of (k, ω0 ) + 0j very intricate if d > 1. In the present paper, we consider Hamiltonian (1.2) with ι = ∞. Following Bourgain’s idea in [Bo5], we put (1.7) into the “integrable” part (1.10) and try to find a symplectic transformation to eliminate (1.5) and (1.6). This leads to solving the homological equation (1.12) of variable coefficients. Thus, we also have to investigate the inverse of the “big” matrix A as in [Bo5]. However, we do not use the technique by Fröhlich and Spencer [F-S]. Observe that A is self-adjoint in 2 (Z2d+2n ). (This fact has been observed by Bourgain [Bo5] in finite dimension settings.) By means of this observation we can prove that the spectra of A are “twisted” with respect to the tangent frequency vector ω, thus the spectra are different from zero if digging out some ω’s of small Lebesgue measure. This implies the existence of A−1 on a Cantor set of large measure. In addition, by choosing Sobolev space as our working space instead of the space with the exponentially weighted norm we do not need the off-diagonal decay of A−1 . More exactly, using the block decomposition technique, we reduce estimating the bound of A−1 to m estimating that of a sub-matrix A−1 1 whose order is C 1 , where C 1 is some constant and m is the number of steps in KAM iterations. Note that A1 is self-adjoint and depends on the parameter vector ω. Using the twist conditions of the eigenvalues of A1 with respect to ω, we can show |λ A1 | ≥ C2−m ,

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where the λ A1 ’s are the eigenvalues of A1 and C2 is a constant. By using the fact that A1 is self-adjoint, we get that −1 ≤ C2m , |||A−1 1 ||| ≤ |λ A1 |

where ||| · ||| is the operator norm in the usual square summable space 2 . Noting that the order of the matrix A1 is C1m ,we have m |||A−1 1 ||| p ≤ C 1 C 2 , mp

where ||| · ||| p is the operator norm in Sobolev space H p . Since the magnitude of m mp the perturbation is 2 in the m th KAM iteration, the bound C1 C2m of A−1 can be m compensated by 2 , that is, C1 C2m · 2 ≤ (4/3) 1, for fixed p > d/2. mp

m

m

(∗)

This inequality guarantees that the KAM procedure can go on repeatedly. It also explains why the constant p can not tend to ∞. In addition, if we take a exponential weight norm, for instance,
||z||a2 = |z j |2 e2a| j| < ∞, for some a > 0, j∈Zd

then we will have aC1 m C2 . |||A−1 1 ||| p ≤ e m

m

This bound can not be compensated by 2 . Therefore we have to work in the Sobolev norm instead of the exponential weight norm. Here it is worthwhile comparing Bourgain’s results [Bo4] with those in the present paper. The existence results of general quasi-periodic solutions for NLW and NLS equations of any dimension d are due to Bourgain [Bo3, Bo4]. In Bourgain’s approach, the separation properties of the resonant sites (i.e., the lattice points (k, j) ∈ Zk ×Zd , where (k, ω0 )+0j is small,) imply the invertibility of the matrix A on a Cantor set of large measure and moreover A−1 is bounded in an exponentially weighted norm. (See Lemmas 19.10 and 20.14 in [Bo4], for the separation properties.) The solutions u(t, θ ) of NLW and NLS equations are analytic in time t and spatial variable θ , due to the exponentially weighted norm. In order to get the bound of A−1 in the exponentially weighted norm, one has to prove that A−1 has off-diagonal decay, which involves the intricate algebraic structure of (k, ω0 ) + 0j . Bourgain’s proof appears profound and subtle. In the present paper, in Sobolev space H p we establish an abstract KAM theorem under the simple growth condition imposed on 0j : 0j ≈ | j|κ , κ > 0.

(1.14)

One may note that κ = 2 for NLS equation and κ = 1 for NLW equation in [Bo4]. Hence our condition (1.14) is weaker than those in [Bo4]. In particular, by using the KAM theorem, we also get the quasi-periodic solutions for NLW and NLS equations of any dimension d. Since we work in Sobolev spaces, the obtained solutions naturally have less regularity: the solutions u(t, ·) ∈ C ω (R, H p (Td )), where C ω denotes the analytic function class. It follows from Sobolev’s embedding theorem that u(t, θ ) is sufficiently

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smooth in variables (t, θ ), if p is taken large enough. However, avoiding completely the necessity of proving the off-diagonal decay of the inverse matrices makes our proof much simpler. In addition, the assumption κ = 2 (κ = 1, respectively) is needed in Lemma 19.10 (Lemma 20.14, respectively) in [Bo4]. It is not clear whether or not the separation properties described by Lemmas 19.10 and 20.14 in [Bo4] hold true under such a weak condition (1.14). In this sense, our KAM theorem applies to wider classes of infinite dimensional Hamiltonian systems. Even in the case d = 1, our KAM theorem has wider applications than the well-known KAM theorem by Kuksin and Pöschel [K1, K2, K3, P1], since the assumption κ ≥ 1 is needed in [K1, K2, K3, P1]. (See Theorem 3.3 below for the details.) In our opinion, the growth rate κ in (1.14) instead of the spatial dimension d of PDEs and the algebraic structure of the normal frequencies is a key point in the KAM theory for Hamiltonian PDEs. Neither Bourgain’s results [Bo1, Bo2, Bo3, Bo4, Bo5] nor the results in the present paper can guarantee the linear stability of the obtained solutions. In order to get the stability, one needs to eliminate the dependence of R zz (x)’s in (1.10) on the angle variable x, that is, to reduce the linearized system of (1.10) to a system of constant coefficients. This seems to imply that, in order to obtain the linear stability of these quasi-periodic solutions, the second Melnikov condition can not be avoided completely. Thus, the linear stability of these quasi-periodic solutions looks like a very challenging problem and all the recently available results take advantage of some extra properties of the model partial differential equations. By observing some algebraic properties of the frequencies 0j = | j|2 of the NLS equation, such as |i0 − 0j | ≥ C > 0, |i| = | j|, i, j ∈ Zd ,

(1.15)

and developing some new techniques, Eliasson and Kuksin [E-K] derived the exciting results of both the existence and the linear stability of KAM tori for nonlinear Schrödinger equations with d ≥ 1. For the semilinear beam equation, the frequencies 0j = | j|4 satisfy (1.15) and the vector field X R 0 of the perturbation term R 0 in (1.2) satisfies X R 0 (z) ∈ H p˜ , p˜ − p > 0, ∀ z ∈ H p .

(1.16)

The inequality p˜ − p > 0 implies the operator X R 0 (·) : H p → H p˜ is smooth (i.e., of negative order in Kuksin’s notion [K1]). Using (1.15), (1.16) and some symmetry of the equation itself, Geng and You [G-Yo] prove the existence and the linear stability of KAM tori for some semilinear beam equation and some non-local nonlinear Schrödiner equation with any d ≥ 1. Independently, it is noted in [Y5] that (1.15) is not necessary if p˜ − p ≥ 557d/2 in (1.16). Therefore the existence and the linear stability of KAM tori are proven for some non-local NLW equations (also for some non-local NLS equations) in [Y5]. It is worth noting that (1.16) does not hold true for the usual NLS equations. From this we can see that Eliasson-Kuksin’s proof about the stability of the invariant tori and quasi-periodic solutions for NLS equations is very profound. Since (1.15) does not hold true for NLW equations, so far the linear stability of the obtained quasi-periodic solutions and invariant tori for NLW equations is still an open problem. The rest of the present paper is organized as follows. In §2, a KAM theorem to deal with Hamiltonian (1.2) with ι = ∞ is given. This theorem does not require the second Melnikov condition. In § 3, the KAM theorem is used to obtain invariant tori and a quasi-periodic solution for the nonlinear wave equation and the nonlinear Schrödinger equation. Sections 4–7 are devoted to the proof of the KAM theorem. The fifth section is the essential part of the present paper.

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103

2. A KAM Theorem 2.1. Some notations. Denote by (2 , || · ||) the usual space of the square summable sequences, and by (L 2 , || · ||) the space of the square integrable functions. By | · | denote the Euclidiean norm. Let ||| · ||| be the operator norm corresponding to || · ||. In all of the present paper, we always denote the dimension of the Laplacian by d and let p be a given positive integer with p > d/2. For a sequence z = (z j ∈ C : j ∈ Zd ), we define its norm as follows:
| j|2 p |z j |2 . (2.1) ||z||2p = j∈Zd

Let H p be the set of all sequences satisfying (2.1). It is easy to see that H p is a Hilbert space with an inner product corresponding to (2.1). (In fact, H p corresponds to the so-called Sobolev space H p by Fourier transform.) Introduce the phase space: P := (Cn /2π Zn ) × Cn × H p × H p ,

(2.2)

where n is a given positive integer. We endow P with a symplectic structure
√ √ d x ∧ dy + −1dz ∧ d z¯ = d x ∧ dy + −1 dz j ∧ d z¯ j , (x, y, z, z¯ ) ∈ P. (2.3) j∈Zd

Let T0n := (Rn /2π Zn ) × {y = 0} × {z = 0} × {¯z = 0} ⊂ P.

(2.4)

Then T0n is a torus in P. Introduce complex neighborhoods of T0n in P: D(s, r ) := {(x, y, z, z¯ ) ∈ P : |Imx| < s, |y| < r 2 , ||z|| p < r, ||¯z || p < r }, D (s, r ) := {(x, y, z, z¯ ) ∈ D(s, r ) : x, y ∈ Rn }, (2.5) where r, s > 0 are constants. Note that x, y are real but z and z¯ are still complex in D (s, r ). For r˜ > 0 we define the weighted phase norms r˜ | W|| p

= |X | +

1 1 1 |Y | + ||Z || p + || Z¯ || p 2 r˜ r˜ r˜

(2.6)

for W = (X, Y, Z , Z¯ ) ∈ P. Let O ⊂ Rn be compact and of positive Lebesgue measure. For a map W : D(s, r ) × O → P, set r˜ | W|| p,D(s,r )×O

:=

sup

(x,ξ )∈D(s,r )×O

r˜ | W (x, ξ )|| p

(2.7)

and L

r˜ | W|| p,D(s,r )×O

:=

sup

D(s,r )×O

r˜ | ∂ξ W (x, ξ )|| p ,

(2.8)

where ∂ξ is the derivative with respect to ξ in the sense of Whitney. Denote by L(H p , H p ) the set of all bounded linear operators from H p to H p and by ||| · ||| p the operator norm.

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For any subset S ⊂ Zd , let us consider a vector u = (u j ∈ C : j ∈ S) and a matrix U = (Ui j ∈ C : i, j ∈ S). We can expand u into  u j, j ∈ S d u˜ = (u˜ j : j ∈ Z ), here u˜ j = 0, j ∈ Zd \ S, and also expand U into U˜ = (U˜ i j : i, j ∈ Zd ), here U˜ i j =



Ui j , i, j ∈ S 0, i or j ∈ Zd \ S.

Define ||u|| p := ||u|| ˜ p and |||U ||| p := |||U˜ ||| p . Similarly define ||u|| := ||u|| ˜ and ˜ |||U ||| := |||U |||. Now for any S1 , S2 ⊂ Zd with S1 ∩ S2 = ∅, and for any bounded linear operator Y = (Yi j : i, j ∈ Zd ) ∈ L(H p , H p ), we can write   Y11 Y12 , Y = Y21 Y22 where Y11 = (Yi j : i, j ∈ S1 ), Y12 = (Yi j : i ∈ S1 , j ∈ S2 ), Y21 = (Yi j : i ∈ S2 , j ∈ S1 ), Y22 = (Yi j : i, j ∈ S2 ). Lemma 2.1. For the relation of |||Y11 ||| p ’s and |||Y ||| p , we have |||Ykl ||| p ≤ |||Y ||| p , k, l ∈ {1, 2}. Proof. For any given p, there are certain weights (whose explicit form does not matter here) vi , w j ∈ R (i, j ∈ Zd ) such that for any Y ∈ L(H p , H p ) we have |||Y ||| p = it suffices to prove this lemma for |||Y |||, where Y = (vi Yi j w j : i, j ∈ Zd ). Therefore,  the 2 operator norm ||| · |||. Let X = Y11 Y12 . Take u ∈ 2 (Zd ) with ||u|| = 1. Then



||X u||2 = ( Yi j u j )2 ≤ ( Yi j u j )2 = ||Y u||2 . i∈S1 j∈Zd

i∈S1 ∪S2 j∈Zd

Thus, |||X ||| ≤ |||Y |||, that is, we have the following “basic inequality”:     Y11 Y12 Y11 Y12 ||| ≤ ||| |||. ( ) : ||| 0 0 Y21 Y22 It is well known that for any bounded linear operator T in 2 , (

) :

|||T ||| = |||T ∗ |||,

where T ∗ is the adjoint of T . (See [p.195, Yos], for example.) Therefore,  ∗  by (

) (by Def.) Y11 0 ∗ ||| |||Y11 ||| → =|||Y11 ||| → =||| 0 0  ∗   ∗   (by ( )) by (

) Y11 Y12 Y11 0 Y11 Y12 ||| = ||| ||| → ≤ ||| ||| → =||| ∗ 0 0 0 0 Y12 0  (by ( )) Y11 Y12 ||| = |||Y |||. → ≤ ||| Y21 Y22 This gives the proof of this lemma for k = l = 1. The remaining proof is similar.

 

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105

The following lemma states the relation between ||| · ||| p and ||| · |||. Lemma 2.2. (i) Let Y = (Y j : j ∈ Zd , | j| ≤ K ) be a vector with some K > 0. Then ||Y || p ≤ K p ||Y || and ||Y || ≤ ||Y || p . (ii) Let Y = (Yi j : i, j ∈ Zd , |i|, | j| ≤ K ) be a matrix. Then |||Y ||| p ≤ K p |||Y ||| and |||Y ||| ≤ |||Y ||| p . Proof. The proofs are trivial, and therefore omitted. In all of this paper, with C or c a universal constant, whose size may be different in different places, these constants might depend on n, d and p. If f ≤ Cg, we write this inequality as f
g, when we do not care about the size of the constant C or c. Similarly, if f ≥ Cg we write f  g. 2.2. Statement of the KAM theorem. For two vectors b, c ∈ Ck or Rk , we write (b, c) = k ∞. We remark that (·, ·) is not inner product in Ck . If k = ∞, j=1 b j c j if k < ∞ we write b, c = j=1 b j c j . Consider an infinite dimensional Hamiltonian in the parameter dependent normal form
N0 = (ω0 (ξ ), y) + 0j (ξ )z j z¯ j , (x, y, z, z¯ ) ∈ P. (2.9) j∈Zd

The tangent frequencies ω0 = (ω10 , . . . , ωn0 ) and the normal frequencies 0j ’s ( j ∈ Zd ) depend on n parameters ξ ∈ O0 ⊂ Rn , where O0 is a given compact set of positive Lebesgue measure. Let 0 (ξ ) = diag(0j (ξ ) : j ∈ Zd ).

(2.10)

N0 = (ω0 (ξ ), y) + 0 (ξ )z, z¯ ,

(2.11)

Let

where z = (z j ∈ C : j ∈ Zd ). The Hamiltonian vector field of N0 is √ √ x˙ = ω0 (ξ ), y˙ = 0, z˙ = −10 (ξ )z, z˙¯ = − −10 (ξ )¯z .

(2.12)

Hence, for each ξ ∈ O0 , there is an invariant n-dimensional torus T0n = Tn × {y = 0} × {z = 0} × {¯z = 0} with a rotational frequencies ω0 (ξ ). The aim of the present paper is to prove the persistence of the torus T0n , for “most” (in the sense of Lebesgue measure) parameter vectors ξ ∈ O0 , under small perturbation R 0 of the Hamiltonian N0 . To this end the following assumptions are required. Assumption A (Multiplicity). Assume that there are constants c1 , c2 > 0 such that for all ξ ∈ O0 , (0j ) ≤ c1 | j|c2 , where we denote by (·) the multiplicity of (·).

(2.13)

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Assumption B (Non-degeneracy). There are two absolute constants c3 , c4 > 0 such that sup |det ∂ξ ω0 (ξ )| ≥ c3 ,

ξ ∈O0

(2.14)

and j

sup |∂ξ ω0 (ξ )| ≤ c4 , j = 0, 1,

ξ ∈O0

(2.15)

where ∂ξ is the derivative with respect to ξ in Whitney’s sense.1 Assumption C (Differentiability of parameters). Assume that both ω0 (ξ ) and 0 (ξ ) are continuously differentiable in ξ ∈ O0 in the sense of Whitney. (Remark. In the following arguments, the differentiability with respect to the parameter vector ξ is always in the sense of Whitney. We will not mention it again.) Assumption D (Growth conditions of normal frequencies). Assume that there exist constants c5 , c6 , c7 > 0 and a constant κ > 0 such that inf 0j ≥ c5 | j|κ + c6 ,

(2.16)

sup |∂ξ 0j | ≤ c7 1,

(2.17)

ξ ∈O0

ξ ∈O0

uniformly for all j. Assumption E (Regularity). Let s0 , r0 be given positive constants. Assume the perturbation term R 0 (x, y, z, z¯ ; ξ ) which is defined on the domain D(s0 , r0 ) × O0 is analytic in the space coordinates and continuously differentiable in ξ ∈ O0 , as well as, for each ξ ∈ O0 its Hamiltonian vector field √ √ X R 0 := (R 0y , −Rx0 , −1∂z R 0 , − −1∂z¯ R 0 )T defines an analytic map X R 0 : D(s0 , r0 ) ⊂ P → P, where T ≡ transpose and ∂z is the 2 -gradient. Also assume that X R 0 is continuously differentiable in ξ ∈ O0 . Assumption F (Reality). For any (x, y, z, z¯ , ξ ) ∈ D (s0 , r0 ) × O0 , the perturbation R 0 is real, that is, R 0 (x, y, z, z¯ , ξ ) = R 0 (x, y, z, z¯ , ξ ), where the bar means complex conjugate. 1 See, for example, [Z] for the derivative in Whitney’s sense.

(2.18)

A KAM Theorem with Application to PDEs

107

Theorem 2.1. Suppose H = N0 + R 0 satisfies Assumptions A–F and the smallness assumption: r0 | X R 0 | p,D(s0 ,r0 )×O0

L

r0 | X R 0 | p,D(s0 ,r0 )×O0

< ,

< 1/3 .

(2.19)

Then there is a sufficiently small 0 = 0 (n, p, d) > 0 such that, for 0 < < 0 , there is a subset O ⊂ O0 with Meas O ≥ (Meas O0 )(1 − O( )), and there are a family of torus embedding  : Tn × O → P and a map ω∗ : O → Rn , where (·, ξ ) and ω∗ (ξ ) is continuously differentiable in ξ ∈ O , such that for each ξ ∈ O the map  restricted to Tn × {ξ } is an analytic embedding of a rational torus with frequencies ω∗ (ξ ) for the Hamiltonian H at ξ . Each embedding is real analytic on Tn × {ξ }, and the following estimates r0 | (x, ξ ) − 0 (x, ξ )|| p

≤ c ,

r0 | ∂ξ ((x, ξ ) − 0 (x, ξ ))|| p

≤ c 1/3 ,

|ω∗ (ξ ) − ω(ξ )| ≤ c , |∂ξ (ω∗ (ξ ) − ω(ξ ))| ≤ c 1/3 , hold true uniformly for x ∈ Tn and ξ ∈ O , where 0 is the trivial embedding Tn × O0 → Tn × {y = 0} × {z = 0} × {¯z = 0}, and c > 0 is a constant depending on n, p and d. 3. Application to Nonlinear Partial Differential Equations 3.1. Application to nonlinear wave equations. Consider the nonlinear wave equation u tt − u + Mσ u + u 3 = 0, θ ∈ Td , d ≥ 1, where u = u(t, θ ) and  = dj=1 ∂θ2j and Mσ is a real Fourier multiplier, Mσ cos( j, θ ) = σ j cos( j, θ ), Mσ sin( j, θ ) = σ j sin( j, θ ), σ j ∈ R, j ∈ Zd .

(3.1)

(3.2)

Pick a set  = {1 , . . . , n } ⊂ Zd . Let Zd = Zd \ {1 , . . . , n }. Following Bourgain [Bo3], we assume  σl = σl , (l = 1, . . . , n) (3.3) σ j = 0, j ∈ Zd . Let λ±j ( j ∈ Zd ) be the eigenvalues of the self-adjoint operator − + Mσ subject to peri± odic b. c. θ ∈ Td , and let φ ± j = φ j (θ ) be the normalized eigenfunctions corresponding to λ±j . Then λ±j

= | j| + σ j = 2

d


jl2 + σ j , j = ( j1 , . . . , jd ) ∈ Zd ,

(3.4)

l=1

and

√ φ +j

2

= √ cos( j, θ ), ( 2π)d

φ− j

√ 2 = √ sin( j, θ ), ( 2π)d

(3.5-1)

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X. Yuan

or in complex form √ 1 φ± e± −1( j,θ) . j = √ ( 2π )d

(3.5-2)

d d Note that {φ ± j : j ∈ Z } is a complete orthogonal system in L 2 (T ). Let
u(t, θ ) = q ±j (t)φ ± j (θ ).

(3.6)

j∈Zd

Inserting (3.6) with (3.5-1) into (3.1), then d q¨ ±j + λ±j q ±j + (u 3 , φ ± j ) L 2 = 0, j ∈ Z , · =

d . dt

(3.7)

 Let q ± = λ±j q ±j and q˙ ±j = p ±j . (We should not confuse these p ±j ’s with the index p j in the Sobolev space H p .) Then ⎧  ± ± ⎪ ⎨ q˙ ± j = λj pj  (3.8) ± 1 q¨ ± = − λ± q ± −  (u 3 , φ ± ). ⎪ p ˙ = ⎩ j j j j ± j ± λ λ j

j

To simplify notation, we do not distinguish + sign and − sign in (3.8). For example, we + − write q ± j as q j . However, we should keep in mind that q j runs over the set {q j , q j }. The system (3.8) is a Hamiltonian system with its symplectic structure dq ∧ dp and Hamiltonian function 1
 λ j ( p 2j + q 2j ) + G(q), (3.9) H = H ( p, q) = 2 d j∈Z

where




G(q) =

G i jkl qi q j qk ql

(3.10)

i, j,k,l∈Zd

and G i jkl =  Let ωl0 =



 1 φi φ j φk φl d x. λi λ j λk λl Td

λl =

 |l |2 + σl , (l = 1, . . . , n),

(3.12)

 | j|2 , j ∈ Zd .

(3.13)

and 0j =

(3.11)



λj =

Introduce a symplectic coordinate change  √ √ pl = (1 + yl )/2 sin xl , ql = (1 + yl )/2 cos xl , l ∈ {1, . . . , n}, z j +¯z j z −¯z (3.14) , p j = √ j √j , j ∈ Zd . qj = √ 2

−1 2

A KAM Theorem with Application to PDEs

To shorten notation, let y˜l =



109

(1 + yl )/2, l = 1, . . . , n.

Then (3.9) is changed into H = H (x, y, z, z¯ ) = (ω0 , y) +




0j z j z¯ j + R 0 ,

(3.15)

j∈Zd

where ω0 = (ω10 , · · · , ωn0 ) and R 0 = R 0 (x, y, z, z¯ )
G i  j k l y˜i y˜ j y˜k y˜l cos xi cos x j cos xk cos xl = i, j,k,l∈{1,...,n}




+

zl + z¯l G i  j k l y˜i y˜ j y˜k cos xi cos x j cos xk √ 2 d

i, j,k∈{1,...,n},l∈Z




+

z k + z¯ k zl + z¯l G i  j kl y˜i y˜ j cos xi cos x j √ √ 2 2 d

i, j∈{1,...,n},k,l∈Z




+

z j + z¯ j z k + z¯ k zl + z¯l G i jkl y˜i cos xi √ √ √ 2 2 2 d

i∈{1,...,n}, j,k,l∈Z

+




z i + z¯ i z j + z¯ j z k + z¯ k zl + z¯l G i jkl √ √ √ √ . 2 2 2 2 d

(3.16)

i, j,k,l∈Z

It follows that for the arguments (x, y, z, z¯ ) ∈ D (s0 , r0 ) with given s0 , r0 > 0, R 0 (x, y, z, z¯ ) = R 0 (x, y, z, z¯ ).

(3.17)

This implies the symmetry Assumption F is fulfilled. By (3.5-1) and (3.11), it is not difficult to verify that for i, j, k, l ∈ Zd , G i jkl = 0, unless i ± j ± k ± l = 0,

(3.18)

H p,

the convolution z
w is

for some combination of plus and minus signs. For w, z ∈ defined by (z
w) j = k∈Zd w j−k z k .

Lemma 3.1. If p > d/2, then ||w
z|| p ≤ c||w|| p ||z|| p for w, z ∈ H p with a constant c depending only on p. Proof. The proof is very elementary. See [p.294–295,P2]. Let s0 = r0 = 1, ω0 = (ω10 , . . . , ωn0 ) and 0 = (0j : j ∈ Zd ). And let the parameter σ = (σ1 , . . . , σn ) run over O0 := [1, 2]d . Take σ as the parameter ξ in Theorem 2.1. Note that R 0 is independent of σ . By (3.18) and Lemma 3.1, we see that R 0 is well defined on D(s0 , r0 ) and Assumption E holds true; moreover, r0 | X R 0 | p,D(s0 ,r0 )×O


,

L

r0 | X R 0 | p,D(s0 ,r0 )×O

= 0.

(3.19)

This implies that (2.19) is fulfilled. It follows from (3.12) that Assumption B (that is, (2.14, 15) ) holds true. By (3.12, 13), we see that Assumptions A, C, D are fulfilled. In particular, (2.16) holds true with κ = 1. Using Theorem 2.1, we have the following theorem.

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Theorem 3.1. There is a subset O0 ⊂ O0 with Meas O0 ≥ (1 − C )Meas O0 such that for any σ ∈ O0 , Eq.(3.1) with small has a rotational invariant torus of frequency vector ω0 = ω0 (σ ). The motion on the torus can be expressed by u(t, θ ) which is quasiperiodic (in time) with frequency ω0 and u(t, ·) : R → H p (Tn ) is an analytic map, and thus the solution u(t, θ ) is, at least, a sufficiently smooth function of (t, θ ) if p is taken large enough.

3.2. Application to nonlinear Schrödinger equations. Consider the nonlinear Schrödinger equation √ −1u t − u + Mσ u + u|u|2 = 0, θ ∈ Td , d ≥ 1, (3.20) where Mσ is a Fourier multiplier √

Mσ e

−1( j,θ)

= σje

√

−1( j,θ)

, σ j ∈ R, j ∈ Zd .

Theorem 3.2. Assume σ j ’s satisfy (3.3). There is a subset O0 ⊂ O0 with Measure O ≥ (1 − C )Measure O such that for any σ = (σ1 , . . . , σn ) ∈ O0 , the nonlinear Schrödinger equation with small has a rotational invariant torus of frequency vector ω0 = ω0 (σ ), where ωl0 = |l |2 + σl , l = 1, . . . , n. The motion on the torus can be expressed by u(t, θ ) which is quasi-periodic (in time) with frequency ω0 and u(t, ·) : R → H p (Tn ) is an analytic map, and thus the solution u(t, θ ) is, at least, a sufficiently smooth function of (t, θ ) if p is taken large enough. Proof. Inserting (3.6) with (3.5-2) into (3.20), then √ ± ± ± 2 d −1q˙ ± j + λ j q j + (u|u| , φ j ) L 2 = 0, j ∈ Z .

(3.21)

As in the proof of Theorem 3.1, we shorten q ± j as q j . We observe that (3.21) is a Hamiltonian system with Hamiltonian function
H= q j q¯ j + G(q, q) ¯ (3.22) j∈Zd

and with symplectic structure G(q, q) ¯ =


i, j,k,l

√ −1 2



j∈Zd

dq j ∧ d q¯ j , where 

G i jkl qi q j q¯k q¯l , G i jkl =

Td

± ± φi± φ ± j φk r φl ,

and where the bar means the complex conjugate. Using (3.5-2) we get  0, ±i ± j ± k ± l = 0 G i jkl = (2π )−d , ±i ± j ± k ± l = 0.

A KAM Theorem with Application to PDEs

111

Therefore G(q, q) ¯ = (2π )−d




qi q j q¯k q¯l .

(3.23)

±i± j±k±l=0

It follows from (3.23) that G(q, q) ¯ = G(q, q) ¯ = G(q, ¯ q), ∀q ∈ H p . Introduce a symplectic coordinate change √  √ ql = (1 + yl )/2 e −1xl , l = 1, . . . , n, q j = z j , j ∈ Zd .

(3.24)

(3.25)

Let q = q (x, y) = (q1 , . . . , qn ) and z = (q j : j ∈ Zd ). Then (3.22) is changed into
H = (ω0 , y) + 0j z j z¯ j + R 0 (x, y, z, z¯ ), (3.26) j∈Zd

where ω0 = (λl : l = 1, . . . , n) and 0j = λ j ( j ∈ Zd ) and R 0 (x, y, z, z¯ ) = G((q , z), (q , z¯ )).

(3.27)

Thus using (3.24) we have R 0 (x, y, z, z¯ ) = G((q , z), (q , z¯ )) = G((q , z), (q , z)) = G((q , z), (q , z)) = G((q , z), (q , z¯ )) = R 0 (x, y, z, z¯ ). This implies the reality Assumption F (that is, (2.18)) is fulfilled. Since  j = λ j = | j|2 + σ j , (2.16) is fulfilled for κ = 2. We omit further details.   3.3. Application to 1-D nonlinear wave equation on the line. In [K1], Kuksin studies nonlinear perturbation of the quantized harmonic oscillator √ u t = −1(−u θθ + (θ 2 + V (θ, a))u + ∇ Ha (u)), u = u(t, θ ), θ ∈ R, u(·, θ ) ∈ L 2 (R), (3.28) where the function V vanishes at θ = ±∞ and a is a parameter vector. The operator −∂θ2 + θ 2 + V has a discrete spectra {λ j (a)}, which obeys Bohr’s quantization law: λ j ∼ C( j + 1/2). Since the normal frequency 0j = λ j in (1.2), the growth rate κ = 1 in (2.16). Using the KAM theorem proved by him, Kuksin [K1] proves that (3.28) possesses many invariant tori and quasi-periodic solutions for typical V , provided that the gradient map ∇ Ha (u) is smooth (i.e., of a negative order in Kuksin notion), in particular,  1 Ha = (3.29) φ(|u
ξ(θ )|2 ; a) d x 2

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X. Yuan

(u
ξ is the convolution with a smooth real-valued function of ξ , vanishing at infinity). Now let us consider the 1-D nonlinear wave equation on the line: u tt − u θθ + (θ 2 + V (θ, a))u + ∇ Ha (u) = 0, u = u(t, θ ), θ ∈ R, u(·, θ ) ∈ L 2 (R). (3.30)  0 In this case, the normal frequency  j = λ j in (1.2). Therefore the growth rate κ = 1/2 in (2.16). This growth rate prevents Kuksin’s KAM theorem2 [K1] from being applied to (3.30). However, using Theorem 2.1 of the present paper, we can prove the following theorem. Theorem 3.3. Assume Ha is of the form (3.29) where φ(θ ) is a real polynomial of θ . Then (3.30) has many invariant tori and quasi-periodic solutions for typical V (θ, a). Proof. According to Weyl’s theory on the singular differential operator, the operator −∂θ2 + θ 2 + V is in the class of the limit-circle. Therefore, the eigenfunctions {ϕ j (θ ) : j = 1, 2, . . .} corresponding to {λ j (a)} are a complete and orthogonal system in L 2 (R). Let
u(t, θ ) = qn (t)ϕn (θ ). j

As in the proofs of Theorems 3.1 and 3.2, (3.30) can be reduced to (1.2). Note κ = 1/2 in (2.16). The proof is finished by using Theorem 2.1. We omit the details. 4. The Linearized Equation 4.1. Unperturbed linear system. Recall that 0j ’s satisfy Assumptions A, C, D. Assume that there is a new tangent frequency vector3 ω satisfying Assumptions B, C. Let O ⊂ O0 be a compact subset with positive Lebesgue measure. In this section, we pick two fixed numbers K − and K , and let s = 1/K − and s  = 4/K . (In the m th KAM step, we will 2 2 choose K − = K m−1 ≈ 2(m−1) /2 and K = K m ≈ 2m /2 .) Let D(s) = {x ∈ Cn /2π Zn : | x| < s}, where  x means the imaginary part of x. Let X N be a linear Hamiltonian vector field with Hamiltonian function:
N = (ω(ξ ), y) + 0j (ξ )z j z¯ j j∈Zd


1
zz 1
z¯ z¯ + B i j (x; ξ )z i z j + B izjz¯ (x; ξ )z i z¯ j + B i j (x; ξ )¯z i z¯ j 2 2 d d d i, j∈Z

i, j∈Z

i, j∈Z

1 1 : = (ω, y) + (0 z, z¯ ) + B zz z, z + B z z¯ z, z¯ + B z¯ z¯ z¯ , z¯ , 2 2

(4.1)

where B i j (x; ξ )’s are analytic in x ∈ D(s) for any fixed ξ ∈ O, and all of ω(ξ ), 0j (ξ ) and Bi j (x; ξ ) are continuously differentiable in ξ ∈ O for fixed x ∈ D(s). Assume Bi j ’s satisfy the following conditions: 2 One may note that Kuksin’s KAM theorem requires κ ≥ 1. (See “spectral asymptotics” on page xiii in [K1] ). 3 Here ω = ω(ξ ) is not necessarily ω0 , due to the well-known phenomenon of the shift of frequency in the KAM iteration.

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113

(1) Reality condition.4 B izzj (x, ξ ) = B iz¯jz¯ (x, ξ ) = B z¯jiz¯ (x, ξ ),

(4.2)

B izjz¯ (x, ξ ) = B izjz¯ (x, ξ ) = B zjiz¯ (x, ξ )

(4.3)

for any (x, ξ ) ∈ Tn × O. (2) Finiteness of Fourier modes. B iuvj (x; ξ ) =




 B iuvj (k)e

√

−1(k,x)

, u, v ∈ {z, z¯ }.

(4.4)

|k|≤K

(3) Boundedness. sup |||B uv (x; ξ )||| p
,

sup |||∂ξ B uv (x; ξ )||| p
1/3 , u, v ∈ {z, z¯ }.

D(s)×O

D(s)×O

(4.5) In particular, it follows from Assumption B that (4) Non-degeneracy.    ∂ω(ξ )   ≥ c9 > 0, ∀ ξ ∈ O. c8 ≥ det ∂ξ 

(4.6)

(This assumption enables us to regard ω instead of ξ as a parameter vector.) 4.2. Split and estimate for small perturbation. We now consider a perturbation ` H = N + R, ` where R` = R(x, y, u; ξ ) is a Hamiltonian defined on D(s, r ) and depends on the parameter ξ ∈ O. Assume that the vector field X R` : D(s, r ) × O → P is real for real argument, analytic in (x, y, z, z¯ ) ∈ D(s, r ) and continuously differentiable in ξ ∈ O. We assume that there are quantities5 ε = ε(r, s, s  , O) and εL = εL (r, s, s  , O) which depend on r, s, s  , O such that r | X R` | p,D(s,r )×O

L

r | X R` | p,D(s,r )×O


ε,


εL , ε < εL 1.

(4.7)

d Z For q = (q j ∈ Z+ : j ∈ Zd ) ∈ ZZ + and z = (z j ∈ C : j ∈ Z ) ∈ C , define  qj
zq = z j , |q| = |q j |. d

j∈Zd

d

j∈Zd

For m = (m 1 , . . . , m n ) ∈ Zn+ and y = (y1 , . . . , yn ) ∈ Cn , define |m| = |m 1 | + . . . + |m n | and y m = y1m 1 · · · ynm n . Let √

R= (4.8) R` mq1 q2 (k)e −1(k,x) y m z q1 z¯ q2 , 2|m|+|q1 |+|q2 |≤2 k∈Zn , d m∈Zn+ ,q1 ,q2 ∈ZZ + 4 This condition implies that N is real for x ∈ Tn , y ∈ Rn and z, z¯ ∈ H p . 5 We will take ε = (4/3)m and ε L = ε 1/3 in the m th KAM iteration step.

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` mq1 q2 (k) are continuously differentiable in ξ ∈ where the Taylor-Fourier coefficients R ` We will approximate R` by O. We see that R is a partial Taylor-Fourier expansion of R. R. Now we give some estimates of R. Lemma 4.1. ∗ r | X R | p,D(s,r )×O ηr | X R


r | X R` | ∗p,D(s,r )×O ≤ ε∗ ,

− X R` | ∗p,D(s,4ηr )×O
η · r | X R` | ∗p,D(s,r )×O
ηε∗ ,

(4.9) (4.10)

for any 0 < η 1, where ∗ = the blank or L, (that is, ε∗ = ε or εL ). Proof. The proof is similar to that of formula (7) of [P1,129]. With this lemma, we decompose R = R 0 + R 1 + R 2 , where R 0 = R x + (R y , y), R 1 = R z , z + R z¯ , z¯ , 1 1 R 2 = R zz z, z + R z z¯ z, z¯ + R z¯ z¯ z¯ , z¯ , 2 2 with R x , R y : D(s)×O → Cn ; R z , R z¯ : D(s)×O → H p ; R zz , R z z¯ , R z¯ z¯ : D(s)×O → L(H p , H p ). For any vector or matrix √
−1(k,x) m q q¯ y z z¯ Y Y = m,q,q¯ (k)e d

k,m∈Zn ;q,q∈ ¯ ZZ +

we introduce the cut-off operator  K as follows: √
−1(k,x) m q q¯ y z z¯ . K Y = Y m,q,q¯ (k)e

(4.11)

d k,m∈Zn ;q,q∈ ¯ ZZ +

|k|≤K

Lemma 4.2. Assume (s − s  )K ≥ | ln η2 | and K η < 1. Let R K = R −  K R. We have ∗ r | X ( K R)| p,D(s  ,r )×O

≤ r | X R | ∗p,D(s,r )×O
ε∗ ,

∗ r | X R K | p,D(s  ,r )×O


ηε∗

(4.12) (4.13)

where ∗ = the blank or L. Proof. The proof of (4.12) is obvious. Let us give the proof of (4.13). Write 1 1 y R K = R Kx + (R K , y) + R Kz , z + R Kz¯ , z¯ + R Kzz z, z + R Kz z¯ z, z¯ + R Kz¯ z¯ z¯ , z¯ . 2 2 y

Note that the terms R Kx , R K , and so on, are analytic in x ∈ D(s) for fixed ξ ∈ O. And observe that √
 R Kx = R Kx (k)e −1(k,x) , . . . . |k|>K

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115

Then by Cauchy’s formula, we have | R Kx (k)| ≤ e−s|k| sup D(s) |R x |, and so on. Thus,
 ∗ ∗ e−(s−s )|k|
K n η2n ε∗ ≤ ηε∗ . r | X R K | p,D(s  ,r )×O ≤ r | X R | p,D(s,r )×O k∈Zn ,|k|>K

  Now we can write R` =  K R + R K + ( R` − R).

(4.14)

Let R =  K R. Hence we can write R = Rx + (R y , y) + Rz , z + Rz¯ , z¯ 1 1 + Rzz z, z + Rz z¯ z, z¯ + Rz¯ z¯ z¯ , z¯ . 2 2

(4.15)

Noting (4.11) we have R∗ =




 R ∗ (k)e

√

−1(k,x)

,

(4.16)

|k|≤K

where ∗ = x, y, z, z¯ , zz, z z¯ , z¯ z¯ . ` the following estimates hold Lemma 4.3. Under the smallness assumption (4.7) on R, true: 2 L |∂x Rx | D(s)×O ≤ r 2 ε, |∂x Rx |L D(s)×O ≤ r ε ,

(4.17)

L |R y | D(s)×O ≤ ε, |R y |L D(s)×O ≤ ε ,

(4.18)

||Ru || p,D(s)×O ≤ r ε, ||Ru ||Lp,D(s)×O ≤ r εL , u ∈ {z, z¯ },

(4.19)

|||Ruv ||| p,D(s)×O ≤ ε, |||Ruv |||Lp,D(s)×O ≤ εL , u, v ∈ {z, z¯ },

(4.20)

where for a function f : D(s) × O → Cn , define | f | D(s)×O :=

sup

(x,ξ )∈D(s)×O

| f (x, ξ )|, | f |L D(s)×O :=

sup

(x,ξ )∈D(s)×O

|∂ξ f (x, ξ )|,

and the other notations are defined similarly. Proof. Consider Rz z¯ . Observe that R z z¯ = ∂z ∂z¯ R|z=¯z =0 . By the generalized Cauchy inequality (see Lemma A.3 in [P1]), |||Rz z¯ ||| p,D(s)×O ≤ |||R z z¯ ||| p,D(s)×O ≤ ≤ r | X R | p,D(s,r )×O < ε.

1 ||∂z¯ R|| p,D(s,r )×O r

The remaining proof is similar to the previous. We omit the details.

(4.21)  

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X. Yuan

Lemma 4.4. Assume R` satisfies the following reality condition: ` ` R(x, y, z, z¯ ; ξ ) = R(x, y, z, z¯ ; ξ ), ∀(x, y, z, z¯ ; ξ ) ∈ D (s, r ) × O.

(4.22)

Let Rizzj , Rizjz¯ , Riz¯jz¯ be the elements of the matrices R zz , R z z¯ , R z¯ z¯ , respectively. Then we have that for real x, R x (x) = R x (x), R y (x) = R y (x),

(4.23)

R z (x) = R z¯ (x),

(4.24)

zz z¯ z¯ Rizzj (x) = R zz ji (x), Ri j (x) = Ri j (x).

(4.25-1)

Rizjz¯ (x) = R zjiz¯ (x), Rizjz¯ (x) = Rizjz¯ (x).

(4.25-2)

Proof. The proof is trivial.

 

Recall R =  K R. Let B zz = B zz + Rzz , B z z¯ = B z z¯ + Rz z¯ , B z¯ z¯ = B z¯ z¯ + Rz¯ z¯ ,

(4.26)

ω=ω+ R y (0),

(4.27)

1 1 N = (ω, y) + (0 z, z¯ ) + B zz z, z + B z z¯ z, z¯ + B z¯ z¯ z¯ , z¯ , 2 2

(4.28)

and

where B z z¯ = B z¯ z . Then by (4.25), (4.2) and (4.3), B izzj (x, ξ ) = B iz¯jz¯ (x, ξ ) = B z¯jiz¯ (x, ξ ), B izjz¯ (x, ξ ) = B izjz¯ (x, ξ ) = B zjiz¯ (x, ξ )

(4.29)

for any x ∈ Rn , ξ ∈ O. Assume ε < and εL < . (In the m th KAM iteration, we will m choose ε = (4/3) . We can see that ε .) By (4.5) and (4.20), we have |||B zz |||∗p , |||B z z¯ |||∗p , |||B z¯ z¯ |||∗p
,

(4.30)

where ∗ = the blank or L. By (4.6, 4.18, 4.27) and ε 1, we get that the new frequency vector ω satisfies:    ∂ω(ξ )   ≥ c11 > 0, ∀ ξ ∈ O.  c10 ≥ det (4.31) ∂ξ  In view of (4.27, 4.28, 4.14, 4.15), R y (0), y) + Rz , z + Rz¯ , z¯ H = N + Rx + (R y −  +(R −  K R) + ( R` − R).

(4.32)

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117

4.3. Derivation of homological equations. The KAM theorem is proven by the usual Newton-type iteration procedure which involves an infinite sequence of symplectic coordinate changes. Each coordinate change is obtained as the time-1 map X tF |t=1 of a Hamiltonian vector field X F . Its generating Hamiltonian F solves the linearized equation {F, N } = Rx + (R y −  R y (0), y) + Rz , z + Rz¯ , z¯ ,

(4.33)

where {·, ·} is a Poisson bracket with respect to the symplectic structure d x ∧ dy + √ x (0) = 0 since it does not affect −1dz ∧ d z¯ . Without loss of generality, we assume R the dynamics. We are now in position to find a solution of Eq. (4.33) and to give some estimates of the solution. To this end, we suppose that F is of the same form as the right-hand of (4.33), that is, F = F 0 + F 1 , where  F 0 = F x + (F y , y), (4.34) F 1 = F z , z + F z¯ , z¯ , x (0) = 0, F y (0) = 0. Let ∂ω = (ω, ∂x ), with F x , F y , F z , F z¯ depending on x, ξ , and F n where (·, ·) is the usual inner product in R . Using (4.34) and (4.28) we can compute the Poisson Bracket {F, N }: {F, N } =

1 1 (F y , ∂x )B zz z, z + (F y , ∂x )B z z¯ z, z¯ + (F y , ∂x )B z¯ z¯ z¯ , z¯ 2 2 +∂ω F x + ∂ω F y , y + ∂ω F z , z + ∂ω F z¯ , z¯ √ + −1(0 F z , z + B z z¯ F z , z + B z¯ z¯ F z , z¯ ) √ − −1(0 F z¯ , z¯ + B z z¯ F z¯ , z¯ + B zz F z¯ , z ). (4.35)

From (4.33,4.35) we derive the homological equations: ∂ω F x =  K R x (x, ξ ),

(4.36)

R y (0), ∂ω F y =  K R y (x, ξ ) − 

(4.37)

  √ √ − −1∂ω F z +  0 F z + ( B z z¯ )F z − ( B zz )F z¯ = − −1 R z (x, ξ ), (4.38-1) √

 √  −1∂ω F z¯ +  0 F z¯ + ( B z z¯ )F z¯ − ( B z¯ z¯ )F z = −1 R z¯ (x, ξ ),

(4.38-2)

where  =  K is defined in (4.11). We should note that if F solves the equations (4.36–4.38), then F satisfies: {F, N } = Rx + (R y −  R y (0), y) + Rz , z + Rz¯ , z¯ 1 1 + (F y , ∂x B zz )z, z + (F y , ∂x B z z¯ )z, z¯ + (F y , ∂x B z¯ z¯ )¯z , z¯ 2  √  2  √  + −1 (1 − )(( B z z¯ )F z ), z + −1 (1 − )(( B z¯ z¯ )F z ), z¯  √   √  − −1 (1 − )(( B zz )F z¯ ), z − −1 (1 − )( B z z¯ )F z¯ ), z¯  √   √  + −1 ((1 − )B z z¯ )F z ), z + −1 ((1 − )B z¯ z¯ )F z ), z¯  √   √  − −1 ((1 − )B zz )F z¯ ), z − −1 ((1 − )B z z¯ )F z¯ , z¯ (4.39)

118

X. Yuan

instead of (4.33). Lemma 4.5. If F in some sub-domain D(s  , r  )×O of D(s, r )×O is the unique solution of the homological equations (4.36–4.38), then F satisfies the reality conditions: F(x, y, z, z¯ ; ξ ) = F(x, y, z, z¯ ; ξ ), ∀ (x, y, z, z¯ ; ξ ) ∈ D (s  , r  ) × O . (F z (x),

(4.40)

F z¯ (x))

Proof. By (4.24, 4.29, 4.38), we see that if solves (4.38), then so does z ¯ z (F (x), F (x)). Using the uniqueness assumption of the solution, we have F z (x) = F z¯ (x). Similarly, we can show that F x (x) = F x (x), F y (x) = F y . Consequently (4.40) holds true.   4.4. Solutions of the homological equations. Proposition 1 (Solution of (4.36)). There is a subset O+1 ⊂ O with MeasO+1 ≥ (MeasO)(1 − K −1 ) such that for ξ ∈ O+1 , |(k, ω(ξ ))| 

1 K n+1

, for all 0 = k ∈ Zn , |k| ≤ K .

(4.41)

Then, on D(s  ) × O+1 , Eq. (4.36) has a solution F x (x, ξ ) which is analytic in x ∈ D(s  ) for ξ fixed and continuously differentiable in ξ for other variables fixed, and which is real for a real argument, such that |∂x F x | D(s  )×O+1
K C r 2 ε, |∂x F x |L
K C r 2 εL . D(s  )×O 1 +

(4.42)

Proof. The existence of the set O+1 is well-known in KAM theory. We omit the proof of the existence. Recall X R : D(s, r ) ⊂ P → P is real analytic in (x, y, z, z¯ ) ∈ D(s, r ) and is continuously differentiable in ξ ∈ O. Expanding ∂x Rx into Fourier series, √
−1(k,x) x ∂ . (4.43) ∂x Rx = x R (k)e 0=k∈Zn ,|k|≤K x Since ∂x Rx is analytic in x ∈ D(s), we get that the Fourier coefficients ∂ x R (k)’s decay exponentially in k, that is, x −s|k| x |∂
e−s|k| r 2 ε, x R (k)|
|∂x R | D(s)×O e

where we have used (4.17). Expanding ∂x F x into Fourier series: √
−1(k,x) x . ∂x F x = ∂ x F (k)e

(4.44)

(4.45)

0=k∈Zn ,|k|≤K

Inserting (4.43, 4.45) into (4.36), we get ∂x F x (x, ξ ) =


0=k∈Zn ,|k|≤K

x √ ∂ x R (k) e −(k,x) . √ −1(k, ω)

By (4.41, 4.44) as well as Lemma A.1, we get that for x ∈ D(s  ),
 ˜ |∂x F x (x, ξ )|
K C r 2 ε e−|k|(s−s )
K C r 2 ε. k∈Zn

Applying ∂ξ to both sides of (4.36) and using a method similar to the above, we can get the second inequality of (4.42).  

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119

Proposition 2 (Solution of (4.37)). On D(s  ) × O+1 , Eq. (4.37) has a solution F y (x, ξ ) which is analytic in x ∈ D(s  ) for ξ fixed and is continuously differentiable in ξ for other variables fixed, and which is real for real argument, such that |∂x F y | D(s  )×O+1
K C r ε, |∂x F y |L
K C r εL . D(s  )×O 1 +

Proof. The proof follows almost exactly the proof of Prop. 1. We omit it.

 

5. Solution of (4.38) This section is the essential part of the present paper. Expand F z (x) and R z (x) into Fourier series. Then solving (4.38) is equivalent to solving the algebraic system ( +   = R,  where ,   will be defined below. Therefore the core part of solving B) F B, R (4.38) is to find the inverse of  +  B and estimate the bound of the inverse. 5.1. Inverse of the matrix using the block decomposition. The matrix  +  B is of infinite dimension. In this sub-section, using the block decomposition technique we will decompose it into a direct sum of 1 +  B11 (of finite dimension) and 2 +  B22 (of infinite dimension), where there is not any small divisor in the inverse of 2 +  B22 . Therefore, finding the inverse of  +  B will be reduced to finding the inverse of 1 +  B11 .  Let B zz (i, j ∈ Zd ) be the elements First of all, let us introduce the notations ,  B, R. ij of matrix B zz . Note that Bizzj is a function of (x, ξ ). We temporarily omit the parameter B zz (k) be the k-Fourier ξ to simplify notation. Regarding B zz as a function of x, we let  coefficient of Bizzj . Let

ij

ij

 Bizzj = (  Bizzj (k − l) : |k|, |l| ≤ K ). Then  Bizzj is a matrix of order K , where K is the cardinality of the set {k ∈ Zn : |k| ≤ K }. Let  B zz be a block matrix whose elements are  B zz (i, j ∈ Zd ), that is, ij

 B zz = (  Bizzj : i, j ∈ Zd ) = (  Bizzj (k − l) : |k|, |l| ≤ K ; i, j ∈ Zd ). Similarly we have  B z z¯ and  B z¯ z¯ . Write F z = F z (x) as a column vector of infinite z z dimension: F (x) = (F j (x) ∈ C : j ∈ Zd ). Let  F jz = (  F jz (k) : |k| ≤ K ) and  F z = ( F jz : j ∈ Zd ), where  F jz (k) is the k th Fourier coefficient of F jz (x). Similarly we R z and  R z¯ . Set have  F z¯ ,   √   z    z B z z¯  = −√ −1 R = F , R  F B =   z ¯ z ¯ F −1 R − B z¯ z¯

 − B zz .  B z z¯

Introduce a tensor product space as follows:



H p := H p ⊗ C K = {z = (z j ∈ C K : j ∈ Zd ) : (||z||∗p )2 =


j

|z j |2 | j|2 p < ∞},

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X. Yuan

where |z j | is the Euclidean norm of z j in C K . By abuse use of notation, we still write || · ||∗p as || · || p . For any u, v ∈ H p , we can regard (u, v)T as a vector in H p ⊕ H p . Define  ||(u, v)T || p = ||u||2p + ||v||2p .

Therefore H p ⊕ H p , is a subspace of 2 ⊕ 2 , where6 2 = 2 ⊗ C K . With those notations, we have


u (k)|2 | j|2 p = u ||2p = u (k)||2p , u ∈ {z, z¯ }, || F |F || F j j∈Zd |k|≤K

|k|≤K

u (k) is k-Fourier coefficient of F u (x). Similarly, where F
||  R u ||2p = ||  R u (k)||2p , u ∈ {z, z¯ }, |k|≤K

and  2p = ||  || F|| F z ||2p + ||  F z¯ ||2p . With these notations, we have the following lemma.  is in H p ⊕ H p with Lemma 5.1. (i) The vector R  p ≤ K n r ε, || R||  Lp ≤ K n r εL . || R||

(5.1)

(ii) The linear operator  B is self-adjoint in 2 ⊕ 2 . (iii) |||  B||| p , |||  B|||Lp ≤ .

(5.2)

Proof. By the first inequality of (4.19), we have ||R u (k)|| p ≤ e−s|k| ||Ru || p,D(s)×O ≤ e−s|k| r ε, for u ∈ {z, z¯ } and |k| ≤ K . Thus, 

 p= || R|| ||R z (k)||2p + ||R z¯ (k)||2p
r ε e−s|k|
K n r ε. |k|≤K

k

 Lp ≤ K n r εL . This Similarly, using the second inequality of (4.19) we can show that || R|| completes the proof of (5.1). By (4.29) we have  B zjiz¯ (l − k),  B z¯jiz¯ (l − k), ∀i, j ∈ Zd , k, l ∈ Zn . Bizjz¯ (k − l) =  Bizzj (k − l) =  Thus the matrix    B z z¯ − B zz  B=  B z z¯ − B z¯ z¯ ⎛ ⎞  Bizjz¯ (k − l) −  Bizzj (k − l) =⎝ : i, j ∈ Zd , k, l ∈ Zn , |k|, |l| ≤ K ⎠   z¯ z¯ z z¯ − Bi j (k − l) Bi j (k − l) is self-adjoint in 2 ⊕ 2 . The proof (5.2) is delayed to the end of Sect. 7. 6 Of course, we can regard  as the usual  . 2 2

 

A KAM Theorem with Application to PDEs

121

Let ± = diag (±(k, ω) + 0j : |k| ≤ K , j ∈ Zd ),  =

 + 0

 0 . −

Expand B zz (x), B z z¯ (x), B z¯ z¯ (x), F z (x), F z¯ (x), R z (x) and R z¯ (x) in (4.38) into Fourier series, we get a lattice equation which reads  = R.  ( +  B) F

(5.3)

Let  M=

1 + K supξ ∈O |ω|

1/κ

c5

,

(5.4)

where c5 , c6 are the constants in (2.16). Write  B = (Bi j (k, l) : |k|, |l| ≤ K , i, j ∈ Zd ). Then we see that Bi j (k, l) is one of  Bizzj (k − l),  Bizjz¯ (k − l) and  Biz¯jz¯ (k − l). Let  B11 = (Bi j (k, l) : |k|, |l| ≤ K , |i|, | j| ≤ M),  B12 = (Bi j (k, l) : |k|, |l| ≤ K , |i| ≤ M, | j| > M),  B21 = (Bi j (k, l) : |k|, |l| ≤ K , |i| > M, | j| ≤ M),  B22 = (Bi j (k, l) : |k|, |l| ≤ K , |i| > M, | j| > M). Then  B=



 B11  B21

  B12 .  B22

And by (5.2) and Lemma 2.1, |||  Bi j ||| p ≤ , |||  Bi j |||Lp ≤ , i, j ∈ {1, 2}.

(5.5)

Let 1 = diag (±(k, ω) + 0j : |k| ≤ K , | j| ≤ M), 2 = diag (±(k, ω) + 0j : |k| ≤ K , | j| > M). Then =

 1 0

 0 . 2

In view of (5.4), | ± (k, ω) + 0j | ≥ 1, for all | j| > M.

(5.6)

Thus by (5.5) and (5.6), we see that there exists the inverse of 2 +  B22 and |||(2 +  B22 )−1 ||| p ≤

∞
j=0

−1  j |||(−1 2 B22 ) ||| p |||2 ||| p
1.

(5.7)

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X. Yuan

Set  Ble f :=



 Brig :=

 0 , E2

(5.8)

 − B12 (Λ2 +  B22 )−1 , E2

(5.9)

E1 −(Λ2 +  B22 )−1  B21 E1 0

where E 1 (E 2 , respectively) is a unit matrix of the same order as that of 1 (2 , respectively). Using (5.7) and (5.5), it is easy to verify that −1 −1 |||  Ble f ||| p , ||| Brig ||| p
1.

(5.10)

11 =  B B11 −  B12 (Λ2 +  B22 )−1  B21 .

(5.11)

Let

Then there exists the inverse of Λ +  B:  11 )−1 −1 (Λ1 + B (Λ +  B)−1 =  Brig 0

 0 −1  Ble f (Λ2 +  B22 )−1

(5.12)

and 11 )−1 |||∗p , |||(Λ +  B)−1 |||∗p
|||(Λ1 + B

(5.13)

11 . where ∗ ≡ the blank or L, provided that there exists the inverse of Λ1 + B 5.2. Twist conditions on the eigenvalues, measure estimates. Now we are in position to 11 . First of all, we would like to point out that the matrix investigate the inverse of Λ1 + B  B11 is of finite order, and the order is bounded by −1

K ∗ := K 2n M d
K 2n+dκ .

(5.14)

11 is also self-adjoint. Secondly, it follows from the self-adjointness of  B that the matrix B Thirdly, by (5.5, 7,11) we have 11 ||| p
, |||B 11 |||Lp
. |||B

(5.15)

Fourthly, since each element of  B is continuously differentiable in ξ ∈ O, the matrix 11 is also continuously differentiable in ξ ∈ O. Without loss of generality, we assume B 11 is non-singular the first entry ω1 of ω is in the interval [1, 2]. Then the matrix 1 + B −1 −1  if and only if ω1 1 + ω1 B11 is non-singular, and 11 )−1 ||| p . 11 )−1 ||| p
|||(ω−1 1 + ω−1 B |||(1 + B 1 1

(5.16)

 : ς1 = 1/ω1 , ς2 = ω2 /ω1 , . . . , ςn = ωn /ω1 .

(5.17)

Let

A KAM Theorem with Application to PDEs

123

Then it is easy to get     det ∂(ς1 , . . . , ςn )  = ω−(n+1)  1. 1  ∂(ω1 , . . . , ωn ) 

(5.18)

Combining (5.18) and (4.31), we can regard ς = (ς1 , . . . , ςn ) as a parameter vector instead of ξ , and we can show that Meas O
Meas (O)
Meas O

(5.19)

11 (ξ(ς ))|||
, |||∂ς B

(5.20)

∂ς 0j = o(1),

(5.21)

as well as

where (2.17) is used in (5.21). Let ς = diag (±(k1 +

n


kl ςl ) + ς1 0j : k = (k1 , . . . , kn ) ∈ Zn , |k| ≤ K , | j| ≤ M).

l=2

(5.22) Take ς as a parameter vector. Then 11 (ξ ) = ς + ς1 B 11 (ξ(ς )) := A1 . ω1−1 1 + ω1−1 B

(5.23)

Since A1 = A1 (ς ) is continuously differentiable (in Whitney’s sense) in ς ∈ (O), it follows from the Whitney extension theorem[p.131, Z] that there exists A1 (ς ) which is continuously differentiable in R such that A1 (ς ) = A1 (ς ) for any ς ∈ (O). Using Lemma A.2, there are continuously differentiable functions µ1 (ς ), . . . , µ K ∗ (ς ) which represent the eigenvalues of A1 for ς ∈ R. In particular, they also represent the eigenvalues of A1 for ς ∈ (O). Moreover, there exists a matrix-valued function U (ς ) of order K ∗ (see (5.14) for K ∗ ), which depends on7 ς , such that for every ς ∈ (O) the following equalities hold: 11 (ξ(ς )) = U (ς )diag(µ1 (ς ), · · · , µ K ∗ (ς ))U ∗ (ς ), Λς + ς1 B

(5.24)

U (ς )(U (ς ))∗ = (U (ς ))∗ U (ς ) = E,

(5.25)

and

where E is the unit matrix of order K ∗ and U ∗ is the conjugate transpose of U . It follows that |||U (ς )||| = |||U ∗ (ς )||| = 1,

(5.26)

7 U (ς ) may not be continuous in ς . See [p.124, Ka]. Fortunately, we do not need the continuity of U (ς ).

124

X. Yuan

where ||| · ||| is the 2 norm of the matrix. Arbitrarily take µ = µ(ς ) ∈ {µ1 (ς ), . . . , µ K ∗ (ς )}. Let φ be the normalized eigenvector corresponding to µ. Using Lemma A.3 and noting (5.20) and (5.21), we get  ∂ς1 µ = ((∂ς1 (Λς + ς1 B 11 (ξ(ς )))φ, φ) = (diag(0j + ς1 ∂ς1 0j : | j| ≤ M, |k| ≤ K )φ, φ) +

   ∂ς1 (ς1  B(ξ(ς ))) φ, φ

≥ min 0j + o(1).

(5.27)

j

Thus, by (2.16), we have ∂ς1 µ ≥ c > 0.

(5.28)

(Ol ) = {ς ∈ (O) : |µl | < 1/(K K ∗ )}, l = 1, . . . , K ∗ .

(5.29)

Let

By (5.28), Meas (Ol )
1/(K K ∗ ). Thus, ∗

Meas

K 

(Ol ) < 1/K .

(5.31)

l=1

Let ∗

(O) = (O) \

K 

(Ol ).

(5.32)

l=1

Therefore, Meas (O) ≥ Meas (O)(1 − O(

1 )) K

(5.33)

and for any ς ∈ (O), |µl (ς )| ≥ 1/K K ∗ .

(5.34)

11 )−1 ||| ≤ |||U ||||||U ∗ ||| max{|µ−1 |} ≤ K K ∗ ≤ K C , |||(ς + ς1 B l

(5.35)

Moreover, l

where C = n + dκ −1 + 1. Returning to the parameter ξ , by (5.19), there is a subset O+2 ⊂ O with Meas O+2 ≥ (Meas O)(1 − C1 K −1 ),

(5.36)

11 )−1 ||| ≤ C2 K C . |||(1 + B

(5.37)

and for any ξ ∈ O+2 ,

A KAM Theorem with Application to PDEs

125

5.3. Solution of (4.38). In this sub-section, using (5.37) we will give the bound of the solution of (4.38). Replacing the 2 operator norm ||| · ||| by ||| · ||| p in (5.37), in view 11 being bounded by K ∗ , using Lemma 2.2 we get of the order of the matrix 1 + B 11 )−1 ||| p ≤ K C (K ∗ ) p ≤ K C˜ , |||(1 + B

(5.38)

where C˜ is a constant depending on n, d, κ, p. By (5.13), ˜

|||( +  B)−1 ||| p ≤ K C , ξ ∈ O+2 .

(5.39)

11 )||| p ≤ K . We have that for any ξ ∈ O+2 , Note |||∂ξ (1 + B 11 )−1 ||| p = |||(1 + B 11 )−1 (∂ξ (1 + B 11 ))(1 + B 11 )−1 ||| p ≤ K 1+2C˜ . |||∂ξ (1 + B Moreover, ˜ |||∂ξ ( +  B)−1 ||| p ≤ K 1+2C .

(5.40)

By (5.1, 3) and (5.39, 5.40),  p ≤ K C¯ r ε, ||∂ξ F||  p ≤ K C¯ r εL , || F||

(5.41)

where C¯ is a constant depending on p, d, n, κ. Recall

 2p = || F|| ||  F z (k)||2p + ||  F z¯ (k)||2p . |k|≤K

(5.42)

|k|≤K

Recall s =

4 . K

(5.43)

Then for u ∈ {z, z¯ }, sup D(s  )×O+2

||F u (x, ξ )||2p =

sup D(s  )×O+2

≤K







||

u (k)e F

√

−1(k,x) 2 || p

|k|≤K

u (k)||2p e2|k|s || F



|k|≤K

≤ K e8




u (k)||2p || F

|k|≤K

 2p ≤ (K C r ε)2 .
e K || F|| 8

(5.44)

That is, ||F u (x, ξ )|| p
K C r ε, u ∈ {z, z¯ }.

(5.45)

||∂ξ F u (x, ξ )|| p
K C r ε, u ∈ {z, z¯ }.

(5.46)

sup D(s  )×O+2

Similarly, sup D(s  )×O+2

Consequently, we have the following proposition.

126

X. Yuan

Proposition 3. There is a subset O+2 ⊂ O with Meas O+2 = (Meas O)(1 − O(1/K )) and there are functions F z , F z¯ : D(s  ) × O+2 → H p which are analytic in x ∈ D(s  ) and are continuously differentiable in ξ ∈ O+2 ; moreover the functions F z , F z¯ solve (4.38) and satisfy the estimates (5.45, 5.46). Remark. In the inequalities (5.45) and (5.46), the domain D(s  ) can be expanded to D(s). In fact, by the homological equation (4.38-2), we get F z¯ (iωs) = F z (iωs)) =  s 0 z = F (0) − e− (s−τ ) [( B z z¯ (iωs))F z (iωs) + ( B z z¯ )F z (iωs) + R z (iωs))]dτ. 0

Let || · || p,s = sup D(s) || · || p . Recall ||B|| p,s = O( ), ||R z || p,s = O(ε). Since F z (x) = F z¯ (x) for real x, we get F z (x) = F z¯ (x) for all x ∈ D(s). Thus ||F z || p,s = ||F z¯ || p,s . Consequently, noting that the entries of 0 are positive and using (5.45), we get ||F z¯ (iωs)|| p ≤ K C ε + ||F z || p,s By (4.38-1), we get ||F z¯ (−iωs)|| p ≤ K C ε + ||F z || p,s . Thus, ||F z || p,s ≤ ||F z¯ (±iωs)|| p ≤ K C ε + ||F z || p,s . This leads to ||F z || p,s ≤ K C ε.

sup D(s)×O+2

||F z (x, ξ )|| p ≤ K C ε.

Applying ∂ξ to both sides of (4.38) and using the method similar to the above, we get sup D(s)×O+2

||∂ξ F z (x, ξ )|| p ≤ K C ε.

6. Symplectic Change of Variables In this section, our procedure is standard and almost the same as that of Sect. 3 in [P1, p. 128–132]. Here we give the outline of the procedure. See [P1] for the details.

A KAM Theorem with Application to PDEs

127

6.1. Coordinate transformation. By Propositions 1–3, we have a Hamiltonian F on D(s  , r ) where F = F x + (F y , y) + F z , z + F z¯ , z and have estimates of F x , F y , F z and F z¯ . Let X F be the vector field corresponding to the Hamiltonian F, that is, √ √ X F = (−∂ y F, ∂x F, −1∂z F, − −1∂z¯ F); here ∂z and ∂ z¯ are the usual 2 -gradients. Recall s = 1/K − and s  = 4/K , and let s  = 3/K , s  = 2/K , s  = 1/K . Let O+ =

2 

j

O+ .

j=1

It follows from Prop.1, 2 and 3 that for (x, y, z, z¯ ; ξ ) ∈ D(s  , r )× ∈ O+ , r| X F| p

1 1 1 |∂x F| + ||∂z F|| p + ||∂z¯ F|| p 2 r r r 1 1 z 1 z¯ y x = |F | + 2 |∂x F | + ||F || p + ||F || p r r r K x 1 z 1 z¯ y ≤ |F | + 2 |F | + ||F || p + ||F || p r r r
K C · ε, = |∂ y F| +

where C is a large constant depending only on n, κ, d and p. That is, r | X F | p,D(s  ,r )×O+


K C ε.

(6.1)


K C εL ,

(6.2)

Similarly, we have L

r | X F | p,D(s  ,r )×O+

where we have used the assumption ε < εL . As in [P1, p.129], we introduce the operator norm r || L||| p

= sup

r | L W|| p

W =0 r | W|| p

.

(6.3)

Using (6.1), (6.2) and the generalized Cauchy inequality (see Lemma A.3 of [P1, p.147]) and the observation that every point in D(s  , r/2) has at least r | · | p distance 1/9K to the boundary of D(s  , r ), we get sup

D(s  ,r/2)×O+

sup

D(s  ,r/2)×O+

r || D X F || p

L

r || D X F || p


K r | X F | p,D(s  ,r )×O+ ≤ K C ε,

(6.4)


K r | X F | Lp,D(s  ,r )×O+ ≤ K C εL ,

(6.5)

where D X F is the differential of X F . Assume that K C ε and K C εL are small enough. (These assumptions will be fulfilled in the following KAM iterations. In fact, in the m th

128

X. Yuan

KAM step, K ≈ C2m /2 and ε = (4/3) . It follows that K C ε 1.) Arbitrarily fix ξ ∈ O+ . By (6.1), the flow X tF of the vector field X F exists on D(s  , r/4) for t ∈ [−1, 1] and takes the domain into D(s  , r/2), and by Lemma A.4 of [P1, p.147], we have 2

t r| X F

m

− id|| p,D(s  ,r/4)×O+
r | X F | p,D(s  ,r )×O+
K C ε

(6.6)

and t r| X F

− id||Lp,D(s  ,r/4)×O+


exp(r || D X F || p,D(s  ,r/2)×O+ ) · r | X F | Lp,D(s  ,r )×O+
(exp(K C ε))K C εL
K C εL ,

(6.7)

for t ∈ [−1, 1]. Furthermore, by the generalized Cauchy’s inequality, t r || D X F

− I||| p,D(s  ,r/8)×O+
K C ε, t ∈ [0, 1]

(6.8)

− I||| Lp,D(s  ,r/8)×O+
K C εL , t ∈ [0, 1].

(6.9)

and t r || D X F

6.2. The new error term. Subjecting H = N + R` to the symplectic transformation  = X tF |t=1 we get the new Hamiltonian H+ := H ◦  = H ◦ X 1F on D(s  , ηr ), where 0 < η 1. By Taylor’s formula H+ = (N + R) ◦ X 1F = (N + R + ( R` − R)) ◦ X 1F = (N + R + R K + ( R` − R)) ◦ X 1F  1 = N − {F, N } + {t{F, N }, F} ◦ X tF dt  +R + 0

0

1

{R, F} ◦ X tF dt + (R K + ( R` − R)) ◦ X 1F .

(6.10)

Recall that (4.39) holds true when F solves (4.36–4.38). Thus, H+ = N+ + R` + ,

(6.11)

where 1 1 N+ = N + (F y , ∂x B zz )z, z + (F y , ∂x B z z¯ )z, z¯ + (F y , ∂x B z¯ z¯ )¯z , z¯ (6.12) 2 2 R` + = R` +1 + R` +2 + R` +3 with

 √   √  R` +1 = −1 (1 − )(( B z z¯ )F z ), z + −1 (1 − )(( B z¯ z¯ )F z ), z¯  √   √  − −1 (1 − )(( B zz )F z¯ ), z − −1 (1 − )( B z z¯ )F z¯ ), z¯  √   √  + −1 ((1 − )B z z¯ )F z ), z + −1 ((1 − )B z¯ z¯ )F z ), z¯  √   √  − −1 ((1 − )B zz )F z¯ ), z − −1 ((1 − )B z z¯ )F z¯ , z¯

(6.13)

(6.14)

A KAM Theorem with Application to PDEs

129

R` +2 = R K ◦ X 1F + ( R` − R) ◦ X 1F  t 3 ` R+ = {(1 + t)R, F} ◦ X tF dt.

(6.15) (6.16)

0

By (4.26, 4.27, 4.28) and (6.12), 1 1 N+ = (ω+ , y) + 0 z, z¯ + B+zz z, z + B+z z¯ z, z¯ + B+z¯ z¯ z¯ , z¯ 2 2

(6.17)

with B+zz = B zz + Rzz + (F y , ∂x (B zz + Rzz )), B+z z¯ = B z z¯ + Rzz + (F y , ∂x (B z z¯ + Rz z¯ )), B+z¯ z¯

=B

z¯ z¯

z¯ z¯

+ R + (F , ∂x (B y

z¯ z¯

(6.18)

z¯ z¯

+ R )),

and y (0). ω+ = ω = ω + F

(6.19)

Hence, the new perturbing vector field is X R` + = X R` 1 + (X 1F )∗ (X R` − X R + R K ) + +



t

0

(X tF )∗ [X (1+t)R , X F ] dt,

where (X tF )∗ is the pull-back of X tF and [·, ·] is the commutator of vector fields. We are now in a position to estimate the new perturbing vector field X R` + . Let Y : D(s  , r ) ⊂ P → P be a vector field on D(s  , r ), depending on the parameter ξ ∈ O+ . Let U = D(s  , ηr ) × O+ and V = D(s  , 2ηr ) × O+ and W = D(s  , 4ηr ) × O+ . By (6.6) and the “proof of estimate (12)” of [P1, p.131–132]8 , we have that t ∗ ηr | (X F ) Y|| p,U




ηr | Y|| p,V

K η2

ηr | Y|| p,W

(6.20)

and t ∗ L ηr | (X F ) Y|| p,U




L

ηr | Y|| p,V

+

·

L

ηr | X F | p,V .

(6.21)

We assume that ε3 K C η−3
1, K C ε
η, K C εη−2
1.

(6.22)

By (4.10) and (6.20, 6.21), 1 ∗ ηr | (X F ) (X R`

− X R )|| p,U


ηr | X R`

− X R | p,V
ηε

(6.23)

εK C η L ε
ηεL , η2

(6.24)

and 1 ∗ ηr | (X F ) (X R` 8 Let a = 0 in (12) of [P1].

− X R )||Lp,U
ηεL +

130

X. Yuan

where we have used K C εη−2
1. Recall that (4.9) holds still true after replacing R by R. By (4.2) and (5.4, 5.5) and using the generalized Cauchy estimate, following [P1, p. 130–131] we get r | [X R(t) ,

X F ]|| p,U
K r | X R| p,V · r | X F | p,V
K r | X R` | p,W · r | X F | p,W

(6.25)


K ε < ηε C 2

and r | [X R(t) ,

X F ]||Lp,U


K r | X R` | Lp,D1 r | X F | p,W + K r | X R` | p,W r | X F | Lp,W C L

L

(6.26)

L


K ε ε + K εε < ηε , C

where we have used K C ε < η. Finally, we have ηr | Y|| p,U

L


η−2 r |Y|| p,U ,

ηr | Y|| p,U


η−2 r |Y||Lp,U ,

(6.27)

for any vector field Y . Note that for any function f : D(s) × O → H p or Rn which is analytic in x ∈ D(s) and is continuously differentiable in ξ ∈ O,
 ||(1 −  K ) f || D(s  )×O ≤ ||  f (k)|| p es |k| ≤ || f || D(s)×O




|k|>K

e

−(s−s  )|k|

(6.28)

|k|>K

≤ || f || D(s)×O · K n ε4 , where we have used (s − s  )K ≥ | ln ε4 |.

(6.29)

Applying (6.27, 6.28) to (6.14), we can easily get ηr | X R` 1 | p,U +


K n η−2 ε4
ηε,

L

ηr | X R` 1 | p,U +


ηεL ,

(6.30)

where (6.22) is used. Collecting all terms above, we then arrive at the estimates r+ | X R` + | p,D(s+ ,r+ )×O+


ηε,

L

r+ | X R` + | p,D(s+ ,r+ )×O+


ηεL ,

(6.31)

where s+ = s  , r+ = ηr.

(6.32)

It follows from (4.22) and (4.40) that the perturbing Hamiltonian R` + satisfies the same ` that is: symmetric condition as R, R` + (x, y, z, z¯ ; ξ ) = R` + (x, y, z, z¯ ; ξ ), ∀ (x, y, z, z¯ ; ξ ) ∈ D (s+ , r+ ) × O+ .

(6.33)

Finally we can easily check that N+ satisfies the same conditions as B, that is, the conditions (4.2–4.6) hold true by replacing B by B+ and replacing ω by ω+ . Here we omit the details.

A KAM Theorem with Application to PDEs

131

7. Iterative Lemma and Proof of the Theorem 7.1. Iterative constants. As usual, the KAM theorem is proven by the Newton-type iteration procedure which involves an infinite sequence of coordinate changes. In order to make our iteration procedure run, we need the following iterative constants: 1. 0 = , l = (4/3) , l = 1, 2, . . .; 1/3 2. ηl = l , l = 0, 1, 2, ..; l

l+3

3. K 0 = 1, K l = 2 j=1 j | ln |l , l = 1, 2, . . ., (thus, K l = 2(l+3)(l+4)/2 | ln |l ); 4. Let s0 > 0 be given, (without loss of generality, we can let s0 = 1.) Let sl = 1/K l−1 , l = 1, 2, . . .; 5. r0 > 0 is given, rl = ηl r0 , l = 1, 2, . . .; 6. D(sl ) = {x ∈ Cn /(2π )n : |x| < sl } ; 7. D(sl , rl ) = {(x, y, z, z¯ ) ∈ P : |x| < sl , |y| < rl2 , ||z|| p < rl , ||¯z || p < rl }. 7.2. Iterative Lemma. Consider a family of Hamiltonian functions Hl (0 ≤ l ≤ m): 1 Hl =(ωl (ξ ), y) + (0 (ξ )z, z¯ ) + Blzz (x, ξ )z, z + Blz z¯ (x, ξ )z, z¯ 2 1 z¯ z¯ ` + Bl (x, ξ )¯z , z¯ + Rl (x, y, z, z¯ , ξ ), 2

(7.1)

where Blzz (x; ξ )’s are analytic in x ∈ D(sl ) for any fixed ξ ∈ Ol , and all of ωl (ξ ), 0 (ξ ) and Blzz (x; ξ )’s (for fixed x ∈ D(sl )) are continuously differentiable in ξ ∈ Ol , and R` l (x, y, z, z¯ , ξ ) is analytic in (x, y, z, z¯ ) ∈ D(sl , rl ) and continuously differentiable in zz d ξ ∈ Ol . Assume Blzz = (Bl,i j : i, j ∈ Z )’s satisfy the following conditions:

(l.1) Reality Condition. For any (x, ξ ) ∈ D(sl ) × Ol , the operators Blzz , Blz z¯ and Blz¯ z¯ satisfy the conditions (4.2) and (4.3) by replacing B by Bl . (l.2) Finiteness of Fourier modes. √
 Bluv (x; ξ ) = Bluv (k)e −1(k,x) , u, v ∈ {z, z¯ }. |k|≤K l

(l.3) Boundedness. sup

D(sl )×Ol

|||Bluv (x; ξ )|||∗p
, u, v ∈ {z, z¯ },

where ∗ ≡ the blank and L. Moreover, we assume the parameter sets Ol ’s satisfy (l.4) O0 ⊃ · · · ⊃ · · · Ol ⊃ · · · Om with Meas Ol ≥ ( Meas O0 )(1 − K l−1 ). (l.5) The map ξ "→ ωl (ξ ) is continuously differentiable in ξ ∈ Ol , and    ∂ωl  j  ≥ c > 0, sup |∂ξ ωl | ≤ c, j = 0, 1. inf det ∂ξ  Ol Ol

132

X. Yuan

(l.6) The perturbation R` l (x, y, z, z¯ ; ξ ) is analytic in the space coordinate domain D(sl , rl ) and continuously differentiable in ξ ∈ Ol , and R` l (x, y, z, z¯ ; ξ ) = R` l (x, y, z, z¯ ; ξ ) ∀ (x, y, z, z¯ ; ξ ) ∈ D (sl , rl ) × Ol ; √ √ moreover, its vector field X R`l := (∂ y R` l , −∂x R` l , −1∂z R` l , − −1∂z¯ R` l )T defines on D(sl , rl ) an analytic map X R`l : D(sl , rl ) ⊂ P → P. (l.7) In addition, the vector field X R`l is analytic in the domain D(sl , rl ) with small norms rr | X R` l | p,D(sl ,rl )×Ol


l ,

L

rl | X R` l | p,D(sl ,rl )×Ol

1/3


l

.

Then there is an absolute positive constant ∗ enough small such that, if 0 < 0 < ∗ , there is a set Om+1 ⊂ Om , and a change of variables m+1 : Dm+1 := D(sm+1 , rm+1 ) × Om+1 → D(sm , rm ) being real for real argument, analytic in (x, y, z, z¯ ) ∈ D(sm+1 , rm+1 ), and continuously differentiable in ξ ∈ Om+1 , as well as that the following estimates hold true: 1/2

rm | m+1

− id|| p,Dm+1
m

rm | m+1

− id||Lp,Dm+1
m .

and 1/4

Furthermore, the new Hamiltonian Hm+1 := Hm ◦ m+1 of the form 1 zz z z¯ Hm+1 =(ωm+1 (ξ ), y) + (0 (ξ )z, z¯ ) + Bm+1 (x, ξ )z, z + Bm+1 (x, ξ )z, z¯ 2 1 z¯ z¯ + Bm+1 (x, ξ )¯z , z¯ + R` m+1 (x, y, z, z¯ , ξ ) 2

(7.2)

satisfies all the above conditions (l.1 − l.7) with l being replaced by m + 1. 7.3. Proof of the Iterative Lemma. As stated as in the iterative lemma, we have a family of Hamiltonian functions Hl ’s (l = 0, 1, . . . , m) which satisfy the conditions (l.1 −l.7). We now consider the Hamiltonian Hm . Let N = Nm , R` = R` m , N+ = Nm+1 and R` + = R` m+1 . Take c a large and absolute constant. Let s = sm , s  = sm+1 , η = ηm , m r = rm = ηm r0 , ε = (4/3) , εL = ε1/3 and O+1 ∪ O+2 = Om+1 . Clearly, ε < εL and ηε < m+1 . In order to use the conclusion of Sect. 6, we need to verify those conditions imposed on s, s  , K − , K and ε. Recall s = 1/K − , s  = 4/K . In the m th KAM step, we take 1

K = K m = 2(m+3)(m+4)/2 | ln |m , η = ηm = m = 3 (4/3) . 1/3

Thus, −1 (s − s  )K = K m K m−1 − 4 = 2m+3 | ln | − 4

m

A KAM Theorem with Application to PDEs

133

and 4| ln ε| = 4| ln |(4/3)m . It is easy to check that 2m+3 | ln | − 4 > 4| ln |(4/3)m . Therefore, (s − s  )K ≥ | ln ε4 |. This verifies the assumption (6.29). Similarly, we can verify the assumptions (s −s  )K ≥ | ln η2 |, K η < 1 in Lemma 4.2 and the assumptions (6.22). We omit the details. By means of the conclusion in Sect. 6, there is a Hamiltonian F = Fm defined on9 D(sm+1 , rm+1 )× Om+1 and a symplectic change of variables m+1 = X tFm |t=1 . Note ηn m = m+1 . It is easy to verify that the conditions (m + 1.1 − 7) are fulfilled. We omit the details. This completes the proof of The Iterative Lemma.   7.4. Proof of Theorem 2.1. The proof is similar to that of [P1]. Here we give an outline. By Assumptions A–F and the smallness assumption in Theorem 2.1, the conditions (l.1 − l.7) in the iterative lemma in Sect. 6.2 are fulfilled with l = 0. Hence the iterative lemma applies to H which is defined in Theorem 2.1. Inductively, we get the following: (i) Domains: for m = 0, 1, 2, . . ., Dm := D(sm , rm ) × Om , Dm+1 ⊂ Dm ; (ii) Coordinate changes:  m = 1 ◦ · · · ◦ m+1 : Dm+1 → D(s0 , r0 ), ; (iii) Hamiltonian functions Hm (m = 0, 1, . . .) satisfy the conditions (l.1 − l.7) with l replaced by m; Let O∞ = ∩∞ m=0 Om , D∞ = ∩Dm . By the same argument as in [P1, pp.134], we conclude that  m , D m , Hm , X Hm , ωm converges uniformly on the domain D∞ , and X H˜ ∞ ◦  ∞ = D ∞ · X ω∞ , where H˜ ∞ := lim H˜ m = (ω∞ (ξ ), y) + 0 (ξ )z, z¯ m→∞

1 z¯ z¯ 1 zz z z¯ (x, ξ )z, z + B∞ (x, ξ )z, z¯ + B∞ (x, ξ )¯z , z¯ , + B∞ 2 2 zz = lim zz here B∞ m→∞ Bm ,…, and X ω∞ is the constant vector field ω∞ on the torus n N T . Thus, T × {0} × {0} is an embedding torus with rotational frequencies ω∞ (ξ ) ∈ ω∞ (O∞ ) of the Hamiltonian H∞ . Returning to the original Hamiltonian H , it has an embedding torus ∞ (Tn × {y = 0} × {z = z¯ = 0}) with frequencies ω∞ (ξ ). This proves the theorem.   Finally, let us finish the proof of (5.2). According to (6.18) we see that, in the m th KAM iteration,  y  y B(x) = Bm (x) = Bm−1 (x) + (Fm (x), ∂x Bm−1 (x)) + Rm (x) + Fm (x), ∂x Rm (x) , 9 Note D m+1 := D(sm+1 , rm+1 ) × Om+1 ⊂ D(sm+1 , rm+1 ) × Om+1 .

134

X. Yuan

where y

|Fm | D(sm )
m K m rm , |Rm | D(S)
m , |||Bm−1 ||| p,D(sm−1 )
, m = 1, 2, . . . , while we have omitted the dependence of all quantities on the parameter ξ . Thus we can write B(x) = B0 (x) +

m


τ (x)

τ =0

τ = ( τ (k − l) : |k|, |l| ≤ K τ ) where  τ (k) is the k th with |||τ ||| p,D(sτ )
τ . Set  Fourier coefficient of τ . Note τ is analytic in D(sτ ). It follows from |||τ ||| p,D(sτ )
τ that τ (k)||| p,D(sτ )
e−|k|sτ τ1/2 ≤ τ1/2 . ||| Let u = (u(l) : |l| ≤ K τ ) with u(l) ∈ H p and ||u||2p = l ||u(l)||2p = 1. Then


τ (k − l)u(l)||2p ≤ τ u||2p =  || || τ2 ≤ K τ2 τ2 . |k|≤K τ

|l|≤K τ

|k|,|l|≤K τ

τ ||| p
K τ τ . Similarly, we have ||| That implies ||| B0 ||| p
K 0 0 = . Thus |||  B||| p ≤

m


K τ τ
.

τ =0

This completes the proof of the first inequality in (5.2). Restoring the dependence of all quantities on ξ and applying ∂ξ to those quantities, the proof of the second inequality in (5.2) can be similarly completed. We omit the details. Final Remark. Assumption E and the smallness condition (2.19) can weaken. Let P p := (Cn /2π Zn ) × Cn × H p × H p . Suppose 0 ≤ p − p˜ ≤ κ, where κ is the growth rate of 0j . (See (2.16) for κ.) For W = (X, Y, Z , Z¯ ) ∈ P p˜ , define r˜ | W|| p˜

= |X | +

1 1 1 |Y | + ||Z || p˜ + || Z¯ || p˜ , 2 r˜ r˜ r˜

r˜ > 0.

For W : D(s, r ) × O ⊂ P p × O → P p˜ , define r˜ | W|| p, p,D(s,r ˜ )×O

:=

sup

(x,ξ )∈D(s,r )×O

r˜ | W (x, ξ )|| p˜

and L

r˜ | W|| p, p,D(s,r ˜ )×O

:=

sup

D(s,r )×O

r˜ |∂ξ W (x, ξ )|| p˜ .

Assumption E* (Regularity). Let s0 , r0 be given positive constants. Assume the perturbation term R 0 (x, y, z, z¯ ; ξ ) which is defined on the domain D(s0 , r0 ) × O0 is analytic

A KAM Theorem with Application to PDEs

135

in the space coordinates and continuously differentiable in ξ ∈ O0 , as well as, for each ξ ∈ O0 its Hamiltonian vector field √ √ X R 0 := (R 0y , −Rx0 , −1∂z R 0 , − −1∂z¯ R 0 )T defines a analytic map X R 0 : D(s0 , r0 ) ⊂ P p → P p˜ , where T ≡ transpose and ∂z is the 2 -gradient. Also assume that X R 0 is continuously differentiable in ξ ∈ O0 , r0 | X R 0 | p, p,D(s ˜ 0 ,r0 )×O0

< ,

L

r0 | X R 0 | p, p,D(s ˜ 0 ,r0 )×O0

< 1/3 .

(2.19*)

Then we get Theorem 2.1*. Replacing Assumption E and (2.19) by Assumption E* and (2.19*), respectively, Theorem 2.1 still holds true. The reason is mainly that (5.7) can be replaced by |||(2 +  B22 )−1 ||| p p˜
1,

(5.7*)

using the growth condition (2.16), where ||| · ||| p p˜ is the operator norm from H p to H p˜ . Theorem 2.1* can be applied to some quasi-linear PDEs, for example, u tt − u + Mσ u + |∂θ u|3 = 0, θ ∈ Td , d ≥ 1,

(3.1*)

√ −1u t − u + Mσ u + u|∂θ u|2 = 0, θ ∈ Td , d ≥ 1.

(3.20*)

We omit the details here. 8. Appendix A. Some Technical Lemmas Lemma A.1. For µ > 0, ν > 0, the following inequality holds true:


1 ν e−2|k|µ |k|ν ≤ ( )ν ν+d (1 + e)d . e µ d

k∈Z

Proof. This lemma can be found in [B-M-S]. Lemma A.2. Consider an n × n complex matrix function Y (ξ ) which depends on the real parameter ξ ∈ R. Let Y (ξ ) be a matrix function satisfying conditions: (i) Y (ξ ) is self-adjoint for every ξ ∈ R; i.e., Y (ξ ) = (Y (ξ ))∗ , where star denotes the conjugate transpose matrix; (ii) Y (ξ ) is continuously differentiable in an interval I of the real variable ξ . Then there exist n continuously differentiable functions µ1 (ξ ), · · · , µn (ξ ) on I that represent the repeated eigenvalues of Y (ξ ). Proof. See [pp.122–124, Ka].

 

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Lemma A.3. Assume Y = Y (ξ ) satisfies the conditions in Lemma A.3. Let µ = µ(ξ ) be any eigenvalue of Y and φ be the normalized eigenfunction corresponding to µ. Then ∂ξ µ = ((∂ξ Y )φ, φ)). Proof. The proof can be found in [Ka, p.125]. Acknowledgement. The author is very grateful to D. Bambusi, H. Eliasson, W. Craig, S. Kuksin, de la Llave, W. M. Wang and C. Zeng for helpful discussions and encouragement, and to the three referees for their useful suggestions.

References [B]

Bambusi, D.: Birkhoff normal form for some nonlinear PDEs. Commun. Math. Phys. 234, 253–285 (2003) [B-G] Bambusi, D., Grébert, B.: Birkhoff normal form for PDEs with Tame modulus. Duke Math. J. 135(3), 507–567 (2006) [B-P] Bambusi, D., Paleari, S.: Families of periodic orbits for resonant PDE’s. J. Nonlinear Science 11, 69–87 (2001) [Ba] Baldi, P.: Quasi-periodic solutions of the equation u tt − u x x + u 3 = f (u). Preprint, 2005 [B-B] Berti, M., Bambusi, D.: A Birkhoff- Lewis type theorem for some Hamiltonian PDEs. SIAM J. Math. Anal. 37(1), 83–102 (2005) [B-Bo] Berti, M., Bolle, P.: Cantor families of periodic solutions for completely resonant nonlinear wave equations. Duke Math. J. 134(2), 359–419 (2006) [Bo-K] Bobenko, A.I., Kuksin, S.: The nonlinear Klein-Gordon equation on an interval as perturbed sine-Gordon equation. Commun. Math. Helv. 70, 63–112 (1995) [Br-K-S] Bricmont, J., Kupiainen, A., Schenkel, A.: Renormalization group and the Melnikov Problems for PDE’s. Commun. Math. Phys., 221, 101–140 (2001) [B-M-S] Bogolyubov, N.N., Mitropolskij, Yu. A., Samojlenko, A.M.: Methods of Accelerated Convergence in Nonlinear Mechanics. New York: Springer-Verlag, 1976 [Russian Original: Kiev: Naukova Dumka, 1969] [Bo1] Bourgain, J.: Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and application to nonlinear pde. Int. Math. Research Notices 11, 475–497 (1994) [Bo2] Bourgain, J.: Periodic solutions of nonlinear wave equations, Harmonic analysis and partial equations. Chicago IL: Chicago Univ. Press, 1999, pp. 69–97 (1999) [Bo3] Bourgain, J., Quasi-periodic solutions of Hamiltonian perturbations for 2D linear Schrödinger equation. Ann. Math. 148, 363–439 (1998) [Bo4] Bourgain, J.: Green’s function estimates for lattice Schrödinger operators and applications. Annals of Mathematics Studies, 158.,Princeton, NJ: Princeton University Press, 2005 [Bo5] Bourgain, J.: On Melnikov’s persistence problem. Math. Res. Lett. 4, 445–458 (1997) [Br] Brézis, H.: Periodic solutions of nonlinear vibrating strings and duality principles. Bull. AMS 8, 409–426 (1983) [C-W] Craig, W., Wayne, C.: Newton’s method and periodic solutions of nonlinear wave equation. Commun. Pure. Appl. Math. 46, 1409–1501 (1993) [Ch-Y] Chen W., Yuan, X., Quasi-periodic solutions of nonlinear wave equation of degeneracy. Preprint [C-Yo] Chierchia, L., You, J.: KAM tori for 1D nonlinear wave equations with periodic boundary conditions. Commun. Math. Phys. 211, 497–525 (2000) [E] Eliasson, L.H.: Of stable invariant tori for Hamiltonian systems. Ann Perturbations Scula Norm. Sup. Pisa CL Sci. 15, 115–147 (1998) [E-K] Eliasson, L.H., Kuksin, S.B.: KAM for non-linear Schroedinger equation. Preprint, 2006 [F-S] Frohlich, J., Spencer, T.: Absence of diffusion in the Anderson tight binding model for large disorder or lower energy. Commun. Math. Phys. 88, 151–184 (1983) [G-Yo] Geng, J., You, J.: A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces. Commun. Math. Phys. 262, 343–372 (2006) [G-M-P] Gentile, G., Mastropietro, V., Procesi, M.: Periodic solutions for completely resonant nonlinear wave equations. Commun. Math. Phys., 256, 437–490 (2005) [Ka] Kato, T.: Perturbation Theory for Linear Operators. (Corrected printing of the second edition) Berlin Heidelberg New York: Springer-Verlag, 1980 [K1] Kuksin, S.B.: Nearly integrable infinite-dimensional Hamiltonian systems. Lecture Notes in Math. 1556. New York: Springer-Verlag, 1993

A KAM Theorem with Application to PDEs

[K2] [K3] [K-P] [L-S] [P1] [P2] [Pr] [W] [Yos] [Y1] [Y2] [Y3] [Y4] [Y5] [Z]

137

Kuksin, S.B.: Elements of a qualitative theory of Hamiltonian PDEs. Doc. Math. J. DMV (Extra Volume ICM) II, 819–829 (1998) Kuksin, S.B.: Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum. Funct. Anal. Appl. 21, 192–205 (1987) Kuksin, S.B., Pöschel, J.: Invariant cantor manifolds of quasiperiodic oscillations for a nonlinear Schrödinger equation. Ann. Math. 143, 149–179 (1996) Lidskij, B.V., Shulman, E.: Periodic solutions of the equation u tt − u x x + u 3 = 0. Funct. Anal. Appl. 22, 332–333 (1988) Pöschel, J.: A KAM-theorem for some nonlinear PDEs. Ann. Scuola Norm. Sup. Pisa, Cl. Sci., IV Ser. 15(23), 119–148 (1996) Pöschel, J.: Quasi-periodic solutions for nonlinear wave equation. Commun. Math. Helv. 71, 269–296 (1996) Procesi, M.: Quasi-periodic solutions for completely resonant nonlinear wave equations in 1D and 2D, Discr. and Cont. Dyn. Syst. 13, 541–552 (2005) Wayne, C.E.: Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory. Commun. Math. Phys. 127, 479–528 (1990) Yosida, K.: Functional Analysis. Berlin Heidelberg New York: Springer-Verlag (1980); reprinted in China by Beijing World Publishing corporation, 1999 Yuan, X.: Invariant Manifold of Hyperbolic-Elliptic Type for Nonlinear Wave Equation. Int. J. Math. Math. Science 18, 1111–1136 (2003) Yuan, X.: Invariant tori of nonlinear wave equations with a given potential. Discrete and continuous dynamical systems 16(3), 615–634 (2006) Yuan, X.: Quasi-periodic solutions of completely resonant nonlinear wave equations. J. Diff. Eq. 230, 213–274 (2006) Yuan, X.: Quasi-periodic solutions of nonlinear Schrödinger equations of higher dimension. J. Diff. Eq. 195, 230–242 (2003) Yuan, X.: A variant of KAM theorem with applications to nonlinear wave equations of higher dimension. Preprint. FDIM 2005-2., see also http://www.ma.utexas.edu/mp− arc, no.06-44 Ziemer, W.P.: Weakly differentiable functions. Graduate Texts in Math., no. 120, Berlin-HeidelbergNew York: Springer Verlag, 1989, reprinted in China by Beijing World Publishing corporation 1999

Communicated by G. Gallavotti

Commun. Math. Phys. 275, 139–162 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0299-y

Communications in

Mathematical Physics

General Adiabatic Evolution with a Gap Condition Alain Joye Institut Fourier, Université de Grenoble 1, BP 74, 38402 St.-Martin d’Hères Cedex, France. E-mail: [email protected] Received: 28 August 2006 / Accepted: 30 January 2007 Published online: 20 July 2007 – © Springer-Verlag 2007

Abstract: We consider the adiabatic regime of two parameters evolution semigroups generated by linear operators that are analytic in time and satisfy the following gap condition for all times: the spectrum of the generator consists in finitely many isolated eigenvalues of finite algebraic multiplicity, away from the rest of the spectrum. The restriction of the generator to the spectral subspace corresponding to the distinguished eigenvalues is not assumed to be diagonalizable. The presence of eigenilpotents in the spectral decomposition of the generator typically leads to solutions which grow exponentially fast in some inverse power of the adiabaticity parameter, even for real spectrum. In turn, this forbids the evolution to follow all instantaneous eigenprojectors of the generator in the adiabatic limit. Making use of superadiabatic renormalization, we construct a different set of time-dependent projectors, close to the instantaneous eigenprojectors of the generator in the adiabatic limit, and an approximation of the evolution semigroup which intertwines exactly between the values of these projectors at the initial and final times. Hence, the evolution semigroup follows the constructed set of projectors in the adiabatic regime, modulo error terms we control.

1. Introduction Singular perturbations of differential equations play an important role in various areas of mathematics and mathematical physics. Such perturbations typically appear when one considers problems that display several different time and/or length scales. In particular, the semiclassical analysis of quantum phenomena and the study of evolution equations in the adiabatic regime lead to singularly perturbed linear differential equations which are the object of many recent works. See for example the monographs [14, 11, 13, 29, 40]. The description of certain non-conservative phenomena with distinct time scales also gives rise to non-autonomous linear evolution equations, which are more general than

140

A. Joye

those stemming from conservative systems, and whose adiabatic regime is of physical relevance, see e.g. [32, 33, 41, 35–37, 2, 3, 1]. The present paper is devoted to the study of general linear evolution equations in the adiabatic limit under some mild spectral conditions on the generator. The chosen set up is sufficiently general to cover most applications where the time dependent generator is characterized by a gap condition on its spectrum. Let us describe informally our result, the precise theorem being formulated in Sect. 2 below. We consider a general linear evolution equation in a Banach space B of the form iε∂t U (t, s) = H (t)U (t, s), U (s, s) = I, s ≤ t ∈ [0, 1]

(1.1)

in the adiabatic limit ε → 0+ , for a time-dependent generator H (t). This equation describes a rescaled non-autonomous evolution generated by a slowly varying linear operator H (t). The evolution operator U (t, s) evidently depends on ε, even though this is not emphasized in the notation. The generator H (t) is assumed to depend analytically on time and to have for any fixed t a spectrum σ (H (t)) divided into two disjoint parts, σ (H (t)) = σ (t) ∪ σ0 (t), where σ (t) consists in a finite number of complex eigenvalues σ (t) = {λ1 (t), λ2 (t), . . . , λn (t)} which remain isolated from one another as t varies in [0, 1]. Moreover, the spectral projector of H (t) associated with σ (t), denoted by P(t), is assumed to be finite dimensional. The part of H (t) which corresponds to the spectral projector P0 (t) associated with σ0 (t) can be unbounded, bounded or zero. In the first case we need to assume H (t) generates a bona fide evolution operator. This spectral assumption, or gap condition, is familiar in the quantum adiabatic context where B is a Hilbert space on which H (t) is further assumed to be self-adjoint, see [10, 25, 30, 7, 1], for example. Note that it is still possible to study the quantum adiabatic limit by altering the gap condition in different ways, as shown in [6, 18, 11, 5, 15, 39, 3, 4]. By contrast to previous studies of similar general problems [12, 32, 28, 23, 1], we do not assume that the restriction of H (t) to the spectral subspace P(t)B is diagonalizable. Such situations take place in the study of open quantum systems by means of phenomenological time-dependent master equations, [35, 36, 41, 37]. We come back to the approach of [35] below. Therefore, for the part H (t)P(t) of the generator, we have a complete spectral decomposition H (t)P(t) =

n


λ j (t)P j (t) + D j (t),

(1.2)

j=1

where the P j (t)’s are eigenprojectors and the D j (t)’s are eigenilpotents associated to the eigenvalue λ j (t) that satisfy n


P j (t) = P(t), P j (t)Pk (t) = δ jk P j (t), and D j (t) = P j (t)D j (t)P j (t).

(1.3)

j=1

In case B is a Hilbert space on which H (t) is self-adjoint or if H (t) is diagonalizable with real simple isolated eigenvalues only, the evolution U (t, s) follows the instantaneous eigenprojectors P j (t) in the adiabatic regime in the sense that U (t, s)P j (s) = P j (t)U (t, s) + O(ε), as ε → 0,

(1.4)

General Adiabatic Evolution with a Gap Condition

141

as shown in [10, 25, 30, 7, 1], and [12, 28, 23], for example. In other words, transitions between different spectral subspaces are suppressed as ε → 0, while the restriction of the evolution within these subspaces dominates the error term. This relation remains true for certain eigenprojectors if the eigenvalues are allowed to have negative imaginary parts, [32, 1]. This fact is also well-known and crucial in the study of the Stokes phenomenon appearing in singularly perturbed differential equations [14]: under analyticity assumptions, one considers certain paths in the complex t-plane, called canonical of dissipative paths, along which an equivalent of (1.4) is true in order to get bounds on, or to compute exponentially small quantities in 1/ε stemming from singularities in the complex t-plane. Such methods are used in [20, 21, 23 and 19], to bound or to compute exponentially small transitions in the adiabatic limit when the relevant eigenvalues are real on the real axis. However, when eigenilpotent are present in the spectral decomposition (1.2), the relation (1.4) cannot hold in general for all eigenprojectors P j (t), even for real-valued eigenvalues λ j (t). Indeed, assuming the eigenvalues are real for the discussion, the soluβ tion U (t, s) generically grows like e D/ε , for some positive D and 0 < β < 1, due to β the presence of eigenilpotents. Hence, the error term in (1.4) becomes of order εe D/ε , which is still smaller than U (t, s). However, the transition amplitudes between spectral subspaces P j (t)U (t, 0)Pk (0) are typically exponentially increasing as ε → 0, rather than vanishing as ε → 0. An explicit example of this fact is provided at the end of the Introduction. We come back to this mechanism below. In this context, our main result reads as follows. We construct a different set of q ∗ (ε) time-dependent projectors P j (t) which approximates the eigenprojectors P j (t) in the adiabatic regime ε → 0. And we show that the evolution U (t, s) can be approx∗ imated up to an error which vanishes as ε → 0 by a simpler evolution, V q (ε) (t, s), ∗ q (ε) which exactly follows the constructed approximations P j (t) of the instantaneous eigenprojectors. In other words, we restore the expected adiabatic behaviour, i.e. suppression of certain transitions, by trading the instantaneous eigenprojectors for other nearby projectors in the limit ε → 0. t When the eigenvalues are complex valued, the “dynamical phases” e−i s λ j (u) du/ε contribute other factors which may be exponentially growing, depending on the imaginary parts of the eigenvalues and which further need to be taken care of as in [32 or 1]. In case H (t) is unbounded, we assume the part H (t)P0 (t) generates a semigroup t bounded by |e−i s λ0 (u) du/ε |, for some function λ0 (t). More precisely, for all j = 1, . . . , n and for any 0 ≤ t ≤ 1, we construct perturbatively a set of projectors close to the spectral projectors of H (t), see Sect. 5, q  (ε)

Pj

q  (ε)

(t) = P j (t) + O(ε), and P0

(t) ≡ I −

n


q  (ε)

Pj

(t).

(1.5)

j=1 

Let W q (ε) (t) be the intertwining operator naturally associated with the projectors q  (ε) Pk (t), k = 0, . . . , n introduced by Kato [25], such that Wq

 (ε)

q  (ε)

(t)Pk

q  (ε)

(0) = Pk

(t)W q

 (ε)

(t), k = 0, . . . , n.

(1.6)

The approximation is then defined by Vq

 (ε)

(t, 0) = W q

 (ε)

(t)q

 (ε)

(t, 0),

(1.7)

142

A. Joye q  (ε)



where q (ε) (t, s) commutes with all the Pk (0), k = 0, . . . , n for any t and satisfies a certain singularly perturbed linear differential equation, see (6.17) below, which q  (ε) describes the effective evolution within the fixed subspaces Pk (0)B. Therefore, the following exact intertwining relation holds: Vq

 (ε)

q  (ε)

(t, 0)Pk

q  (ε)

(0) = Pk

(t)V q

 (ε)

(t, 0), k = 0, . . . , n.

(1.8)

Introducing ω(t) = maxk=0,...,n λk (t) to control the norm of the “dynamical phases”, we prove the existence of κ > 0 such that for any 0 ≤ t ≤ 1, U (t, 0) = V q

 (ε)

(t, 0) + O(te−κ/ε e

t



t 0

ω(u) du/ε

),

(1.9)

β

where V q (ε) (t, 0) = O(e 0 ω(u) du/ε e D/ε ), for some D ≥ 0 and 0 < β < 1. Note that the first term always dominates the exponentially smaller error term, and moreover, that this error term tends to zero as ε → 0 for times up to T > 0, of order one, T such that 0 ω(u) du = κ. The latter property ensures that the transition amplitudes q  (ε)

q  (ε)

(t)U (t, 0)Pk (0), j = k, vanish in the adiabatic limit, provided 0 ≤ t ≤ T . P j In case B is a Hilbert space and H (t) is self-adjoint, both the evolution and its approximation are unitary and D can be chosen equal to zero. The intertwining identity (1.8) q  (ε) and (1.9) show that the transitions between the different subspaces P j (0)B are exponentially small in ε, while the transitions between the spectral subspaces of H are of  order ε. Constructions leading to approximations V q (ε) of this type with exponentially small error term go under the name superadiabtic renormalization, according to the terminology coined by Berry [9], in this quantum adiabatic context. The first general rigorous construction of this type appears in [31], but we shall use that of [22]. The statement (1.9) is thus very similar to the Adiabatic Theorem of quantum mechanics [25, 30, 7, 1]... and, more precisely, to the subsequent exponentially accurate versions in an analytic context provided in [21, 31, 22, 24, 16, 17]... or variants thereof. However, while the improvement of the error term in (1.9) from O(ε) to O(e−κ/ε ) by considering q  (ε) Pj in place of P j in the adiabatic context is just that, improvement, in case there are q  (ε)

non-zero nilpotents in the decomposition (1.2), it becomes necessary to consider P j and achieve exponential accuracy to get a result on the vanishing of transition amplitudes between certain subspaces. q  (ε) (0), This can be understood as follows. As ε (t, 0) commutes with all the Pk k = 0, . . . , n for any t we can write q

 (ε)

(t, 0) =

n


q  (ε)

Pk

(0)q

 (ε)

k=0

q  (ε)

(t, 0)Pk

(0) ≡

n


q  (ε)

k

(t, 0).

(1.10)

k=0

q  (ε)

q  (ε)

The operator  j (t, 0) describing the evolution within the fixed subspaces P j satisfies for j ≥ 1, q  (ε)

iε∂t  j

q  (ε)

(t, 0) = (λ j (t)P j

q  (ε)  j (0, 0)

=

q  (ε) Pj (0),

q  (ε)

 j (t, ε) + O(ε)) (0) + D j

(t, 0),

(0)B

(1.11)

General Adiabatic Evolution with a Gap Condition

143

 j (t, ε) denotes the nilpotent D  j (t, ε) = W q  (ε) −1 (t)D j (t)W q  (ε) (t). We can where D write q  (ε)

j

i

(t, 0) = e− ε

t 0

λ j (u)du

q  (ε)

j

(t, 0),

(1.12)

q  (ε)

where the operator  j is essentially generated by a nilpotent. Such adiabatic evolutions generated by perturbations of analytic nilpotents are studied in Sect. 4. We show q  (ε) that  j typically grows when ε → 0 as q  (ε)

j q  (ε)

βj

(t, 0)  ec/ε , with 0 < β j < 1,

(1.13) β

whereas  j (t, 0) remains bounded as ε → 0 iff D j (t) ≡ 0. The growth in e1/ε , 0 < β < 1, of adiabatic evolutions generated by certain nilpotents is already present in the works [42] and [38]. Hence, to compensate the exponential growth in 1/εβ j of the q  (ε)  j (t, 0)’s which induces transitions between the instantaneous eigenspaces of the same order, see the example below, it is necessary to push the estimates to exponential q  (ε) order, see (1.9), by trading the P j ’s for the P j . This requires analyticity of the data, see Sect. 5. Analyticity is also essential in Sect. 4 where the properties of nilpotent generators and the adiabatic evolutions they generate are studied. Let us finally comment on the paper [35]. It addresses, at a theoretical physics level, the evolution of master equations describing open quantum systems in which the components of the Lindblad generator are slowly varying functions of time. Mathematically, this corresponds to a particular case of problem (1.1) with a generator containing nilpotents in its decomposition (1.2) and for which ω(t) ≡ 0. The authors argue under certain implicit conditions on the evolution, that it is possible to approximate U (t, 0) by some operator V ε (t, 0) which satisfies the intertwining relation (1.8) with the instantaq  (ε) neous projectors P j (t) in place of the approximate projectors P j (t) = P j (t) + O(ε). The authors recognize that such a statement cannot be true generically, and we confirm their conclusion. The statement does hold, however, under the hypotheses of [1], that is when the nilpotent part of the generator in the corresponding subspace P j (t)B is absent, together with an a priori bound on the evolution (see also remark iii) at the end of the section). It also holds when the considered spectral subspace P j (t)B is always decoupled from the others, an example of this sort is indeed provided in [35]. Otherwise the error term becomes too large due to the growth (1.13). The paper is organized as follows. We close the introduction by the example alluded to above and then provide the precise hypotheses and the mathematical statement corresponding to our main result. The rest of the paper is devoted to the proof of it. The main steps are in Sect. 4 which studies adiabatic evolutions generated by (perturbations of) analytic nilpotents. The iterative scheme providing the adiabatic renormalization of [22] is briefly recalled in Sect. 5. The approximation and its properties are presented in Sect. 6.

1.1. About the effect of nilpotents. We consider here an explicitly solvable model defined by a simple generator with two real valued distinct eigenvalues possessing a nilpotent in its spectral decomposition. We show that this nilpotent induces exponentially increasing

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transitions (in 1/εβ , β < 1) between the instantaneous eigenspaces, thereby underlying the necessity to use superadiabatic renormalization to achieve our result. We also q ∗ (ε) identify the approximated projectors P j that the evolution follows. Let H be a constant 3 × 3 matrix in a canonical basis {e1 , e2 , e3 } defined by ⎛ ⎞ 0 a 0 H = ⎝0 0 0⎠ , (1.14) 0 0 1 and let L be another constant 3 × 3 matrix defined by ⎛ ⎞ 0 0 −k L = ⎝−k 0 0 ⎠ , 0 0 0

(1.15)

where the non-zero scalars a, k will be chosen later on. We set S(t) := e−it L , H (t) := S(t)H S −1 (t),

(1.16)

and consider the adiabatic evolution U (t, 0) defined for any t ∈ [0, 1] by iεU (t, 0) = H (t)U (t, 0), U (0, 0) = I.

(1.17)

The spectrum of H (t) is {0, 1} and its decomposition reads H (t) = S(t)(0P0 + D0 + 1P1 )S −1 (t) ≡ 0P0 (t) + D0 (t) + 1P1 (t),

(1.18)

where P0 = e1 e1 | + e2 e2 |, P1 = e3 e3 | and D0 = a e1 e2 |. Here { e j |} j=1,2,3 denotes the adjoint basis of {e j } j=1,2,3 . The operator (t) := S −1 (t)U (t, 0) satisfies iε (t) = (H − εL) (t), ⇒ (t) = e−it (H −εL)/ε .

(1.19)

The matrix H − εL is now diagonalizable and its spectrum is √ √ (1.20) {1, + εak, − εak} ≡ {1, λ+ (ε), −λ+ (ε)} ≡ {1, λ+ (ε), λ− (ε)}, √ where · denotes any branch of the square root function. The corresponding spectral projectors are denoted by P1 (ε), P+ (ε) and P− (ε) and they are given by ⎛ εk ⎞ 0 0 1−εak ε2 k 2 ⎠ , P1 (ε) = ⎝0 0 (1.21) 1−εak 0 0 1 ⎞ ⎛ λ (ε) λ (ε)εk +

− (ε) ⎜ λ+ (ε)−λ εk P+ (ε) = ⎝ λ+ (ε)−λ− (ε) 0

a λ+ (ε)−λ− (ε) λ+ (ε) λ+ (ε)−λ− (ε)

0

+

(λ+ (ε)−λ− (ε))(λ+ (ε)−1) ⎟ ε2 k 2 ⎠, (λ+ (ε)−λ− (ε))(λ+ (ε)−1)

(1.22)

0

and P− (ε) has√ the same expression as P+ (ε) with indices + and − exchanged. Note that P± (ε)  ±a/ εak as ε → 0, whereas the projectors P1 (ε) = P1 + O(ε) P0 (ε) = P+ (ε) + P− (ε) = P0 + O(ε)

(1.23) (1.24)

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145

admit expansions in powers of ε. Hence U (t, 0) = S(t)(e−it/ε P1 (ε) + e−itλ+ (ε)/ε P+ (ε) + e−itλ− (ε)/ε P− (ε)), so that, as ε → 0,



√ √

a (e−it εak/ε − eit εak/ε )

, U (t, 0) 

√ 2 εak

(1.25)

(1.26)

which diverges, whatever the nonzero value of ak √ is. We now choose ak < 0 and λ± (ε) = ±i ε|ak| ∈ iR, for definiteness. Since Pk (t)U (t, 0)P j (0) = S(t)Pk (t)P j , j, k ∈ {0, 1}, where S(t) is independent of ε, it is enough to compute Pk (t)P j to get the behaviour in ε of the transitions between the corresponding instantaneous subspaces. We get for t > 0 and P1 = e3 e3 |, ⎛ ⎞ εk − 2 √ √ ⎜ 3/2 2 ⎟ P0 (t)P1 = et |ak|/ ε ⎝ iε√ k ⎠ e3 |(1 + O(ε1/2 )), as ε → 0, (1.27) 2 |ak| 0 (1.28) P1 (t)P0 ≡ 0. The first formula thus implies that the evolution U (t, 0) does not follow the instantaneous eigenprojector P1 (t), whereas the second formula simply reflects the non-generic fact that P0 is invariant under H − εL in our example, see the remarks below. The model being explicitly solvable, we can readily identify the approximated projectors the evolution follows. Setting for j = 0, 1, P j∗ (t, ε) := S(t)P j (ε)S −1 (t) = P j (t) + O(ε),

(1.29)

we compute by means of (1.25), U (t, 0)P j∗ (0, ε) = P j∗ (t, ε)U (t, 0).

(1.30)

Thus the evolution U (t, 0) exactly follows the projectors (1.29) whereas the transition √ from P1 (0) to P0 (t) are exponentially large in 1/ ε. Remarks. i) If the product ak ∈ C\R+ , a similar result holds. We took ak < 0 for simplicity. If the product ak is positive, the transition does vanish in the limit ε → 0. This is due to the fact that the spectral projector P0 corresponding to the unperturbed eigenvalue 0 of H is of dimension 2. For the natural generalization of this example with dim P0 = d, d > 2, the following holds. Generically, the splitting of the unique eigenvalue zero of the nilpotent P0 H by a perturbation of order ε yields d perturbed eigenvalues λ j (ε)  αε1/d e j2iπ/d , j = 0, . . . , d − 1, α ∈ C, see [26]. Hence, one of them has a non vanishing imaginary part that produces exponentially growing contributions as ε → 0. ii) As already mentioned, this example is non-generic in the sense that P0 is invariant under (t), see (1.28). The choice of non-generic L (1.15) was made to keep the formulas simple. However, as should be clear from the analysis, a generic choice for L implies an exponential increase as ε → 0 for both P1 (t)P0 and P0 (t)P1 , when ak ∈ C\R+ .

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iii) The real unperturbed eigenvalues 0 and 1 can be replaced by any different complex numbers λ0 and λ1 without difficulty. The main consequence is that the exponents in (1.25) have to be changed according to λ± (ε) → λ0 + λ± (ε) and 1 → λ1 . One can assume without loss that λ j ≤ 0, j = 0, 1. Observe that if λ0 is real and λ1 < 0, conclusions similar to (1.27) can be drawn. In case λ0 < 0 and λ1 is real, the transition P0 (t)P1 is of order ε, ε → 0. This is a case where the results of [1] apply, since the evolution (1.25) becomes uniformly bounded in ε due to the exponential decay stemming from λ0 < 0. 2. Main Result Let us specify here our hypotheses and state our result. Let a > 0 and Sa = {z ∈ C | dist(z, [0, 1]) < a}. H1. Let {H (z)}z∈ S¯a be a family of closed operators densely defined on a common domain D of a Banach space B and for any ϕ ∈ D, the map z → H (z)ϕ is analytic in Sa . As a consequence, the resolvent R(z, λ) = (H (z) − λ)−1 is locally analytic in z for λ ∈ ρ(H (z)), where ρ(H (z)) denotes the resolvent set of H (z). H2. For t ∈ [0, 1], the spectrum of H (t) is of the form σ (H (t)) = σ (t) ∪ σ0 (t), and there exists G > 0 such that inf dist(σ (t), σ0 (t)) ≥ G.

t∈[0,1]

Moreover, σ (t) = {λ1 (t), λ2 (t), . . . , λn (t)}, where λ j (t), j = 1, . . . , n, n < ∞, are eigenvalues of constant multiplicity m j < ∞ such that inf dist(λ j (t), λk (t)) ≥ G.

t∈[0,1] j =k

Let  j ∈ ρ(H (t)) be a loop encircling λ j (t) only. The finite dimensional spectral projectors corresponding to the eigenvalues λ j (t) are given by 1 P j (t) = − 2πi

 j

R(t, λ)dλ and we set P0 (t) = I −

n


P j (t) ≡ I − P(t). (2.1)

j=1

The loop  j can be chosen locally independent of t. It is a classical perturbative fact, see [26], that H2 also holds for the spectrum of H (z) with z ∈ Sa , provided a is small enough, and that the eigenvalues are analytic functions in Sa . By this we mean that the inf t∈[0,1] can be replaced by inf t∈Sa in H2. Hence, (2.1) also holds for z ∈ Sa and z → Pk (z) is analytic in Sa , for k = 0, . . . , n. Consequently, the eigenilpotents given by D j (z) = (H (z) − λ j (z))P j (z) are analytic in Sa as well. We now state a technical hypothesis needed to deal with evolution operators generated by unbounded generators. In case one works with bounded operators only, this hypothesis is not necessary. H3. Let H0 (t) = P0 (t)H (t)P0 (t). There exists a C 1 complex valued function t → λ0 (t) such that for all t ∈ [0, 1], H0 (t) + λ0 (t) generates a contraction semigroup and 0 ∈ ρ((H0 (t) + λ0 (t)). In other words, H3 says that the solution T (s) = e−iλ0 (t)s e−i H0 (t)s to the strong equation on D i∂s T (s) = (H0 (t) + λ0 (t))T (s) satisfies T (s) ≤ 1, for all s ≥ 0. By

General Adiabatic Evolution with a Gap Condition

147

Hille-Yoshida’s Theorem, H3 is equivalent to the following spectral condition for any t ∈ [0, 1]: [0, ∞) ⊂ ρ(−i H0 (t) − iλ0 (t)) and (i H0 (t) + iλ0 (t) + λ)−1  ≤

1 , ∀λ > 0. λ (2.2)

This hypothesis implies that the equation iε∂t U0 (t, s)ϕ = H0 (t)U0 (t, s)ϕ, U0 (s, s)ϕ = ϕ, s ≤ t ∈ [0, 1], ∀ϕ ∈ D, (2.3) defines a unique strongly continuous two-parameter evolution operator U0 (t, s). It means that U0 (t, s) is uniformly bounded, strongly continuous in the triangle 0 ≤ s ≤ t ≤ 1 and satisfies the relation U0 (t, s)U0 (s, r ) = U0 (t, r ) for any 0 ≤ r ≤ s ≤ t ≤ 1. Moreover, U0 (t, s) maps D into D, also satisfies iε∂s U0 (t, s)ϕ = −U0 (t, s)H0 (s)ϕ, ∀ϕ ∈ D,

(2.4)

and is such that H0 (t)U0 (t, s)(H0 (s) + λ0 )−1 is bounded and continuous in the triangle 0 ≤ s ≤ t ≤ 1. Moreover, see [34], Thm X.70., the following bound holds: U0 (t, s) ≤ e

t

λ0 (u) du/ε

s

, ∀s ≤ t ∈ [0, 1].

(2.5)

Since H (t) = H0 (t) + P(t)H (t)P(t), where P(t)H (t)P(t) is bounded and analytic in t, Hypothesis H3 also implies existence and uniqueness of a bona fide evolution operator U (t, s) associated with the equation iε∂t U (t, s)ϕ = H (t)U (t, s)ϕ, U (s, s)ϕ = ϕ, s ≤ t ∈ [0, 1], ∀ϕ ∈ D, (2.6) see [27], Thm 3.6, 3.7 and 3.11. Theorem 2.1. Assume H1, H2 and H3 and consider U (t, 0) defined by (2.6). For  q  (ε) k = 0, . . . , n, let Pk (t) = Pk (t)+ O(ε) be defined by (5.7), (5.11) and V q (ε) (t, 0) =   W q (ε) (t)q (ε) (t, 0) given by (6.1), (6.10), (6.12) and (5.11). Define ω(t) = maxk=0,...,n λk (t). Then, there exists a constant κ > 0 such that for any 0 ≤ t ≤ 1, e−

t 0

ω(u) du/ε

q ∗ (ε)

U (t, 0)Pk

+O(te−κ/ε sup e−

s 0

(0) = e− ω(u) du/ε

t

V

0

ω(u) du/ε

q ∗ (ε)

0≤s≤t

Vq

∗ (ε)

q ∗ (ε)

(s, 0)Pk

q ∗ (ε)

(t, 0)Pk

(0)

(0)),

with Vq

 (ε)

q  (ε)

(t, 0)Pk

q  (ε)

(0) = Pk

(t)V q

 (ε)

(t, 0), k = 0, . . . , n.

Moreover, for all k ≥ 0 there exists 0 ≤ βk < 1, ck > 0, and dk ≥ 0, with d0 = 0, such that V q

 (ε)

q  (ε)

(t, 0)Pk

βk

(0) ≤ ck edk /ε e

t 0

λk (u)du/ε

with d j = 0, if and only if D j (t) ≡ 0, j ∈ {1, . . . , n}, in (1.2). As a direct

,

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Corollary 2.1. Under the hypotheses of Theorem 2.1, there exists κ > 0, 0 < β < 1, and D ≥ 0 such that U (t, 0) = V q where V q

∗ (ε)

(t, 0) = O(e

t 0

∗ (ε)

(t, 0) + O(te−κ/ε e

t 0

ω(u) du/ε

),

ω(u) du/ε e D/εβ ).

Remarks. 0) The equivalent results hold if the initial time 0 is replaced by any 0 ≤ s ≤ t, mutatis mutandis. See Subsect. 6.1. i) As  is obvious from the formulation, the natural operators to control are t t ∗ − ω(u) du/ε − ω(u) du/ε q 0 0 e U (t, 0) and e V (ε) (t, 0). ii) As in particular cases of Theorem 2.1, we recover the results of [12, 32, 28, 23, 1]. iii) In case κ is sufficiently large, the different components of the leading order term have amplitudes whose instantaneous exponential decay or growth rates in 1/ε may change with time. More precisely, assume that 

t



t

λk (u) du >

0

ω(u) du − κ, ∀t ∈ [0, 1], and ∀k = 0, . . . , n. (2.7)

0

This can be achieved by perturbating weakly a generator for which all λk are real valued, for example. Then, for any initial condition ϕ=

n


q  (ε)

Pk

(0)ϕ ≡

k=0

n


ϕk (ε) ∈ D,

(2.8)

k=0

we get U (t, 0)ϕ =

n


e−i

t 0

λk (u)du/ε

q  (ε)

k

(t, 0)ϕk (ε) + O(te−κ/ε e

t 0

ω(u) du/ε

), (2.9)

k=0

where the error term is exponentially smaller than the leading terms. Each term of the sum  t decays or grows as ε → 0 with an instantaneous  texponential rate given by  0 λk (u)du/ε. Depending on the functions t →  0 λk (u)du/ε, the index of the component which is the most significant may vary with time. iv) In case all λk (t) are real, k = 0, . . . , n, and H (t) is diagonalizable, we can take dk = 0 for all k = 0, . . . , n, and ω(t) ≡ 0. The evolution U and its approxima∗ tion V q (ε) are then uniformly bounded in ε and differ by an error of order e−κ/ε . Theorem 2.1 thus generalizes Thm 2.4 in [23] in the sense that we allow permanently degenerate eigenvalues λ j (t), whereas they were assumed to be simple in [23].

3. Preliminary Estimates We start by recalling a perturbation formula for evolution operators that we will use several times in the sequel.

General Adiabatic Evolution with a Gap Condition

149

Let {A(t)}t∈[0,1] be a densely defined family of linear operators on a common domain D of a Banach space B, and assume t → A(t) is strongly continuous. Let B(t) be linear, bounded and strongly continuous in t ∈ [0, 1]. Assume there exist two-parameter evolution operators T (t, s) and S(t, s) associated with the equations i∂t T (t, s)ϕ = A(t)T (t, s)ϕ, T (s, s) = I, i∂t S(t, s)ϕ = (A(t) + B(t))S(t, s)ϕ, S(s, s) = I,

(3.1) (3.2)

for all ϕ ∈ D and s ≤ t ∈ [0, 1]. Then, for any ϕ ∈ D, and any r ≤ s ≤ t ∈ [0, 1], i∂s (T (t, s)S(s, r )ϕ) = T (t, s)B(s)S(s, r )ϕ, so that by integration on s between r and t,  t S(t, r )ϕ = T (t, r )ϕ − i dsT (t, s)B(s)S(s, r )ϕ.

(3.3)

(3.4)

r

Iterating this formula, we deduce the representation   t  s1
n (−i) ds1 ds2 · · · S(t, r ) = n≥0

r

r

sn−1

dsn

r

× T (t, s1 )B(s1 )T (s1 , s2 )B(s2 ) · · · B(sn )T (sn , r ).

(3.5)

Further assuming that T (t, s) satisfies the bound T (t, s) ≤ Me

t s

ω(u)du

,

(3.6)

for a constant M and a real valued integrable function u → ω(u), we get from (3.5) S(t, s) ≤ Me

t r

(ω(u)+MB(u)) du

.

(3.7)

As a first application of (3.7), we get from (2.5) a first estimate on U (t, s) that we will improve later on U (t, s) ≤ e

t s

(λ0 (u)+P(u)H (u)P(u)) du/ε

.

(3.8)

4. Nilpotent Generators For later purposes, we study here the adiabatic evolution generated by an analytic nilpotent, in a finite dimensional space. We assume N1. For any z ∈ Sa , N (z) is an analytic nilpotent valued operator in a linear space B of finite dimension such that for a fixed integer d ≥ 0, N (z)d ≡ 0. The detailed analysis of the properties of analytic nilpotent matrices is performed in Sect. 5 of the book [8]. It is shown in particular that such operators have the following structure. For any nilpotent N (z) satisfying N1 in Sa , there exists a finite set of points Z 0 ⊂ Sa , with a < a, and, there exists a family of invertible operators {S(z)}z∈Sa \Z 0 such that for any z ∈ Sa \ Z 0 , N (z) = S −1 (z)N S(z)

(4.1)

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with S(z) and S −1 (z) meromorphic in Sa and regular in Sa \ Z 0 . The set Z 0 where N (z) is not similar to the constant nilpotent N is called the set of weakly splitting points of N (z). At these points, the range and kernel of N (z) change. We consider Y (t, s), defined as the solution to ε∂t Y (t, s) = N (t)Y (t, s), Y (s, s) = I, ∀s, t ∈ [0, 1],

(4.2)

and estimate the way Y (t, s) depends on ε, as ε → 0. Note that we don’t need to impose s ≤ t since we deal with bounded generators. In case N is constant, with N d−1 = 0, Y (t, s) = e(t−s)N /ε behaves polynomially in 1/ε, i.e. like ((t − s)/ε)d−1 , as ε → 0. When N (t) is not constant, one may expect that Y (t, s) explodes less fast than ec/ε , which is the worst behavior as ε → 0 for bounded generators. In such cases, however, Y (t, s) grows typically faster than polynomially in 1/ε, as the following example shows. For N (z) given by

t −1 , (4.3) N (t) = 2 −t t we get that the solution Y (t, 0) to (4.2) reads ⎛   cosh √t ε     √ Y (t, 0) = ⎝ t cosh √t ε − ε sinh √t ε

⎞   − √1ε sinh √t ε    ⎠, (4.4) cosh √t ε − √t ε sinh √t ε

√

which behaves as et/ ε , when ε → 0. The growth is nevertheless slower than exponential in 1/ε. We show that the characteristic behaviour of Y generated by an analytic nilpotent operator is similar. For later purposes, we actually consider generators given by an order ε perturbation of a nilpotent. Proposition 4.1. Suppose the nilpotent N (t) satisfies N1 and let {A(t)}t∈[0,1] be a C 0 family of operators on B. Then, there exist c > 0 and 0 < β < 1 such that the solution Y (t, s) of ε∂t Y (t, s) = (N (t) + ε A(t))Y (t, s), Y (s, s) = I, ∀s, t ∈ [0, 1],

(4.5)

satisfies uniformly in t, s ∈ [0, 1], β

Y (t, s) ≤ cec/ε . Remarks. i) Asymptotic expansions as ε → 0 of solutions to such equations are derived in [42, 38], in the neighbourhood of points which are not weakly splitting points for N (z). ii) In case both 0 and 1 are not weakly splitting points and A is analytic, it is possible to take β = (d − 1)/d, which is the optimal exponent, see the example. As we shall not need such improvements, we don’t give a proof. iii) The adiabatic evolution generated by an analytic nilpotent does not have to grow exponentially fast in 1/εβ , as ε → 0. Consider for example (4.3) and (4.4) along the imaginary t-axis. However, such evolutions cannot be uniformly bounded in ε, as the next lemma shows, under slightly stronger conditions. iv) It is actually enough to assume t → A(t) is uniformly bounded on [0, 1].

General Adiabatic Evolution with a Gap Condition

151

Lemma 4.1. Assume {N (t)}t∈[0,1] is a C 1 family of nilpotents and {A(t)}t∈[0,1] is a C 0 family of operators on B. Consider Y (t, s) the solution to (4.5). Then sup Y (t, s) < ∞ ⇐⇒ N (u) ≡ 0 ∀s ≤ u ≤ t. ε>0 s≤t

Proof of Proposition 4.1. The proof consists in two steps. First we prove the result for generators with more structure and then, making use of the results of Sect. 5 in [8] on the detailed structure of analytic nilpotents, we extend it to the general case. Lemma 4.2. Assume N (t) = S −1 (t)N S(t), where N satisfies N d = 0 and where {S(t)}t∈[0,1] is a C 1 family of invertible operators. Let {A(t)}t∈[0,1] be a C 0 family of operators and set B(t) = S(t)A(t)S −1 (t) + S (t)S −1 (t). Then, there exists c > 0 such that the solution Y (t, s) of (4.5) satisfies Y (t, s) ≤ S −1 (t)S(s)

c ε(d−1)/d

e

t s

(1+cB(u))du/ε(d−1)/d

, ∀s ≤ t ∈ [0, 1].

Remarks. 0) The constant c depends on N only.  s i) If s ≥ t, the same estimate holds with t B(u)du in the exponent. ii) This lemma also holds in infinite dimension. Proof of Lemma 4.1. Let Z (t, s) = S(t)Y (t, s)S −1 (s). This operator satisfies by construction ε∂t Z (t, s) = (N + ε B(t))Z (t, s), Z (s, s) = I, ∀s, t ∈ [0, 1].

(4.6)

Let us compare Z (t, s) with Z 0 (t, s) = e N (t−s)/ε , s, t ∈ [0, 1]

(4.7)

by means of (3.5). We get Z (t, r ) =


 n≥0 r



t

ds1 r

s1

 ds2 · · ·

sn−1

dsn r

× Z 0 (t, s1 )B(s1 )Z 0 (s1 , s2 )B(s2 ) · · · B(sn )Z 0 (sn , r ).

(4.8)

Consider now Z δ (s) = e(N −δ)s , for δ > 0.

(4.9)

This operator is such that there exists a c > 0, which depends on N only, such that Z δ (s) ≤ c/δ d−1 ∀s ≥ 0, and 0 < δ ≤ 1.

(4.10)

Indeed, on the one hand, we have for s ≥ s0 , with s0 large enough Z δ (s) ≤ K e−δs s d−1 , where K is some constant which depends on N only. Maximizing over s ≥ 0, we get d−1 e−δs s d−1 ≤ e1−d (d−1) . On the other hand, for all 0 ≤ s ≤ s0 , we have Z δ (s) ≤ δ d−1 es0 N  , so that if 0 < δ ≤ 1, (4.10) holds with c = max(es0 N  , K ((d − 1)/e)d−1 ). Coming back to (4.8) in which we make use of the relation Z 0 (t, s) = Z δ ((t − s)/ε)eδ(t−s)/ε ,

(4.11)

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A. Joye

and (4.10), we get Z (t, r ) ≤ e

δ(t−r )/ε






t

ds1

=

 ds2 · · ·

r

n≥0 r

≤

s1

sn−1

dsn r

×Z δ ((t − s1 )/ε)B(s1 )Z δ ((s1 − s2 )/ε)B(s2 ) · · · B(sn )Z δ ((sn − r )/ε) t ceδ(t−r )/ε
(c B(s) ds/δ d−1 )n r

δ d−1

n!

n≥0

ceδ(t−r )/ε c  t B(s) ds/δ d−1 e r . δ d−1

(4.12)

The left-hand side is independent of δ, which we can choose as δ = ε1/d , so that we eventually get Z (t, r ) ≤

c

ε

e (d−1)/d

t r

(1+cB(s)) ds/ε(d−1)/d

,

(4.13)

from which the result follows.   Let us go on with the proof of the proposition. If Z 0 ∩ [0, 1] = ∅, Lemma (4.2) applies and Proposition 4.1 holds. If not, there exist a finite set of real points {0 ≤ t1 < t2 < · · · < tm ≤ 1} and a finite set of integers { p j } j=1,...,m such that max(S(t), S −1 (t), S (t)S −1 (t)) = O(1/(t − t j ) p j ), as t → t j . (4.14) Since Y is an evolution operator, we can split the integration range in finitely many intervals, so that it is enough to control Y (t, s) for s ≤ t ∈ [v, w] ⊂ R, where [v, w] contains one singular point only. Call this singular point t0 and the corresponding integer p0 . Assume to start with that v < t0 < w. Let δ > 0 be small enough and v ≤ s < t0 < t ≤ w so that we can write Y (t, s) = Y (t, t0 + δ)Y (t0 + δ, t0 − δ)Y (t0 − δ, s).

(4.15)

The first and last terms of the right-hand side can be estimated by Lemma 4.2, whereas we get for the middle term 1

Y (t0 + δ, t0 − δ) ≤ e ε

 t0 +δ t0 −δ

N (u)+ε A(u)du

.

(4.16)

Altogether this yields Y (t, s) ≤ c2 S −1 (t)S(t0 + δ)S −1 (t0 − δ)S(s)/ε2(d−1)/d ×e

(

t s

0 −δ

+

t

t0 +δ )(1+cB(u))du/ε

(d−1)/d

e

+ 1ε

 t0 +δ t0 −δ

N (u)+ε A(u)du

. (4.17)

By (4.14), there exists a constant c (that may change from line to line) which in dependent of ε such that the pre-exponential factors are bounded by c/δ 2 p0 . Also, since N (t) is C 1 and A(t) is C 0 on [0, 1],  t0 +δ  t N (u) + ε A(u) du ≤ cδ and B(u) du ≤ c/δ p0 (4.18) t0 −δ

t0 +δ

General Adiabatic Evolution with a Gap Condition

and similarly for

 t0 −δ s

153

B(u). Hence, Y (t, s) satisfies the bound

Y (t, s) ≤ ce

c(

1 δ p0 ε(d−1)/d

+ εδ )

/(δ 2 p0 ε2(d−1)/d ).

(4.19)

1

Choosing δ = δ(ε) = ε d( p0 +1) in order to balance the contributions in the exponent, we get with a suitable constant c, Y (t, s) ≤ cec/ε Picking

( p0 +1)d−1 ( p0 +1)d

( p0 +1)d−1 ( p0 +1)d

/ε

2(d( p0 +1)−1) ( p0 +1)d

.

(4.20)

< β0 < 1, we get for yet another constant c, β0

Y (t, s) ≤ cec/ε .

(4.21)

A similar analysis yields the same result in case t0 = u or t0 = w. As there are only finitely many weakly splitting points to take care of, taking for β < 1 the largest of the β j , for j = 1, . . . , m, we get the result.   Remarks. i) The proof is valid in arbitrary dimension, assuming only (4.14) at a finite number of points. ii) The exponents pi > 0 in (4.14) need not be integers. Proof of Lemma 4.2. Let Y (t, s) be a solution to (4.5) and assume N (u) ≡ 0 for all t s ≤ u ≤ t. Then Y (t, s) ≤ e s  A(u)| du , which shows one implication. We prove the reverse implication by contradiction. Assume there exists u 0 ∈ [s, t] such that the nilpotent N (u 0 ) = 0 and Y (t, s) ≤ c, uniformly as ε → 0, for all 0 ≤ s ≤ t ≤ 1. We compare Y (t, s) with Z 0 (t, s) = e N (u 0 )(t−s)/ε

(4.22)

and get the following estimate from (3.4) and (3.5): Z 0 (t, s) ≤ cec

t s

N (u)−N (u 0 )+ε A(u) du/ε

.

By Taylor’s formula, there exists a δ > 0 such that t − s ≤ δ implies  t N (u) − N (u 0 ) + ε A(u) du ≤ cδ(δ + ε),

(4.23)

(4.24)

s

for another constant c. Hence, if t − s ≤ δ, with δ small enough, Z 0 (t, s) ≤ cecδ

2 /ε

,

(4.25)

for some c. On the other hand, if t − s = δ and ε << δ, we have for some c, Z 0 (t, s) = c(δ/ε)d−1 .

(4.26)

Thus, by letting δ and ε tend to zero in such a way that δ 2 << ε << δ, we get a contradiction between (4.25) and (4.26), which finishes the proof of the statement.

154

A. Joye

5. Iterative Scheme ∗

We present here the iterative construction which leads to the construction of V q (ε) (t, s) developed in [22], to which we refer the reader for proofs and more details. The first general construction of this kind is to be found in [31]. Assume H1 and H2 with a > 0 small enough so that H2 holds in Sa . By perturbation theory in z ∈ Sa , if z 0 ∈ Sa and  j ∈ ρ(H (z 0 )), j = 1, . . . , n are simple loops encircling the eigenvalues λ j (z 0 ), there exists r > 0 such that for any z ∈ B(z 0 , r ), where B(z 0 , r ) is an open disc of radius r centered at z 0 ,  j ∈ ρ(H (z)). For z ∈ B(z 0 , r ), we set  1 P j (z) = − (H (z) − λ)−1 dλ ≡ P j0 (z), P0 (z) = P00 (z), (5.1) 2πi  j K (z) = i 0

n




Pk0 (z)Pk0 (z).

(5.2)

k=0

The operator K 0 is bounded, analytic and we define the closed operator H 1 (z) = H (z) − εK 0 (z) on D.

(5.3)

For ε small enough, the gap hypothesis H2 holds for all z ∈ B(z 0 , r ), and we set for ε small enough  n
1 P j1 (z) = − (H 1 (z) − λ)−1 dλ, P01 = I − P j1 (z), (5.4) 2πi  j j=1

K 1 (z) = i

n




Pk1 (z)Pk1 (z).

(5.5)

k=0

Note that H 1 , Pk1 , k = 0, · · · , n, and K 1 are ε-dependent and strongly analytic in B(z 0 , r ). We define inductively, for ε small enough, the following hierachy of operators for q ≥ 1: H q (z) = H (z) − εK q−1 (z),  n
1 q q q q −1 (H (z) − λ) dλ, P0 = I − P j (z), P j (z) = − 2πi  j

(5.6) (5.7)

j=1

K (z) = i q

n


q

q

Pk (z)Pk (z).

(5.8)

k=0

It is proven among other things in [22], see also [23], that the following holds: Proposition 5.1. There exists ε0 > 0, b > 0 and g > 0 such that for all q ≤ q ∗ (ε) ≡ [g/ε] and all z ∈ B(z 0 , r ), K q (z) is analytic in Sa , and q ε , (5.9) K q (z) − K q−1 (z) ≤ bq! eg (5.10) K q (z) ≤ b.

General Adiabatic Evolution with a Gap Condition

155

Remarks. i) As a corollary, for q = q ∗ (ε) = [g/ε],

(5.11)

we get the exponential estimate K q

∗ (ε)

(z) − K q

∗ (ε)−1

(z) ≤ eb e−g/ε .

(5.12)

ii) The values of ε0 and g which determine the exponential decay above only depend on sup z∈B(z 0 ,r ) λ∈∪nj=1  j

(H (z) − λ)−1  < ∞,

see [22] for explicit constants. iii) Since Sa is compact, at the expense of decreasing the value of a, we can assume that Proposition 5.1 holds for any z ∈ Sa , with uniform constants g, ε0 and b. Before we go on, let us recall a few facts from perturbation theory applied to our setting, that will be needed in the sequel. Assume q ≤ q ∗ (ε) and let λ ∈ ∪nj=1  j ⊂ ρ(H (z 0 )) and z ∈ B(z 0 , r ). We can write for ε < ε0 , (H q (z) − λ)−1 = (H (z) − εK q−1 (z) − λ)−1 = (H (z) − λ)−1 + ε(H (z) − λ)−1 K q−1 (z)(H q (z) − λ)−1 = (I − ε(H (z) − λ)−1 K q−1 (z))−1 (H (z) − λ)−1 . (5.13) Hence, for any j = 1, . . . , n, q

P j (z) = P j (z) −



ε 2πi

j

(H (z) − λ)−1 K q−1 (z)(H q (z) − λ)−1 dλ

q

= P j (z) − ε R j (z)

(5.14)

is analytic in z and the remainder is of order ε, together with all its derivatives. Moreover, making use of (H (z) − λ)−1 = (H (z) − λ0 )−1 (I − (λ − λ0 )(H (z) − λ)−1 )

(5.15)

for λ0 in ρ(H (z)), we can write q

q

H (z)P j (z) = H (z)P j (z) + εF j (z),

(5.16)

q

where F j (z) given by H (z)(H (z) − λ0 )−1

 j

(I − (λ − λ0 )(H (z) − λ)−1 )K q−1 (z)(H q (z) − λ)−1

dλ . 2πi (5.17)

The identity H (z)(H (z) − λ0 )−1 = I + λ0 (H (z) − λ0 )−1 , q F j (z)

is uniformly bounded as ε → 0 and analytic. shows that As a consequence, we have

(5.18)

156

A. Joye q

Lemma 5.1. Let F j be defined by (5.17). Then H q (z)P j (z) = H (z)P j (z) + ε(F j (z) − K q−1 (z)P j (z)),

q

q

(5.19)

q H q (z)P0 (z)

q K q−1 (z)P0 (z)),

(5.20)

q

where F0 (z) = −

q

n

=

q H0 (z) + ε(F0 (z) −

q

j=1

F j (z).

6. The Approximation Let q ≤ q ∗ (ε) and consider V q , defined as the solution to iε∂t V q (t, s)ϕ = (H q (t) + εK q (t))V q (t, s)ϕ, ϕ ∈ D, V q (s, s) = I, 0 ≤ s ≤ t ≤ 1.

(6.1)

As H q = H − εK q−1 we get that H q (t) + εK q (t) = H0 (t) +

n


P j (t)H (t)P j (t) + ε(K q (t) − K q−1 (t))

(6.2)

j=1

is a bounded, smooth perturbation of H0 (t). The results of [27] guarantee the existence and uniqueness of the solution to (6.1). Moreover, as is well known [26, 27], V q further satisfies q

q

V q (t, s)Pk (s) = Pk (t)V q (t, s), ∀k = 0, . . . , n, 0 ≤ s ≤ t ≤ 1.

(6.3)

In order to show by means of (3.7) that V q , with q = q ∗ (ε), is a good approximation of U , we need to control the behaviour of the norm of V q as ε → 0. We split V q into q components within the spectral subspaces of Pk . Set q

q

Vk (t, s) = V q (t, s)Pk (s) s.t. V q (t, s) =

n


q

Vk (t, s).

(6.4)

k=1 q

Since the projectors {Pk (s)}k=0,...,n have norms uniformly bounded from above and below in s ∈ [0, 1] and ε > 0, there exists a positive constant γ such that γ −1 max Vk (t, s) ≤ V q (t, s) ≤ γ max Vk (t, s). q

q

k=0,...,n

k=0,...,n

(6.5)

We have Proposition 6.1. There exist constants Ck > 0, k = 0, 1, . . . , n, d j ≥ 0 and 0 < β j < 1, j = 1, . . . , n such that for all ε < ε0 , and all q ≤ q ∗ (ε), V0 (t, s) = V q (t, s)P0 (s) ≤ C0 e q

q

t s

λ0 (u)du/ε βj

V j (t, s) = V q (t, s)P j (s) ≤ C j ed j /ε e q

q

t s

,

λ j (u)du/ε

Moreover, (6.7) holds with d j = 0 if and only if D j (t) ≡ 0 in (1.2).

(6.6) .

(6.7)

General Adiabatic Evolution with a Gap Condition

157 q

Proof of Proposition 6.1. We first consider V0 (t, s), the part of V q corresponding to q the infinite dimensional subspace P0 . Because of (6.3), it satisfies for 0 ≤ s ≤ t ≤ 1 and any ϕ ∈ D, q

q

q

q

q

iε∂t V0 (t, s)ϕ = ((H q (t) + εK q (t))P0 (t))V0 (t, s)ϕ, V0 (s, s) = P0 (s). (6.8) q

Lemma 5.1 shows that the generator of V0 (t, s) is equal to H0 (t) plus a smooth bounded q q perturbation of order ε. We can thus compare V0 (t, s) and U (t, s)P0 (s) by means of q (3.7). The fact that the initial condition is P0 (s) instead of the identity simply multiplies q the estimate by P0 (s), so that we get V q (t, s)P0 (s) ≤ P0 (s)e q

q

t s

λ0 (u)du/ε

C0 ≤ e

t s

λ0 (u)du/ε

C0 ,

(6.9)

where C0 and C0 = C0 sup s∈[0,1] P0 (s) are uniform in ε. ε>0 The control of the remaining components is conveniently done by taking advantage of the intertwining relation (6.3) as follows. Let W q be the bounded operator satisfying the equation q

i W q (t) = K q (t)W q (t), W q (0) = I.

(6.10)

This operator enjoys a certain number of properties. As K q is smooth and bounded, the q solution is given by a convergent Dyson series, and W q (t) interwines between Pk (0) q and Pk (t). Moreover, W q and its inverse map D into D, see [21]. Finally, by regular perturbation theory and Proposition 5.1, K q = K 0 + O(ε) so that sup W q (t)±1  < ∞.

(6.11)

t∈[0,1] 0<ε<1

Therefore, the bounded operator defined by q (t, s) = W q (t)−1 V q (t, s)W q (s), 0 ≤ s ≤ t ≤ 1

(6.12)

satisfies by construction q

[q (t, s), Pk (0)] ≡ 0, ∀ k = 0, . . . , n ∀ 0 ≤ s ≤ t ≤ 1.

(6.13)

We can thus view q

q

 j (t, s) = q (t, s)P j (0), q 0 (t, s)

=

q

j = 1, . . . , n,

q (t, s)P0 (0)

(6.14) (6.15)

as operators in the finite dimensional Banach spaces P j (0)B, for j ≥ 1 and in the infinite dimensional Banach space P0 (0)B. Moreover, thanks to (6.11), there exists a constant C such that, uniformly in 0 ≤ s ≤ t ≤ 1 and ε > 0, C −1 Vk (t, s) ≤ k (t, s) ≤ CVk (t, s), k = 0, . . . , n. q

q

q

(6.16)

158

A. Joye q

The operator  j (t, s) satisfies for any ϕ ∈ D, iε∂t  j (t, s)ϕ = W q (t)−1 H q (t)V q (t, s)W q (s)P j (0) j (t, s) q

q

q

= P j (0)W q (t)−1 H q (t)P j (t)W q (t))P j (0) j (t, s)ϕ q

q

q

q

q (t)q (t, s)ϕ, ≡H j j

(6.17)

q (t) is bounded, see Lemma 5.1. In a sense, q (t, s) describes where the generator H j j q q the evolution within the spectral subspaces. Let us further compute with P j = (P j )2 and (5.14), q

q

q

q

H q (t)P j (t) = P j (t)(H (t)P j (t) + ε(F j (t) − K q−1 (t))P j (t) q

q

q

= P j (t)(λ j (t)P j (t) + D j (t) + ε(F j (t) − K q−1 (t))P j (t) q

q

q

= λ j (t)P j (t) + P j (t)D j (t)P j (t) q

q

q

q

+ε P j (t)(λ j (t)R j (t) + F j (t) − K q−1 (t))P j (t) q

q

q

≡ P j (t)(λ j (t) + D j (t))P j (t) + ε J j (t).

(6.18)

The last term is bounded, analytic in t and of order ε. We will deal with it perturbatively. q Equations (6.18) suggest to decompose  j (t, s), j = 1, . . . , n, as  j (t, s) = e−i q

q

q

t s

λ j (u) du/ε

q

 j (t, s),

(6.19)

q

where  j (t, s) : P j (0)B → P j (0)B satisfies iε∂t  j (t, s) = P j (0)W q (t)−1 (D j (t) + ε J j (t))W q (t)P j (0)  j (t, s), q

q

q

q

q

q

q

 j (s, s) = P j (0),

(6.20)

where, in the leading part of the generator,  j (t) = W q (t)−1 D j (t)W q (t) D

(6.21)

 j (t)m j = 0, with m j = dim P j (t). However, the restricis analytic and nilpotent with D q q   j (t)P q (0), is not nilpotent. Nevertheless,  q (t, s) tion of D j (t) to P j (0)B, P j (0) D j j satisfies the same type of estimates an evolution generated by a perturbed analytic nilpotent does: q

Lemma 6.1. Let  j (t, s) be defined by (6.19), for j = 1, . . . , n. Then, there exist 0 < β j < 1 and d j ≥ 0, c j > 0 such that βj

 j (t, s) ≤ c j ed j /ε . q

Moreover, the estimate holds with d j = 0 if and only if D j (t) ≡ 0 in (1.2).

(6.22)

General Adiabatic Evolution with a Gap Condition

159 q

Proof of Lemma 6.1. Equations (5.14) and (1.3) allow to get rid of the projectors P j (0) in (6.20) up to an error of order ε, q  j (t)P q (0) = W q (t)−1 P q (t)D j (t)P q (t)W q (t) = D  j (t) + εL q (t), P j (0) D j j j j

where

 q q q L j (t) = −W q (t)−1 R j (t)D j (t)P j (t) + P j (t)D j (t)R j (t)  q q − ε R j (t)D j (t)R j (t) W q (t)

(6.23)

(6.24)

is analytic and of order ε0 . Since W q (t)±1 is analytic and uniformly bounded, the nilpo j (t) satisfies N1 uniformly in ε > 0, and (6.20) and (6.24) show that the generator tent D q of  j (t, s) satisfies the hypotheses of Proposition 4.1, which yields the estimate. The last statement stems from Lemma 4.1.    

It remains to gather (6.16), (6.19) and Lemma 6.1 to end the proof of Proposition 6.1.

6.1. End of the proof. Given Proposition 6.1, we can finish the proof of our main statement as follows. Applying (3.4) to U and V q , we get  t q U (t, r ) = V (t, r ) + i V q (t, s)(K q (s) − K q−1 (s))U (s, r ) ds. (6.25) r

Let t → ω(t) be the continuous function defined by ω(t) = max λk (t).

(6.26)

k=0,...,n

Applying (6.25) to Pk (r ) and multiplication by e− q

e−

t r

ω(u) du/ε



≤

t

e−

t s

t r

ω(s) ds/ε

gives with (6.4),

q

(U (t, r ) − V q (t, r ))Pk (r ) ω(u) du/ε

V q (t, s)(K q (s) − K q−1 (s))

r

 s q × e− r ω(u) du/ε (U (s, r ) − V q (s, r ))Pk (r )  s q +e− r ω(u) du/ε Vk (s, r ) ds.

(6.27)

Proposition 6.1 and the definition of ω(t) yield for any 0 ≤ r ≤ s ≤ 1, e−

s r

ω(u) du/ε

βk

Vk (s, r ) ≤ Ck edk /ε , q

(with d0 = 0).

(6.28)

Further taking q = q ∗ (ε), (5.12), (6.5) show the existence of constants B > 0 and 0 < κ < g such that e−

t s

ω(u) du/ε

Vq

∗ (ε)

(t, s)(K q

∗ (ε)

(s) − K q

∗ (ε)−1

β

(s)) ≤ ebCe D/ε e−g/ε ≤ Be−κ/ε . (6.29)

160

A. Joye

Hence, we get using 0 ≤ t − s ≤ 1, e−

t r

ω(u) du/ε

q ∗ (ε)

∗

(U (t, r ) − V q (ε) (t, r ))Pk (r )  t s q ∗ (ε) ≤ Be−κ/ε e− r ω(u) du/ε Vk (s, r ) ds r

+Be−κ/ε sup e−

s r

ω(u) du/ε

r ≤s≤t

(U (s, r ) − V q

∗ (ε)

q ∗ (ε)

(s, r ))Pk

(r ),

(6.30) from which we deduce that if ε is so small that Be−κ/ε < 1/2, sup e−

s

ω(u) du/ε

r

r ≤s≤t

(U (s, r ) − V q

≤ 2Be−κ/ε (t − r ) sup e−

s r

∗ (ε)

q ∗ (ε)

(s, r ))Pk

ω(u) du/ε

r ≤s≤t

q ∗ (ε)

Vk

(r )

(s, r ).

(6.31)

In particular, our main result follows. For ε small enough, for any 0 ≤ r ≤ t ≤ 1, and for all k = 0, . . . , n, e−

t r

ω(u) du/ε

q ∗ (ε)

U (t, s)Pk

(r ) = e−

t r

ω(u) du/ε

× sup e−

q ∗ (ε)

Vk

s r

(t, r ) + O((t − r )e−κ/ε

ω(u) du/ε

r ≤s≤t

q ∗ (ε)

Vk

(s, r )).

(6.32)

∗

We chose to estimate the difference U − V q (ε) applied on the projectors, because the q ∗ (ε) norms of the different components Vk vary with k. Of course, (6.32) also holds with q ∗ (ε)

q ∗ (ε)

∗

removed and V q (ε) in place of Vk . Making further use of (6.28) in the error term of (6.32), we get (lowering the value of 0 < κ < g) Pk

U (t, r ) = V q where V q

∗ (ε)

(t, r ) = O(e

t r

∗ (ε)

(t, r ) + O((t − r )e−κ/ε e

ω(u) du/ε e D/εβ ),

t r

ω(u) du/ε

),

(6.33)

for some 0 < β < 1, and D ≥ 0.  

Acknowledgement. The author would like to thank C. Ogabi, R. Rebolledo and D. Spehner for useful discussions.

References 1. Abou-Salem, W.: On the quasi-static evolution of nonequilibrium steady states. To appear in Ann. Henri Poincaré, DOI: 10.1007/s00023-006-0316-z, 2007 2. Abou-Salem, W.: “Nonequilibrium quantum statistical mechanics and thermodynamics”. PhD Thesis, ETH 16187, (2005) 3. Abou-Salem, W., Fröhlich, J.: Adiabatic Theorems and Reversible Isothermal Processes. Lett. Math. Phys. 72, 153–163 (2005) 4. Abou-Salem, W., Fröhlich, J.: Adiabatic Theorems for Quantum Resonances. To appear in Commun. Math. Phys., DOI: 10.1007/s00220-007-0198-z, 2007 5. Avron, J.E., Elgart, A.: Adiabatic Theorem without a Gap Condition. Commun. Math. Phys. 203, 445– 463 (1999) 6. Avron, J.E., Howland, J.S., Simon, B.: Adiabatic theorems for dense point spectrum. Commun. Math. Phys. 128, 497–507 (1990)

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7. Avron, J.E., Seiler, R., Yaffe, L.G.: Adiabatic Theorems and Applications to the Quantum Hall Effect. Commun. Math. Phys. 110, 33–49 (1987) 8. Baumgärtel, H.: “Analytic Perturbation Theory for Matrices and Operators”. Operator Theory: Advances and Applications. Vol. 15, Basel-Boston: Birkhäuser, 1985 9. Berry, M.V.: Histories of adiabatic quantum transitions. Proc. Roy. Soc. Lond. A 429, 61–72 (1990) 10. Born, M., Fock, V.: Beweis des Adiabatensatzes. Zeit. f. Physik 51, 165–180 (1928) 11. Bornemann, F.: “Homogeneization in Time of Singularly Perturbed Mechanical Systems”. Lecture Notes in Mathematics 1687, Heidelberg: Springer, 1998 12. Boutet de Monvel, A., Nenciu, G.: “On the Theory of Adiabatic Invariants for Linear Hamiltonian Systems”. Comptes Rendus Acad. Sci. Paris A 310, 807–810 (1990) 13. Dimassi, M., Sjöstrand, J.: “Spectral asymptotics in the semi-classical limit”. London Math. Soc. Lecture Note Ser. 268, Cambridge: Cambridge University Press, 1999 14. Fedoryuk, M.: “Méthodes Asymptotiques pour les Equations différentielles ordinaires linéaires”. Moscou: Mir, 1987 15. Guérin, S., Joye, A., Jauslin, H., Monti, F.: Adiabatic Evolution for Systems with Infinitely many Eigenvalue Crossings. J. Math. Phys. 40, 5456–5472 (1999) 16. Hagedorn, G., Joye, A.: Elementary Exponential Error Estimates for the Adiabatic Approximation. J. Math. Anal. Appl. 267, 235–246 (2002) 17. Hagedorn, G., Joye, A.: Recent Results on Non-Adiabatic Transitions in Quantum Mechanics. In: Recent Advances in Differential Equations and Mathematical Physics. AMS Contemporary Mathematics Series 412, Ed. by N. Chernov, Y. Karpeshina, I. Knowles, R. Lewis, R. Weikard. Providence, RI: Amer. Math. Soc., 2006, p. 183–198 18. Joye, A.: An Adiabatic Theorem for Singularly Perturbed Hamiltonians. Ann. Inst. H. Poincaré Phys. Théor. 63, 231–250 (1995) 19. Joye, A.: Exponential Asymptotics in a Singular Limit for n-Level Scattering Systems. SIAM J. Math. Anal. 28, 669–703 (1997) 20. Joye, A., Kunz, H., Pfister, C.-E.: Exponential Decay and Geometric Aspect of Transition Probabilities in the Adiabatic Limit. Ann. Phys. 208, 299–332 (1991) 21. Joye, A., Pfister, C.-E.: Exponentially small adiabatic invariant for the Schrödinger equation. Commun. Math. Phys. 140, 15–41 (1991) 22. Joye, A., Pfister, C.-E.: Superadiabatic evolution and adiabatic transition probability between two non-degenerate levels isolated in the spectrum. J. Math. Phys. 34, 454–479 (1993) 23. Joye, A., Pfister, C.-E.: Semiclassical Asymptotics Beyond all Orders for Simple Scattering Systems. SIAM J. Math. Anal. 26, 944–977 (1995) 24. Joye, A., Pfister, C.-E.: Exponential Estimates in Adiabatic Quantum Evolution. In: XIIth International Congress of Mathematical Physics, ICMP ’97, De Wit, Bracken, Gould, Pearce, eds, Cambridge, MA: International Press, 1999, p. 309–315 25. Kato, T.: On the Adiabatic Theorem of Quantum Mechanics. J. Phys. Soc. Japan 5, 435–439 (1950) 26. Kato, T.: Perturbation Theory for Linear Operators. Berlin-Heidelberg-New York: Springer-Verlag, 1980 27. Krein, S.G.: Linear Differential Equations in Banach Space. Providence, RI: Amer. Math. Soc. 1971 28. Martin, Ph.A., Nenciu, G.: Semi-classical Inelastic S-Matrix for One-dimensional N-States Systems. Rev. Math. Phys. 7, 193–242 (1995) 29. Martinez, A.: An Introduction to Semiclassical and Microlocal Analysis. New York: Springer-Verlag UTX Series, 2002 30. Nenciu, G.: On the adiabatic theorem of quantum mechanics. J. Phys. A 13, L15–L18 (1980) 31. Nenciu, G.: Linear adiabatic theory. Exponential Estimates. Commun. Math. Phys. 152, 479–496 (1993) 32. Nenciu, G., Rasche, G.: On the adiabatic theorem for nonself-adjoint Hamiltonians. J. Phys. A 25, 5741– 5751 (1992) 33. Kvitsinsky, A., Putterman, S.: Adiabatic evolution of an irreversible two level system. J. Math. Phys. 32, 1403–1407 (1991) 34. Reed, M., Simon, B.: “Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, Self-Adjointness”. London-New York: Academic Press, 1975 35. Sarandy, M.S., Lidar, D.A.: Adiabatic approximation in open quantum systems. Phys. Rev. A 71, 012331 (2005) 36. Sarandy, M.S., Lidar, D.A.: Adiabatic Quantum Computation in Open Quantum Systems. Phys. Rev. Lett. 95, 250503 (2005) 37. Sarandy, M.S., Lidar, D.A.: Abelian and non-Abelian geometric phases in adiabatic open quantum systems. Phys. Rev. A 73, 062101 (2006) 38. Stengle, G.: A new basis for asymptotic solution of differential equations containing one or several parameters. Trans. AMS 156, 1–43 (1971) 39. Teufel, S.: A note on the adiabatic theorem without gap condition. Lett. Math. Phys. 58, 261–266 (2001)

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40. Teufel, S.: “Adiabatic perturbation theory in quantum dynamics”. Lecture Notes in Mathematics 1821, Berlin-Heidelberg-New York: Springer-Verlag, 2003 41. Thunstrom, P., Aberg, J., Sjoqvist, E.: Adiabatic Approximation for weakly open systems. Phys. Rev. A 72, 022328 (2005) 42. Turrittin, H.L.: Asymptotic expansions of solutions of systems of ordinary differential equations containing a parameter. In: “Contributions to the theory of nonlinear oscillations. Volume II”, Lefschetz, S. ed., Annals of mathematics studies (029), Princeton, NJ: University Press, 1965 Communicated by H. Spohn

Commun. Math. Phys. 275, 163–186 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0301-8

Communications in

Mathematical Physics

Homogenized Dynamics of Stochastic Partial Differential Equations with Dynamical Boundary Conditions Wei Wang1 , Jinqiao Duan2 1 Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100080, China.

E-mail: [email protected]

2 Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA.

E-mail: [email protected] Received: 13 September 2006 / Accepted: 2 March 2007 Published online: 21 July 2007 – © Springer-Verlag 2007

Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday Abstract: A microscopic heterogeneous system under random influence is considered. The randomness enters the system at physical boundary of small scale obstacles as well as at the interior of the physical medium. This system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes (obstacles or heterogeneities), together with random dynamical boundary conditions on the boundaries of these small holes. A homogenized macroscopic model for this microscopic heterogeneous stochastic system is derived. This homogenized effective model is a new stochastic partial differential equation defined on a unified domain without small holes, with a static boundary condition only. In fact, the random dynamical boundary conditions are homogenized out, but the impact of random forces on the small holes’ boundaries is quantified as an extra stochastic term in the homogenized stochastic partial differential equation. Moreover, the validity of the homogenized model is justified by showing that the solutions of the microscopic model converge to those of the effective macroscopic model in probability distribution, as the size of small holes diminishes to zero. 1. Introduction Stochastic effects in the multiscale modeling of complex phenomena have drawn more and more attention in many areas such as material science [10], climate dynamics [27], chemistry and biology [21, 51]. Stochastic partial differential equations (SPDEs or stochastic PDEs) arise naturally as mathematical models for multiscale systems under random influences. The need to include stochastic effects in mathematical modeling of some realistic complex behaviors has become widely recognized in science and engineering. But implementing this approach poses some challenges both in mathematical theory and computation [44, 16, 26, 27, 51, 43]. The addition of stochastic terms to mathematical models has led to interesting new mathematical problems at the interface of dynamical systems, partial differential equations, scientific computing, and probability theory.

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Sometimes, noise affects a complex system not only inside the physical medium but also at the physical boundary. Such random boundary conditions arise in the modeling of, for example, the air-sea interactions on the ocean surface [42], heat transfer in a solid in contact with a fluid [31], chemical reactor theory [32], and colloid and interface chemistry [56]. Random boundary conditions may be static or dynamical. The static boundary conditions, such as Dirichlet or Neumann boundary conditions, are not involved with time derivatives of the system state variables. On the contrary, the dynamical boundary conditions contain such time derivatives. Randomness in such boundary conditions are often due to various fluctuations. In this paper we consider a microscopic heterogeneous system, modeled by a SPDE with random dynamical boundary condition, in a medium which exhibits small-scale spatial heterogeneities or obstacles. One example of such microscopic systems of interest is composite materials containing microscopic holes (i.e., cavities), under the impact of random fluctuations in the domain and on the surface of the holes [28, 35]. A motivation for such a model is based on the consideration that the interaction between the atoms of the different compositions in a composite material causes the thermal noise when the scale of the heterogeneity scale is small. A similar consideration appears also in a microscopic stochastic lattice model [6] for a composite material. Here the microscopic structure is perturbed by random effect and the complicated interactions on the boundary of the holes is dynamically and randomly evolving. The heterogeneity scale is assumed to be much smaller than the macroscopic scale, i.e., we assume that the heterogeneities are evenly distributed. From a mathematical point of view, one can assume that microscopic heterogeneities (holes) are periodically placed in the media. This spatial periodicity with small period can be represented by a small positive parameter
(i.e., the period). In fact we work on the spatial domain D
, obtained by removing S
, a collection of small holes of size
, periodically distributed in a fixed domain D. When taking
→ 0, the holes inside domain D are smaller and smaller and their numbers go to ∞. This signifies that the heterogeneities are finer and finer. In other words, we consider a spatially extended system with state variable u
, where stochastic effects are taken into account both in the model equation and in the boundary conditions, defined on a domain perforated with small scale holes. Specifically, we study a class of stochastic partial differential equations driven by white noise on a perforated domain with random dynamical boundary conditions:
 du
(t, x) = u
(t, x) + f (t, x, u
, ∇u
) dt + g1 (t, x)dW1 (t, x) in D
× (0, T ),
∂u (t, x) 

2 du
(t, x) = − −
bu
(t, x) dt +
g2 (t, x)dW2 (t, x) ∂ν
on ∂ S
× (0, T ). This model will be described in more detail in the next section. The goal is to derive a homogenized effective equation, which is a new stochastic partial differential equation (see Theorems 5.1, 6.1, 6.2 and 6.3), for the above microscopic heterogenous system, by homogenization techniques in the sense of probability. Homogenization theory has been developed for deterministic systems, and compactness discussion for the solutions {u
}
in some function space is a key step in various homogenization approaches [12]. However, due to the appearance of the stochastic terms in the above microscopic system considered in this paper, such a compactness result does not hold for this stochastic system. Fortunately the compactness in the sense of probability,

Homogenized Dynamics of SPDES

165

that is, the tightness of the distributions for {u
}, still holds. So one appropriate way is to homogenize the stochastic system in the sense of probability. It is shown that the solution u
of the microscopic or heterogeneous system converges to that of the macroscopic or homogenized system as
↓ 0 in probability distribution. This means that the distribution of {u
}
weakly converges, in some appropriate space, to the distribution of a stochastic process which solves the macroscopic effective equation. It is interesting to note that, for the above system with random dynamical boundary conditions, the random force on the boundary of microscopic scale holes leads, in the homogenization limit, to a random force distributed all over the physical domain D, even when the model equation itself contains no stochastic influence in the domain; see Remark 5.2 in §5. We could also say that the impact of small scale random dynamical boundary conditions is quantified or carried over to the homogenized model as an extra random forcing. Therefore, the homogenized effective model is a new stochastic partial differential equation, defined on a unified domain without holes. In the present paper, the two-scale convergence techniques are employed in our approach. The two-scale convergence method is an important method in homogenization theory which is a formal mathematic procedure for deriving macroscopic models from microscopic systems. The two-scale convergence method contains more information than the usual weak convergence method; see [2] or §4. Moreover by use of the twoscale convergence, we do not need the extension operator as introduced in [13]. Partial differential equations (PDEs) with dynamical boundary conditions have been studied recently in, for example, [4, 20, 22, 23, 25, 47] and references therein. The parabolic SPDEs with noise in the static Neumann boundary conditions have also been considered in [16, 17, 36]. In [11], the authors have studied well-posedness of the SPDEs with random dynamical boundary conditions. One of the present authors, with collaborators, has considered [18, 57] dynamical issues of SPDEs with random dynamical boundary conditions. The homogenization problem for the deterministic systems defined in perforated domains or in other heterogeneous media has been investigated in, for example, [8, 39, 40, 46, 48] for heat transfer in a composite material, [8, 13, 15] for the wave propagation in a composite material and [34, 38] for the fluid flow in a porous media. For a systematic introduction in homogenization in the deterministic context, see [12, 28, 35, 45]. In [47], the effective macroscopic dynamics of a deterministic partial differential equation with deterministic dynamical boundary condition on the microscopic heterogeneity boundary is studied. Recently there are also works on homogenization of partial differential equations (PDEs) in the random context; see [28, 29, 37, 41] for PDEs with random coefficients, and [7, 28, 58, 59] for PDEs in randomly perforated domains. A basic assumption in these works is the ergodic hypotheses on the random coefficients, for the passing of the limit as
→ 0. Note that the microscopic models in these works are partial differential equations with random coefficients, so-called random partial differential equations (random PDEs) [9, 29, 30, 34, 41, 53], instead of stochastic PDEs — PDEs with noises — in the present paper; see also [52]. Another novelty in the present paper is that the microscopic system is under the influence of random dynamical boundary conditions. We first consider the linear system and then present results about nonlinear systems with special nonlinear terms. This paper is organized as follows. The problem formulation is stated in §2. Section 3 is devoted to basic properties of the microscopic heterogeneous system, and some knowledge to be used in our approach is introduced

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in §4. The homogenized effective macroscopic model for the linear system is derived in §5. In the last section, homogenized effective macroscopic models are obtained for three types of nonlinear systems.

2. Problem formulation Let the physical medium D be an open bounded domain in Rn , n ≥ 2, with smooth boundary ∂ D, and let
> 0 be a small parameter. Let Y = [0, l1 ) × [0, l2 ) × · · · × [0, ln ) be a representative elementary cell in Rn and S an open subset of Y with smooth boundary ∂ S, such that S ⊂ Y . The elementary cell Y and the small cavity or hole S inside it are used to model small scale obstacles or heterogeneities in a physical medium D. Write l = (l1 , l2 , . . . , ln ). Define
S = {
y : y ∈ S}. Denote by S
,k the translated image of
S by kl, k ∈ Z n , kl = (k1l1 , k2 l2 , . . . , kn ln ). And let S
be the set all the holes contained in D and D
= D\S
. Then D
is a periodically perforated domain with holes of the same size as period
. We remark that the holes are assumed to have no intersection with the boundary ∂ D, which implies that ∂ D
= ∂ D ∪ ∂ S
. See Fig. 1 for the case n = 2. This assumption is only needed to avoid technicalities and the results of our paper will remain valid without this assumption [3]. In the sequel we use the notations Y ∗ = Y \S, ϑ =

|Y ∗ | |Y |

with |Y | and |Y ∗ | the Lebesgue measure of Y and Y ∗ respectively. Denote by χ the indicator function, which takes value 1 on Y ∗ and value 0 on Y \ Y ∗ . In particular, let χ A be the indicator function of A ⊂ Rn . Also denote by v˜ the zero extension to the whole D for any function v defined on D
:  v˜ =

v on D
, 0 on S
.

Now for T > 0 fixed final time, we consider the following Itô type nonautonomous stochastic partial differential equation defined on the perforated domain D
in Rn ,
 du
(t, x) = u
(t, x) + f (t, x, u
, ∇u
) dt + g1 (t, x)dW1 (t, x) in D
× (0, T ),

Fig. 1. Geometric setup in R2

(2.1)

Homogenized Dynamics of SPDES

167


∂u (t, x) 

2 du
(t, x) = − −
bu
(t, x) dt +
g2 (t, x)dW2 (t, x) ∂ν
on ∂ S
× (0, T ), u
(t, x) = 0 on ∂ D × (0, T ), u
(0, x) = u 0 (x) in D
,

(2.2) (2.3) (2.4)

where b is a real constant, f : [0, T ] × D × R × Rn → R satisfies some property which will be described later and ν
is the exterior unit normal vector on the boundary ∂ S
, v0 ∈ L 2 (∂ S
) and u 0 ∈ L 2 (D). Moreover, W1 (t, x) and W2 (t, x) are mutually independent L 2 (D) valued Wiener processes on a complete probability space (, F, P) with a canonical filtration (Ft )t≥0 . Denote by Q 1 and Q 2 the covariance operators of W1 and W2 respectively. Here we assume that gi (t, x) ∈ L(L 2 (D)), i = 1, 2 and that there is a positive constant C T independent of
such that

gi (t, ·) 2 Qi := L2

∞ 

1

gi Q i2 e j 2L 2 (D) ≤ C T , i = 1, 2, t ∈ [0, T ],

(2.5)

j=1

where {e j }∞ j=1 are eigenvectors of operator − on D with Dirichlet boundary condition and they form an orthonormal basis of L 2 (D). Here L(L 2 (D)) denotes the space of bounded linear operators on L 2 (D) and L2Q i = L2Q i (H ) denotes the space of Hilbert-Schmidt operators related to the trace operator Q i [16]. We also denote by E the expectation operator with respect to P. Let S be a Banach space and S be the strong dual space of S. We recall the definitions and some properties of weak convergence and weak∗ convergence [54]. Definition 2.1. A sequence {sn } in S is said to converge weakly to s ∈ S if ∀s ∈ S , lim (s , sn )S ,S = (s , s)S ,S ,

n→∞

which is written as sn  s weakly in S. Note that (s , s) denotes the value of the continuous linear functional s at the point s. Lemma 2.2 (Eberlein-Shmulyan). Assume that S is reflexive and let {sn } be a bounded sequence in S. Then there exists a subsequence {sn k } and s ∈ S such that sn k  s weakly in S as k → ∞. If all the weak convergent subsequence of {sn } has the same limit s, then the whole sequence {sn } weakly converges to s. Definition 2.3. A sequence {sn } in S is said to converge weakly ∗ to s ∈ S if ∀s ∈ S, lim (sn , s)S ,S = (s , s)S ,S

n→∞

which is written as sn  s weakly ∗ in S . Lemma 2.4. Assume that the dual space S is reflexive and let {sn } be a bounded sequence in S . Then there exists a subsequence {sn k } and s ∈ S such that sn k  s weakly ∗ in S as k → ∞. If all the weakly ∗ convergent subsequence of {sn } has the same limit s , then the whole sequence {sn } weakly ∗ converges to s . In the following, for a fixed T > 0, we always denote by C T a constant independent of
. And denote by DT the set [0, T ] × D.

168

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3. Basic Properties of the Microscopic Model In this section we will present some estimates for solutions of the microscopic model (2.1), and then discuss the tightness of the distributions of the solution processes in some appropriate space. We focus our argument in the case of linear microscopic systems, where the term f is independent of u
and ∇u
and f (·, ·) ∈ L 2 (0, T ; L 2 (D)). Then we briefly extend this to the case of nonlinear microscopic systems with Lipschitz nonlinearities. Define by H
1 (D
) the space of elements of H 1 (D
) which vanish on ∂ D. Denote by −1 H
(D
) the dual space of H
1 (D
) with the usual norm and let γ
: H 1 (D
) → L 2 (∂ S
) be the trace operator with respect to ∂ S
which is continuous [49]. We also denote that −1

1

1

H 2 (∂ S
) = γ
(H 1 (D
)) and let H
2 (D
) be the dual space of H
2 (D
). Introduce the following function spaces: 1   X
1 = (u, v) ∈ H
1 (D
) × H
2 (∂ S
) : v =
γ
u

and   X
0 = L 2 (D
) × L 2
(∂ S
) with the usual product and norm. Define an operator B
on the space H
1 (D
) as B
u =

∂u +
bu, u ∈ H
1 (D
). ∂n

(3.1)

Now we define the operator A
on D(A
) = {(u, v) ∈ X
1 : (−u, R
B
u) ∈ X
0 }, where R
is the restriction to ∂ S
, as 1 A
z = (−u, R
B
u), z = (u, v) ∈ D(A
).
Associated with the operator A
, we introduce the bilinear form on X
1 ,   a
(z, z¯ ) = ∇u∇ ud ¯ x +
b γ
(u)γ
(u)ds ¯ D


(3.2)

(3.3)

S


with z = (u, v), z¯ = (u, ¯ v) ¯ ∈ X
1 . Because |γ
(u)|2L 2 (∂ S ) ≤ C(S
)|u|2H 1 (D ) , we see


that there is M > 0, independent of
, such that a
(z, z¯ ) ≤ M|u| H
1 (D
) |u| ¯ H
1 (D
) and the following coercive property of a holds: ¯
|2 0 , z ∈ X
1 a
(z, z) ≥ α|z| ¯ 2X 1 − β|z X





(3.4)

for some constants α, ¯ β¯ > 0 which are also independent of
. Write the C0 -semigroup generated by operator −A
as S
(t). Then the system (2.1)–(2.4) can be rewritten as the following abstract stochastic evolutionary equation dz
(t, x) = [−A
z
(t, x) + F
(t, x)]dt + G
(t, x)dW (t, x), z
(0) = z 0 ,

(3.5)

Homogenized Dynamics of SPDES

169

where F
(t, x) = ( f (t, x), 0)t , G
(t, x)dW (t) = (g1 (t, x)dW1 (t, x), g2 (t, x)dW2 (t, x))t and z 0 = (u 0 , v0 ). And the solution of (3.5) can be written in the mild sense  z
(t) = S
(t)z 0 +



t

S
(t − s)F
(s)ds +

0

t 0

S
(t − s)G
(s)dW (s).

(3.6)

Moreover, the variational formulation is  T D


0

=−

u˙
ϕd xdt +
2  T 0

D


0

∂ S


 T +


 T 0

∂ S


∇u
∇ϕd xdt +

u˙
ϕd xdt +
b  T 0

 T 0

f ϕd xdt + D


u
ϕd xdt

∂ S
 T 0

g1 ϕ W˙ 1 d xdt D


g2 ϕ W˙ 2 d xdt

(3.7)

d for ϕ(t, x) ∈ C0∞ ([0, T ] × D
). Here˙denotes dt . For the well-posedness of system (3.5) we have the following result.

Theorem 3.1 (Global well-posedness of microscopic model). Assume that (2.5) holds  for T > 0. If z 0 = (u 0 , v0 ) is a F0 , B(X
0 ) -measurable random variable, then the  system (3.5) has a unique mild solution z
∈ L 2 , C(0, T ; X
0 )∩ L 2 (0, T ; X
1 ) , which is also a weak solution in the following sense:  (z
(t), φ) X
1 = (z 0 , φ) X
1 +

t 0

 (−A
z
(s), φ) X
1 ds +

t 0

 (F
, φ) X
1 ds +

t 0

(G
dW, φ) X
1 (3.8)

for t ∈ [0, T ) and φ ∈ X
1 . Moreover if z 0 is independent of W (t) with E|z 0 |2X 0 < ∞,
then  t E|z
(t)|2X 0 + E |z
(s)|2X 1 ds ≤ (1 + E|z 0 |2X 0 )C T , for t ∈ [0, T ] (3.9)


and E



sup

t∈[0,T ]




0

|z
(t)|2X 0







 ≤ 1 + E|z 0 |2X 0 + E




T 0

|z
(s)|2X 1 ds C T .


(3.10)

Proof. By the assumption (2.5), we have

G
(t, x) 2 Q = g1 (t, x) 2 Q 1 + g2 (t, x) 2 Q 2 < ∞. L2

L2

L2

Then the classical result [16] yields the local existence of z
. By applying the stochastic Fubini theorem [16], it can be verified that the local mild solution is also a weak solution. Now we give the following a priori estimates which yield the existence of weak solution on [0, T ] for any T > 0.

170

W. Wang, J. Duan

Applying the Itô formula to |z
|2X 0 , we derive


d|z
(t)|2X 0


+ 2(A
z
, z
) X
0 dt = 2(F
(t, x), z
) X
0 dt + 2(G
(t, x)dW (t), z
) X
0 + |G
(t, x)|2 Q dt.

(3.11)

L2

By the coercivity (3.4) of a
(·, ·), integrating (3.11) with respect to t yields  t 2 |z
(t)| X 0 + 2α¯ |z
(s)|2X 1 ds

0  t 2 2 ¯ ≤ |z 0 | X 0 + |F
| L 2 (0,T ;X 0 ) + (2β + 1) |z
(s)|2X 0 ds


0  t  t +2 (G
(s)dW (s), z
(s)) X
0 + |G
(s)|2 Q ds. 0

L2

0

Taking expectation on both sides of the above inequality yields  t 2 E|z
(t)| X 0 + 2αE ¯ |z
(s)|2X 1 ds

0  t  t ≤ E|z 0 |2X 0 + |F
|2L 2 (0,T ;X 0 ) + (2β¯ + 1) E|z
(s)|2X 0 ds + |G
(s)|2 Q ds.








0

L2

0

Then the Gronwall lemma gives the estimate (3.9). Notice that, by Lemma 7.2 in [16],  T

 t

2



E sup

S
(t − s)G
(s, x)ds 0 ≤ C T |G
(s)|2 Q ds. t∈[0,T ]

X


0

L2

0

Therefore by the assumption on f and (3.6) we have the estimate (3.10). The proof is hence complete.   By the above result and the definition of z
we have the following corollary. Corollary 3.2. Assume the conditions in Theorem 3.1. Then for t ∈ [0, T ], we have  t   E |u
(t)|2L 2 (D ) +
2 |γ
u
(t)|2L 2 (∂ S ) + E |u
(t)|2H 1 (D ) +
2 |γ
u
(t)|2H 1/2 (∂ S ) ds








0







≤ (1 + E|z 0 |2X 0 )C T

(3.12)




and E



sup |u
(t)|2L 2 (D ) +
2 |γ
u
(t)|2L 2 (∂ S

t∈[0,T ]





)



≤ (1 + E|z 0 |2X 0 )C T .


(3.13)

We recall a probability concept. Let z be a random variable taking values in a Banach space S, namely, z :  → S. Denote by L(z) the distribution (or law) of z. In fact, L(z) is a Borel probability measure on S defined as [16] L(z)(A) = P{ω : z(ω) ∈ A}, for every event (i.e., a Borel set) A in the Borel σ −algebra B(S), which is the smallest σ −algebra containing all open balls in S.

Homogenized Dynamics of SPDES

171

As stated in §1, for the SPDE (2.1) we aim at deriving an effective equation in the sense of probability. A solution u
may be regarded as a random variable taking values in L 2 (0, T ; L 2 (D
)). So for a solution u
of (2.1)–(2.4) defined on [0, T ], we focus on the behavior of distribution of u
in L 2 (0, T ; L 2 (D
)) as
→ 0. For this purpose, the tightness [19] of distributions is necessary. Note that the function space changes with
, which is a difficulty for obtaining the tightness of distributions. Thus we will treat {L(u
)}
>0 as a family of distributions on L 2 (0, T ; L 2 (D)) by extending u
to the whole domain D. Recall that the distribution (or law) of u˜
is defined as: L(u˜
)(A) = P{ω : u˜
(·, ·, ω) ∈ A} for the Borel set A in L 2 (0, T ; L 2 (D)). First we define the following spaces which will be used in our approach. For the Banach space U and p > 1, define W 1, p (0, T ; U ) as the space of functions h ∈ L p (0, T ; U ) such that

dh p



p p |h|W 1, p (0,T ;U ) = |h| L p (0,T ;U ) + p < ∞. dt L (0,T ;U ) And for any α ∈ (0, 1), define W α, p (0, T ; U ) as the space of function h ∈ L p (0, T ; U ) such that  T T p |h(t) − h(s)|U p p dsdt < ∞. |h|W α, p (0,T ;U ) = |h| L p (0,T ;U ) + |t − s|1+αp 0 0 For ρ ∈ (0, 1), we denote by C ρ (0, T ; U ) the space of functions h : [0, T ] → X that are Hölder continuous with exponent ρ.  Theorem 3.3 (Tightness of distributions). Assume that z 0 = (u 0 , v0 ), is a F0 , B(X
0 ) measurable random variable which is independent of W (t) with E|z 0 |2X 0 < ∞. Then


for any T > 0, (L(u˜
))
, the distributions of (u˜
)
, are tight in L 2 (0, T ; L 2 (D)) ∩ C(0, T ; H −1 (D)). Proof. Denote the projection (u, v) → u by P. By the result of Corollary 3.2, E|u
|2L 2 (0,T ;H 1 (D


Write z
(t) as



z
(t) = z
(0) −

t 0

 A
z
(s)ds +

0

t


))

≤ CT .

(3.14) 

F
(s, x)ds +

0

t

G
(s, x)dW (s).

Then by (3.3) and (3.8), when (h, 0) ∈ X
1 , we have the following estimate, for some positive constant C > 0 independent of
,  t  t







A
z
(s)ds + P F
(s, x)ds, h 2

−P

L (D )
0 0

 t

 t 





≤

f (s, x), h L 2 (D ) ds

a(P z
(s), h)ds +


0 0  t  t ≤C |u
(s)| H
1 (D
) ds + | f (s)| L 2 (D) ds |h| H 1 (D
) . 0

0

0

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W. Wang, J. Duan

Thus we have  t 



A
z
(s)ds + P E − P 0

Let M
(s, t) = have

t

t 0

2

F
(s, x)ds

W 1,2 (0,T ;H −1 (D
))

≤ CT .

(3.15)

G
(s, x)dW (s). By Lemma 7.2 of [16] and the Hölder inequality, we

s

  t 2 E|P M
(s, t)|4L 2 (D ) ≤ E|P M
(s, t)|4L 2 (D) ≤ cE |g1 (τ )|2 Q 1 dτ
L2 s  t ≤ K (t − s) E|g1 (τ )|4 Q 1 dτ L2

s



≤ K (t − s)

2

for t ∈ [s, T ], and for positive constants K and K independent of
, s and t. Therefore  T E |P M
(0, t)|4L 2 (D ) dt ≤ C T (3.16)


0

and for α ∈ ( 41 , 21 ), 

T



T

E 0

|P M
(0, t) − P M
(0, s)|4L 2 (D |t − s|1+4α

0


)

dsdt ≤ C T .

(3.17)

Combining the estimates (3.14)–(3.17) with the Chebyshev inequality [16, 19], it is clear that for any δ > 0 there is a bounded set with X =

L 2 (0, T ;

H
1 (D)) ∩



Kδ ⊂ X W 1,2 (0, T ;

H −1 (D)) + W α,4 (0, T ; L 2 (D)) , such that

P{u˜
∈ K δ } > 1 − δ. Moreover by the compact embedding L 2 (0, T ; H 1 (D)) ∩ W 1,2 (0, T ; H −1 (D)) ⊂ L 2 (0, T ; L 2 (D)) ∩ C(0, T ; H −1 (D)) and L 2 (0, T ; H 1 (D)) ∩ W α,4 (0, T ; L 2 (D)) ⊂ L 2 (0, T ; L 2 (D)) ∩ C(0, T ; H −1 (D)), we conclude that K δ is compact in L 2 (0, T ; L 2 (D))∩C(0, T ; H −1 (D)). Thus {L(u˜
)}
is tight in L 2 (0, T ; L 2 (D)) ∩ C(0, T ; H −1 (D)). The proof is complete.   Remark 3.4. When f = f (t, x, u
) is nonlinear (i.e., it depends on u
) but is also globally Lipschitz in u
, the results in Theorem 3.1 and Corollary 3.2 still hold. For example, see [11] for such SPDEs with stochastic dynamical boundary conditions. Moreover, by the Lipschitz property, we have | f (t, x, u
)| L 2 (D) ≤ C T . Hence a similar analysis as in the proof of Theorem 3.3 yields the tightness of the distribution for u
in this globally Lipschitz nonlinear case. This fact will be used in the beginning of §6 to get the homogenized effective model when f = f (t, x, u
) is globally Lipschitz nonlinear. In fact, in §6, we will also derive homogenized effective models for three types of nonlinearities f = f (t, x, u
) that are not globally Lipschitz in u
.

Homogenized Dynamics of SPDES

173

4. Two-Scale Convergence and Some Preliminary Results In this section we present some basic results about the two-scale convergence [2, 12]. In the following we denote by C ∞ per (Y ) the space of infinitely differentiable functions n 1 (Y ) the completion of in R that are periodic in Y . We also denote by L 2per (Y ) or H per ∞ 2 1 C per (Y ), in the usual norm of L (Y ) or H (Y ), respectively. We also introduce the 1 (Y )/R, which is the space of the equivalent classes of u ∈ H 1 (Y ) under space H per per the following equivalent relation u ∼ v ⇔ u − v = constant. Definition 4.1. A sequence of functions u
(t, x) in L 2 (DT ) is said to be two-scale convergent to a limit u(t, x, y) ∈ L 2 (DT × Y ), if for any function ϕ(x, y) ∈ C0∞ (DT , C ∞ per (Y )),    1 x u
(t, x)ϕ(t, x, )d xdt = u 0 (t, x, y)ϕ(t, x, y)d yd xdt. lim
→0 DT
|Y | DT Y 2−s

This two-scale convergence is written as u
−→ u. The following result ensures the existence of the two-scale limit and for the proof see [2, 12]. Lemma 4.2. Let u
be a bounded sequence in L 2 (DT ). Then there exist a function u ∈ L 2 (DT × Y ) and a subsequence u
k with
k → 0 as k → ∞ such that u
k two-scale converges to u. Remark 4.3. Taking ϕ independent of y in the definition of two-scale convergence, then 2−s

u
−→ u implies that u
weakly converges to its spatial average:  1 u
(t, x)  u(t, ¯ x) = u(t, x, y)dy. |Y | Y So, we see that, for a given bounded sequence L 2 (DT ), the two-scale limit u(t, x, y) contains more information than the weak limit u(t, x): u gives some knowledge on the periodic oscillations of u
, while u¯ is just the average with respect to y. Another advantage of the usage of two-scale convergence is that we do not need an extending operator such as in [13, 15] in the homogenization procedure. For more properties of two-scale convergence we refer to [2]. The following result is useful when considering two-scale convergence of the product of two convergent sequences, see [2, 12]. Lemma 4.4. Let v
be a sequence in L 2 (DT ) that two-scale converges to a limit v(x, y) ∈ L 2 (DT × Y ). Further assume that    1 2 |v
(t, x)| d xdt = |v(t, x, y)|2 d yd xdt. (4.1) lim
→0 DT |Y | DT Y Then, for any sequence u
∈ L 2 (DT ), which two-scale converges to a limit u ∈ L 2 (DT × Y ), we have the weak convergence of the product u
v
:  1 u(·, ·, y)v(·, ·, y)dy, as
→ 0 in L 2 (DT ). u
v
 |Y | Y

174

W. Wang, J. Duan

Remark 4.5. Condition (4.1) always holds for a sequence of functions ϕ(t, x, x/
), with ϕ(t, x, y) ∈ L 2 (DT ; C per (Y )). Such functions v
are called admissible test functions. With the additional condition (4.1), the two-scale convergence of v
is also called strong two-scale convergence [2]. Let u
be a sequence of functions defined on [0, T ] × D
which is bounded in L 2 (0, T ; H
1 (D
)). Then we have the following result concerning the two-scale limit of ∇x u
; for the proof see [2]. the bounded sequences u˜
and
1 (Y )) and a Lemma 4.6. There exist u(t, x) ∈ H01 (DT ), u 1 (t, x, y) ∈ L 2 (DT ; H per subsequence u
k with
k → 0 as k → ∞, such that 2−s

u˜
k (t, x) −→ χ (y)u(t, x), k → ∞ and 2−s
∇x u
k −→ χ (y)[∇x u(t, x) + ∇ y u 1 (t, x, y)],

where χ (y) is the indicator function of Y \Y ∗ ).

Y∗

k → ∞,

(which takes value 1 on Y ∗ and value 0 on

Since we consider the dynamical boundary condition, the technique of transforming the surface integrals into the volume integrals is useful in our approach. For this we follow the method of [55] (see also [14]) for the nonhomogeneous Neumann boundary problem for an elliptic equation. For h ∈ H −1/2 (∂ S) and Y -periodic, define  1 h = ∗ h(x)d x. |Y | ∂ S Also define λh = Thus, in particular 1 =

|∂ S| |Y ∗ |

1 h, 1 H −1/2 ,H 1/2 = ϑh . |Y |

and λ := λ1 =

|∂ S| , |Y |

(4.2)

where | · | denotes Lebesgue measure. For h ∈ L 2 (∂ S) and Y -periodic, define λ
h ∈ H −1 (D) as  x h( )ϕ(x)d x, for ϕ ∈ H01 (D). λ
h , ϕ =

∂ S
Then we have the following result about the convergence of the integral on the boundary. Lemma 4.7. Let ϕ
be a sequence in H01 (D) such that ϕ
 ϕ weakly in H01 (D) as
→ 0. Then  ϕd x, as
→ 0. λ
h , ϕ
| D
 → λh D

For the proof we refer to [14].

Homogenized Dynamics of SPDES

175

5. Homogenized Macroscopic Model In this section we derive the effective macroscopic model for the original model (2.1), by the two-scale convergence approach. We first obtain a two-scale limiting model. Then the homogenized macroscopic model is obtained by exploiting the relation between the weak limit and the two-scale limit. By the proof of Theorem 3.3 for any δ > 0 there is a bounded closed set K δ ⊂ X which is compact in L 2 (0, T ; L 2 (D)) such that P{u˜
∈ K δ } > 1 − δ. Then the Prohorov theorem and the Skorohod embedding theorem ([16]) assure that for any sequence {
j } j with
j → 0 as j → ∞, there exist a subsequence {
j (k) }, random variables {u ∗
j (k) } ⊂ L 2 (0, T ; H
j (k) ) and u ∗ ∈ L 2 (0, T ; H ) defined on a new probability space (∗ , F ∗ , P∗ ), such that L(u˜ ∗
j (k) ) = L(u˜
j (k) ) and u˜ ∗
j (k) → u ∗ in L 2 (0, T ; H ) as k → ∞, . Moreover u ∗
solves system (2.1)–(2.4) with W replaced by the for almost all ω ∈  j (k) Wiener process Wk∗ defined on probability space (∗ , F ∗ , P∗ ) with the same distribution as W . In the following, we will determine the limiting equation (homogenized effective equation) that u ∗ satisfies and the limiting equation is independent of
. After this is done we see that L(u˜
) weakly converges to L(u ∗ ) as
↓ 0. For u
in set K δ , by Lemma 4.6 there is u(t, x) ∈ H01 (DT ) and u 1 (t, x, y) ∈ 1 T ; H per (Y )) such that

L 2 (D

2−s

u˜
j (t, x) −→ χ (y)u(t, x) and 2−s
∇x u
j −→ χ (y)[∇x u(t, x) + ∇ y u 1 (t, x, y)].

Then by Remark 4.3, 1 u˜
j (t, x)  |Y |

 χ (y)u(t, x)dy = ϑu(t, x), weakly in L 2 (DT ). Y

In fact by the compactness of K δ , the above convergence is strong in L 2 (DT ). In the following, we will determine the limiting equation, which is a two-scale system that u and u 1 satisfy. Then the limiting equation (homogenized effective equation) that u 0 satisfies can be easily obtained by the relation between weak limit and the two-scale limit. Define a new probability space (δ , Fδ , Pδ ) as δ = {ω ∈  : u˜
(ω) ∈ K δ }, Fδ = {F ∩ δ : F ∈ F},

176

W. Wang, J. Duan

and Pδ (F) =

P(F ∩ δ ) , for F ∈ Fδ . P(δ )

Denote by Eδ the expectation operator with respect to Pδ . Now we restrict the system on the probability space (δ , Fδ , Pδ ). Replace the test function ϕ in (3.7) by ϕ
(t, x) = φ(t, x) +
(t, x, x/
) with φ(t, x) ∈ C0∞ (DT ) and (t, x, y) ∈ C0∞ (DT ; C ∞ per (Y )). We will consider the terms in (3.7) respectively . By the choice of ϕ
and noticing that χ D
 ϑ, weakly∗ in L ∞ (D), we have 





T

f (t, x)ϕ
(t, x)d xdt =

D


0



 D

0



T

→ϑ

T

χ D
f (t, x)ϕ
(t, x)d xdt

f (t, x)φ(t, x)d xdt,
→ 0.

0

(5.1)

D

And by the condition (2.5), 

T





0

D


g1 (t, x)ϕ
(t, x)d xd W1 (t) =





0



T

→ϑ

T

D

χ D
g1 (t, x)ϕ
(t, x)d xd W1 (t)

g1 (t, x)φ(t, x)d xd W1 (t),
→ 0 in L 2 ().

0

(5.2)

D

Integrating by parts and noticing that u˜
converges strongly to ϑu(t, x) in L 2 (DT ), 

T 0



 D


u˙
(t, x)ϕ
(t, x)d xdt = −



T

=−

 D

0



T

→−

0

˙ x)d xdt −
u˜
(t, x)φ(t,



0

T

˙ x)d xdt = ϑu(t, x)φ(t, D

 D
 T 0



0

T

u
(t, x)ϕ˙
(t, x)d xdt  

x ˙ x, )d xdt u˜
(t, x)(t,
D ϑ u(t, ˙ x)φ(t, x)d xdt. D

By the choice of ϕ
, x 2−s ∇x φ(t, x) + ∇ y (t, x, ) −→ ∇x φ(t, x) + ∇ y (t, x, y),
→ 0
and x lim ∇x φ(t, x) + ∇ y (t, x, ) [L 2 (DT )]n


2 1



=

∇x φ(t, x) + ∇ y (t, x, y) d yd xdt. |Y | DT ×Y


→0

(5.3)

Homogenized Dynamics of SPDES

177

Hence by Theorem 4.4, we have  T  T x ∇u
(t, x)∇ϕ
(t, x)d xdt = ∇u
(t, x)(∇x φ(t, x) + ∇ y (t, x, ))d xdt
0 D
0 D
 T 
(t, x)(∇x φ + ∇ y (t, x, x ))d xdt = ∇u
0 D  T     1 → χ (y) ∇x u(x) + ∇ y u 1 (x, y) ∇x φ(t, x) + ∇ y (t, x, y) d yd xdt |Y | 0 D Y  T     1 ∇x u(x) + ∇ y u 1 (x, y) ∇x φ(t, x) +∇ y (t, x, y) d yd xdt. (5.4) = |Y | 0 D Y ∗ Now we consider the integrals on the boundary. First for a fixed T > 0, it is easy to see that  T
2 u˙
(t, x)ϕ
(t, x)d xdt ∂ S


0



= −
2



∂ S


  = −
λ
1 , And then



T


b =

0 T



u
(t, x)ϕ˙
(t, x)dtd x



u˜
(t, x)ϕ˙
(t, x)dt

0

 D


→ 0,
→ 0.

(5.5)

 u
(t, x)ϕ
(t, x)d xdt

∂ S


0

T

λ
1 ,



T 0

 → bϑλ 0



u˜
(t, x)ϕ
(t, x)dt

T



 D


u(t, x)φ(t, x)d xdt,
→ 0.

(5.6)

D

By the same method as above and the condition (2.5) we have the limit of the stochastic integral on the boundary  T
g2 (t, x)ϕ
(t, x)d xd W2 (t) 0

∂ S
T



→λ



0

g2 (t, x)φ(t, x)d xd W2 (t),
→ 0, in L 2 ().

(5.7)

D

Combining the above analysis in (5.1)–(5.7) and by the density argument we have  T ϑ u(t, ˙ x)φ(t, x)d xdt 0

D

1 =− |Y |

T

 

T



 

0

−bϑλ  T

0

0

D

+ϑ

D

Y∗

   ∇x u(x) + ∇ y u 1 (x, y) ∇x φ(t, x) + ∇ y (t, x, y) d xdt 

u(t, x)φ(t, x)d xdt + ϑ D

g1 (t, x)φ(t, x)d xd W1 (t) + λ

T

 f (t, x)φ(t, x)d xdt

D 0  T

g2 (t, x)φ(t, x)d xd W2 (t)

0

D

(5.8)

178

W. Wang, J. Duan

1 (Y )/R). Integrating by parts, we see that for any φ ∈ H01 (DT ) and  ∈ L 2 (DT ; H per (5.8) is the variational problem of the following two-scale homogenized system:   ϑdu = − divx A(∇x u) − bϑλ1 u + ϑ f dt + ϑg1 dW1 (t) + λg2 dW2 (t), (5.9)

[∇x u + ∇u 1 ] · ν = 0, on ∂Y ∗ − ∂Y,

(5.10)

∂Y ∗

− ∂Y and where ν is the unit exterior norm vector on  1 A(∇x u) = [∇x u(t, x) + ∇ y u 1 (t, x, y)]dy, |Y | Y ∗ with u 1 satisfying the following integral equation:  [∇x u + ∇ y u 1 ]∇ y dy = 0, u 1 is Y − periodic, Y∗

(5.11)

(5.12)

1 (Y )). The problem (5.12) has a unique solution for any fixed for any  ∈ H01 (DT ; H per u, and so A(∇x u) is well-defined. Furthermore A(∇x u) satisfies

and

A(ξ1 ) − A(ξ2 ), ξ1 − ξ2  L 2 (D),L 2 (D) ≥ α ξ1 − ξ2 2L 2 (D)

(5.13)

|A(ξ ), ξ  L 2 (D),L 2 (D) | ≤ β ξ 2L 2 (D)

(5.14)

with some α, β > 0 and any ξ , ξ1 , ξ2 ∈ H01 (D). For more detailed properties of A(∇u) and (5.12) we refer to [24]. Then by the classical theory of the SPDEs, [16], (5.9)–(5.10) is well-posed. In fact A(∇u) can be transformed to the classical homogenized matrix by u 1 (t, x, y) =

n  ∂u(t, x) i=1

∂ xi

(wi (y) − ei y),

(5.15)

n is the canonical basis of Rn and wi is the solution of the following cell where {ei }i=1 problem (problem defined on the spatial elementary cell):

 y wi (y) = 0 in Y ∗ , wi − ei y is Y − periodic, ∂wi = 0 on ∂ S. ∂ν

(5.16) (5.17) (5.18)

Then a simple calculation yields A(∇u) = A∗ ∇u with A∗ = (Ai∗j ) being the classical homogenized matrix defined as  1 wi (y)w j (y)dy. Ai∗j = |Y | Y ∗

(5.19)

Then the above two-scale system (5.9) is equivalent to the following homogenized system:    ϑdu = − divx A∗ ∇x u − bϑλu + ϑ f dt + ϑg1 dW1 (t) + λg2 dW2 (t). (5.20)

Homogenized Dynamics of SPDES

179

Let U (t, x) = ϑu(t, x). We thus have the limiting homogenized equation,    dU = − ϑ −1 divx A∗ ∇x U −bλU + ϑ f dt + ϑg1 dW1 (t) + λg2 dW2 (t). (5.21) And then u ∗ , we have mentioned at the beginning of this section, satisfies (5.21) with W = (W1 , W2 ) replaced by a Wiener process W ∗ with the same distribution as W . By the classical existence result [16], the homogenized model (5.21) is well-posed. We formulate the main result of this section as follows. Theorem 5.1 (Homogenized macroscopic model). Assume that (2.5) holds. Let u
be the solution of (2.1)–(2.4). Then for any fixed T > 0, the distribution L(u˜
) converges weakly to µ in L 2 (0, T ; H ) as
↓ 0, with µ being the distribution of U , which is the solution of the following homogenized effective equation:    dU = − ϑ −1 divx A∗ ∇x U − bλU + ϑ f dt + ϑg1 dW1 (t) + λg2 dW2 (t), (5.22) with the boundary condition U = 0 on ∂ D, the initial condition U (0) = u 0 /ϑ and the effective matrix A∗ = (Ai∗j ) being determined by (5.19). Moreover, the constant coefficient ϑ =

|Y ∗ | |Y |

is defined in the beginning of §2 and λ =

|∂ S| |Y |

is defined in (4.2).

Proof. Noticing the arbitrariness of the choice of δ, this is a direct result of the analysis of the first part in this section by the Skorohod theorem and the L 2 (δ )−convergence of u˜
on (δ , Fδ , Pδ ).   Remark 5.2. It is interesting to note the following fact. Even when the original microscopic model equation (2.1) is a deterministic PDE (i.e., g1 = 0), the homogenized macroscopic model (5.22) is still a stochastic PDE, due to the impact of random dynamical interactions on the boundary of small scale heterogeneities. Remark 5.3. For the macroscopic system (5.22), we see that the fast scale random fluctuations on the boundary is recognized or quantified in the homogenized equation, through the µ1 g2 dW2 (t) term. The effect of random boundary evolution is thus felt by the homogenized system on the whole domain. 6. Homogenized Macroscopic Dynamics for Nonlinear Microscopic Systems In this section, we derive homogenized macroscopic model for the microscopic system (2.1)–(2.4), when the nonlinearity f is either globally Lipschitz, or non-globally Lipschitz. As Remark 3.4 has pointed out, if f is a globally Lipschitz nonlinear function of u
all the estimates in § 3 hold. In fact, similar results in § 5 on the homogenized model also hold. In fact for f satisfying f (t, x, 0) = 0 and | f (t, x, u 1 ) − f (t, x, u 2 )| ≤ L|u 1 − u 2 | for any t ∈ R, x ∈ D and u 1 , u 2 ∈ R with some positive constant L. Since u˜
→ ϑu strongly in L 2 (0, T ; L 2 (D)) and by the Lipschitz property of f (t, x, u) with respect to u, f (t, x, u˜
(t, x)) → f (t, x, u(t, x)) strongly in L 2 (0, T ; L 2 (D)). Equation (5.1) still holds for f (t, x, u
). Then we can obtain the same effective macroscopic system as (5.22) with nonlinearity f = f (t, x, U ):    dU = − ϑ −1 divx A∗ ∇x U − bλU + ϑ f (t, x, U ) dt +ϑg1 dW1 (t) + λg2 dW2 (t). (6.1)

180

W. Wang, J. Duan

For the rest of this section, we consider three types of nonlinear systems with f being a non-global-Lipschitz nonlinear function in u
. The difficulty is at passing the limit
→ 0 in the nonlinear term. These three types of nonlinearity include: Polynomial nonlinearity; nonlinear term that is sublinear; and nonlinearity that contains a gradient term ∇u
. We look at these nonlinearities case by case, and only highlight the difference with the analysis in §5. Case 1: Polynomial nonlinearity. First we suppose f is in the following form: f (t, x, u) = −a(t, x)|u| p u

(6.2)

with 0 < a0 ≤ a(t, x) ≤ a1 for t ∈ [0, ∞), x ∈ D. And p satisfies the following condition: 2 p≤ , if n ≥ 3; p ∈ R, if n = 2. (6.3) n−2 For this case we need the following Weak convergence lemma from Lions [33]: Let Q be a bounded region in R×Rn . For any given functions g
and g in L p (Q) (1 < p < ∞), if |g
| L p (Q) ≤ C, g
→ g in Q almost everywhere for some positive constant C, then g
 g weakly in L p (Q). Noticing that F
(t, x, z
) = ( f (t, x, u
), 0) and (F
(t, x, z
), z
) X
0 ≤ 0, the results in Theorem 3.1 can be obtained by the same method as in the proof of Theorem 3.1. Moreover by the assumption (6.3), | f (t, x, u
)| L 2 (DT ) ≤ C T , which by the analysis of Theorem 3.3, yields the tightness of the distribution of u˜
. Now we pass the limit
→ 0 in f (t, x, u˜
). In fact, noticing that u˜
converges strongly to ϑu in L 2 (0, T ; L 2 (D)), by the above weak convergence lemma with g
= f (t, x, u˜
) and p = 2, f (t, x, u˜
) converges weakly to f (t, x, ϑu) in L 2 (DT ). Therefore by the analysis for the linear system in §5, we have the following result. Theorem 6.1. Assume that (2.5) holds. Let u
be the solution of (2.1)–(2.4) with nonlinear term f being (6.2). Then for any fixed T > 0, the distribution L(u˜
) converges weakly to µ in L 2 (0, T ; H ) as
↓ 0, with µ being the distribution of U , which is the solution of the following homogenized effective equation:    dU = − ϑ −1 divx A∗ ∇x U − bλU + ϑ f (t, x, U ) dt +ϑg1 dW1 (t) + λg2 dW2 (t), (6.4) with the boundary condition U = 0 on ∂ D, the initial condition U (0) = u 0 /ϑ, and the effective matrix A∗ = (Ai∗j ) being determined by (5.19). Moreover, the constant coefficient ϑ =

|Y ∗ | |Y |

is defined in the beginning of §2 and λ =

|∂ S| |Y |

is defined in (4.2).

Case 2: Nonlinear term that is sublinear. More generally, we consider f : [0, T ] × D × R → R a measurable function which is continuous in (x, ξ ) ∈ D × R for almost all t ∈ [0, T ] and satisfies    (6.5) f (t, x, ξ1 ) − f (t, x, ξ2 ) ξ1 − ξ2 ≥ 0 for t ≥ 0, x ∈ D and ξ1 , ξ2 ∈ R. Moreover, we assume that f is sublinear, | f (t, x, ξ )| ≤ g(t)(1 + |ξ |), ξ ∈ R, t ≥ 0,

(6.6)

Homogenized Dynamics of SPDES

181

∞ [0, ∞). Notice that under the assumption (6.5) and (6.6), f may not be where g ∈ L loc a Lipschitz function. By the assumption (6.6) we can also have the tightness of the distributions of u˜
and also conclude that χ D
f (t, x, u˜
) two-scale converges to a function denoted by f 0 (t, x, y) ∈ L 2 (DT × Y ). In the following we need to identity f 0 (t, x, y). Let φ ∈ C0∞ (DT ) and ψ ∈ C0∞ (DT ; C ∞ per (Y )). And for κ > 0 let

x ξ
(t, x) = φ(t, x) + κψ(t, x, ).


(6.7)

Then by the assumption (6.5) we have  T    f (t, x, u
) − f (t, x, ξ
) u
− ξ
d xdt 0≤ D
0    x   f (t, x, u˜
) − f (t, x, ξ
) u˜
− ξ
d xdt = χ
DT 

= I
= I1,
− I2,
− I3,
+ I4,
with



x f (t, x, u˜
)u˜
d xdt χ
DT  
→0 1 −→ f 0 (t, x, y)ϑu(t, x)d yd xdt, |Y | DT Y

I1,
=

(6.8)



x f (t, x, u˜
)ξ
d xdt χ
DT    
→0 1 −→ f 0 (t, x, y) φ(t, x) + κψ(t, x, y) d yd xdt, |Y | DT Y

I2,
=

(6.9)



x f (t, x, ξ
)u˜
d xdt χ
DT  
→0 1 −→ χ (y) f (t, x, φ(t, x) + κψ(t, x, y)) |Y | DT Y ϑu(t, x)d yd xdt,

I3,
=

and

(6.10)



x f (t, x, ξ
)ξ
d xdt χ
DT  
→0 1 −→ χ (y) f (t, x, φ(t, x) + κψ(t, x, y)) |Y | DT Y   × φ(t, x) + κψ(t, x, y) d yd xdt.

I4,
=

(6.11)

In (6.8)–(6.11) we have used the fact of strong two-scale convergence of χ ( x
) and f (t, x, ξ
), and the strong convergence of u
to ϑu.

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W. Wang, J. Duan

Now we have lim I
 =


→0



DT



  f 0 (t, x, y)−χ (y) f (t, x, φ +λψ) ϑu(t, x)−φ(t, x)−κψ d yd xdt ≥ 0,

Y

for any φ ∈ C0∞ (DT ) and ψ ∈ C0∞ (DT ; C per (Y )). Letting φ → ϑu, dividing the above formula by κ on both sides of the above formula and letting κ → 0 yields     f 0 (t, x, y) − χ (y) f (t, x, ϑu) ψd yd xdt ≤ 0 DT

for any ψ ∈

Y

C0∞ (DT ; C per (Y )),

which means

f 0 (t, x, y) = χ (y) f (t, x, ϑu). Then by the similar analysis for linear systems in §5, we have the following homogenized model. Theorem 6.2. Assume that (2.5) holds. Let u
be the solution of (2.1)–(2.4) with nonlinear term f satisfying (6.5) and (6.6). Then for any fixed T > 0, the distribution L(u˜
) converges weakly to µ in L 2 (0, T ; H ) as
↓ 0, with µ being the distribution of U , which is the solution of the following homogenized effective equation:    dU = − ϑ −1 divx A∗ ∇x U − bλU + ϑ f (t, x, U ) dt +ϑg1 dW1 (t) + λg2 dW2 (t), (6.12) with the boundary condition U = 0 on ∂ D, the initial condition U (0) = u 0 /ϑ, and the effective matrix A∗ = (Ai∗j ) being determined by (5.19). Moreover, the constant coefficient ϑ =

|Y ∗ | |Y |

is defined in the beginning of §2 and λ =

|∂ S| |Y |

is defined in (4.2).

Case 3: Nonlinearity that contains a gradient term. We next consider f in the following form containing a gradient term, f (t, x, u, ∇u) = h(t, x, u) · ∇u,

(6.13)

where h(t, x, u) = (h 1 (t, x, u), . . . , h n (t, x, u)) and each h i : [0, T ] × D × R → R, i = 1, . . . , n, is continuous with respect to u and h(·, ·, u(·, ·)) ∈ L 2 (0, T ; L 2 (D)) for u ∈ L 2 (0, T ; L 2 (D)). Moreover assume that h satisfies (1) |h(t, x, u) · ∇u, v L 2 | ≤ C0 |∇u| L 2 |v| L 2 with some positive constant C0 . (2) |h i (t, x, ξ1 ) − h i (t, x, ξ2 )| ≤ k|ξ1 − ξ2 | for ξ1 , ξ2 ∈ R, i = 1, . . . , n and k is a positive constant. Now we have





(F
(t, x, z
), z
) X 0 = (h(t, x, u
) · ∇u
, u
) L 2 ≤ C0 |z
| X 0 |z
| X 1 .




(6.14)

By applying the Itô formula to |z
|2X 0 , we obtain


d|z
(t)|2X 0 + 2(A
z
, z
) X
0 dt = 2(F
(t, x, z
), z
) X
0 dt + 2(G
(t, x)dW (t), z
) X
0 +


|G
(t, x)|2 Q dt. L2

(6.15)

Homogenized Dynamics of SPDES

183

By (6.14), coercivity (3.4) of a
(·, ·) and the Cauchy inequality, integrating (6.15) with respect to t yields  t 2 |z
(s)|2X 1 ds |z
(t)| X 0 + α¯

0  t 2 ¯ ≤ |z 0 | X 0 + (2β + 1 (α)) ¯ |z
(s)|2X 0 ds +

0  t  t 2 (G
(s)dW (s), z
(s)) X
0 + |G
(s)|2 Q ds, 0

0

L2

where 1 is a positive constant depending on α. ¯ Then by the Gronwall lemma we see that (3.9) and (3.10) hold. Moreover, the fact |h(t, x, u) · ∇u| L 2 ≤ C0 |z
| X
1 , together with the Hölder inequality yields  t  t

2



E − P A
z
(s)ds + P F
(s, x, z
)ds

0

0

W 1,2 (0,T ;H −1 (D
))

≤ CT ,

(6.16)

where P is defined in Theorem 3.3. Then by the same discussion of Theorem 3.3, we have the tightness of the distributions of u˜
. Now we pass the limit
→ 0 in the nonlinear term f (t, x, u
, ∇u
). In fact we restrict the system on (δ , Fδ , Pδ ). By the assumption (2) on h and the fact that u˜
strongly converges to ϑu in L 2 (DT ), we have 
  2 h t, x, u˜
(t, x) − h t, x, ϑu(t, x) d xdt = 0. lim
→0 DT

For any ψ ∈ C0∞ (DT ),  h(t, x, u˜
) ·
∇u
ψd xdt DT 
   h t, x, u˜
− h t, x, ϑu ·
∇u
ψd xdt = DT   + h t, x, ϑu ·
∇u
ψd xdt DT     
→0 1 −→ h t, x, ϑu · χ (y) ∇x u + ∇ y u 1 ψd yd xdt. |Y | DT Y

(6.17)

Combining with the analysis for the linear system in §5, we have the following result. Theorem 6.3. Assume that (2.5) holds. Let u
be the solution of (2.1)–(2.4) with nonlinear term (6.13). Then for any fixed T > 0, the distribution L(u˜
) converges weakly to µ in L 2 (0, T ; H ) as
↓ 0, with µ being the distribution of U = ϑu which satisfies the following homogenized effective equation:    dU = − ϑ −1 divx A∗ ∇x U − bλU + f ∗ (t, x, U, ∇x U ) dt +ϑg1 dW1 (t) + λg2 dW2 (t), (6.18)

184

W. Wang, J. Duan

where the boundary condition U = 0 on ∂ D, the initial condition U (0) = u 0 /ϑ, the effective matrix A∗ = (Ai∗j ) is determined by (5.19) and f ∗ is the following spatial average:     1 f ∗ (t, x, U, ∇x U ) = h t, x, U · χ (y) ϑ −1 ∇x U + ∇ y u 1 dy, |Y | Y with u 1 being given by (5.15) and χ (y) the indicator function of Y ∗ . Moreover, the ∗ constant coefficient ϑ = |Y|Y || is defined in the beginning of §2 and λ = |∂|YS|| is defined in (4.2). Remark 6.4. All the results in this paper hold when  is replaced by a more general strong elliptic operator div(A
∇u), where A
is Y −periodic and satisfies the strong ellipticity condition. Acknowledgements. A part of this work was done while J. Duan was visiting the Ennio De Giorgi Center of Mathematical Research (www.crm.sns.it), Pisa, Italy. J. Duan would like to thank Giuseppe Da Prato and Franco Flandoli for their financial support and hospitality. This work was partly supported by the NSF Grants DMS-0209326 & OCE-0620539 and the Outstanding Overseas Chinese Scholars Fund of the Chinese Academy of Sciences.

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18. Duan, J., Gao, H., Schmalfuss, B.: Stochastic Dynamics of a Coupled Atmosphere-Ocean Model. Stochastics and Dynamics 2, 357–380 (2002) 19. Dudley, R.M.: Real Analysis and Probability. Cambridge: Cambridge Univ. Press, 2002 20. Duncan, T.E., Maslowski, B., Pasik-Duncan, B.: Ergodic boundary/point control of stochastic semilinear systems. SIAM J Control Optim. 36, 1020–1047 (1998) 21. E, W., Li, X., Vanden-Eijnden, E.: Some recent progress in multiscale modeling. In: Multiscale modeling and simulation, Lect. Notes Comput. Sci. Eng. 39, Berlin: Springer, 2004, pp. 3–21 22. Escher, J.: Quasilinear parabolic systems with dynamical boundary. Comm. Part. Diff. Eq. 18, 1309–1364 (1993) 23. Escher, J.: On the qualitative behavior of some semilinear parabolic problem. Diff. and Integ. Eq. 8(2), 247–267 (1995) 24. Fusco, N., Moscariello, G.: On the homogenization of quasilinear divergence structure operators. Ann. Math. Pura Appl. 164(4), 1–13 (1987) 25. Hintermann, T.: Evolution equations with dynamic boundary conditions. Proc. Roy. Soc. Edinburgh Sect. A 113, 43–60 (1989) 26. Huang, Z., Yan, J.: Introduction to Infinite Dimensional Stochastic Analysis. Beijing/New York:Science Press/Kluwer Academic Pub.,1997 27. Imkeller, P., Monahan, A. (eds.): Stochastic Climate Dynamics, a Special Issue in the journal Stochastics and Dynamics, Vol. 2, No. 3 (2002) 28. Jikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of Differential Operators and Integral Functionals. Berlin: Springer-Verlag, 1994 29. Kleptsyna, M.L., Piatnitski, A.L.: Homogenization of a random non-stationary convection-diffusion problem. Russ. Math. Surve. 57, 729–751 (2002) 30. Kushner, H.J., Huang, H.: Limits for parabolic partial differential equations with wide band stochastic coefficients and an application to filtering theory. Stochastics 14(2), 115–148 (1985) 31. Langer, R.E.: A problem in diffusion or in the flow of heat for a solid in contact with a fluid. Tohoku Math. J. 35, 260–275 (1932) 32. Lapidus, L., Amundson, N.(eds.): Chemical Reactor Theory, Englewood Cliffs, NJ: Prentice-Hall, 1977 33. Lions, J.L.: Quelques methodes ´ de resolution ´ des problèmes non lineaires. ´ Paris: Dunod, 1969 34. Lions, P.L., Masmoudi, N.: Homogenization of the Euler system in a 2D porous medium. J. Math. Pures Appl. 84, 1–20 (2005) 35. Marchenko, V.A., Khruslov, Ya. E.: Homogenization of partial differential equations. Boston: Birkhauser, 2006 36. Maslowski, B.: Stability of semilinear equations with boundary and pointwise noise. Annali Scuola Normale Superiore di Pisa Scienze Fisiche E Matematiche 22, 55–93 (1995) 37. Maso, G.D., Modica, L.: Nonlinear stochastic homogenization and ergodic theory. J. Rei. Ang. Math. B. 368, 27–42 (1986) 38. Mikelic, ´ A., Paloi, L.: Homogenization of the invisicid incompressible fluid flow through a 2D porous medium. Proc. Amer. Math. Soc. 127, 2019–2028 (1999) 39. Nandakumaran, A.K., Rajesh, M.: Homogenization of a parabolic equation in a perforated domain with Neumann boundary condition. Proc. Indian Acad. Sci. (Math. Sci.) 112, 195–207 (2002) 40. Nandakumaran, A.K., Rajesh, M.: Homogenization of a parabolic equation in a perforated domain with Dirichlet boundary condition. Proc. Indian Acad. Sci. (Math. Sci.) 112, 425–439 (2002) 41. Pardoux, E., Piatnitski, A.L.: Homogenization of a nonlinear random parabolic partial differential equation. Stochastic Process Appl. 104, 1–27 (2003) 42. Peixoto, J.P., Oort, A.H.: Physics of Climate. New York: Springer, 1992 43. Rockner, M.: Introduction to Stochastic Partial Differential Equations. Preprint 2006, to appear as text notes in Math. 1905, Springer, 2007 44. Rozovskii, B.L.: Stochastic Evolution Equations. Boston: Kluwer Academic Publishers,1990 45. Sanchez-Palencia, E.: Non Homogeneous Media and Vibration Theory. Lecture Notes in Physics 127, Berlin: Springer-Verlag, 1980 46. Souza, J., Kist, A.: Homogenization and correctors results for a nonlinear reaction-diffusion equation in domains with small holes. The 7th Workshop on Partial Differential Equations II Mat. Contemp. 23, 161–183 (2002) 47. Timofte, C.: Homogenization results for parabolic problems with dynamical boundary conditions. Romanian Rep. Phys. 56, 131–140 (2004) 48. Taghite, M.B., Taous, K., Maurice, G.: Heat equations in a perforated composite plate: Influence of a coating. Int J. Eng. Sci. 40, 1611–1645 (2002) 49. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. Amsterdam: North-Holland, 1978 50. Watanabe, H.: Averaging and fluctuations for parabolic equations with rapidly oscillating random coefficients. Prob. Theory & Related Fields 77, 359–378 (1988)

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51. Waymire, E., Duan, J.(eds.): Probability and Partial Differential Equations in Modern Applied Mathematics. IMA Volume 140, New York: Springer-Verlag, 2005 52. Wang, W., Cao, D., Duan, J.: Effective macroscopic dynamics of stochastic partial differential equations in perforated domains. SIAM J. Math. Anal. 38, 1508–1527 (2007) 53. Wright, S.: Time-dependent Stokes flow through a randomly perforated porous medium. Asymptot. Anal. 23(3-4), 257–272 (2000) 54. Yosida, K.: Functional Analysis. 5th ed., Berlin:Springer-Verlag, 1978 55. Vanninathan, M.: Homogenization of eigenvalues problems in perforated domains. Proc. Indian Acad. Sci. 90, 239–271 (1981) 56. Vold, R., Vold, M.: Colloid and Interface Chemistry. Reading MA: Addison-Wesley, 1983 57. Yang, D., Duan, J.: An impact of stochastic dynamic boundary conditions on the evolution of the Cahn-Hilliard system. Stoch. Anal. and Appl. 25(3), 613–639 (2007) 58. Zhikov, V.V.: On homogenization in random perforated domains of general type. Matem. Zametki 53, 41–58 (1993) 59. Zhikov, V.V.: On homogenization of nonlinear variational problems in perforated domains. Russ. J Math. Phys. 2, 393–408 (1994) Communicated by P. Constantin

Commun. Math. Phys. 275, 187–208 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0302-7

Communications in

Mathematical Physics

Dimers on Surface Graphs and Spin Structures. I David Cimasoni, Nicolai Reshetikhin Mathematics Department University of California, Berkeley, CA 94720, USA. E-mail: [email protected]; [email protected] Received: 28 September 2006 / Accepted: 9 January 2007 Published online: 25 July 2007 – © Springer-Verlag 2007

Abstract: Partition functions for dimers on closed oriented surfaces are known to be alternating sums of Pfaffians of Kasteleyn matrices. In this paper, we obtain the formula for the coefficients in terms of discrete spin structures.

1. Introduction Dimer models on graphs have a long history in statistical mechanics [6, 15]. States in dimer models are perfect matchings between vertices of the graph where only adjacent vertices are matched. The probability of a state is determined by assigning weights to edges. Dimer models also have many interesting mathematical aspects involving combinatorics, probability theory [11, 3], real algebraic geometry [10, 9], etc.... One of the remarkable facts about dimer models is that the partition function can be written as a linear combination of 22g Pfaffians of N × N matrices, where N is the number of vertices in the graph and g the genus of a surface where the graph can be embedded. The matrices in the Pfaffian formula for the dimer partition function are called Kasteleyn matrices. They involve certain orientations of edges of the graph known as Kasteleyn orientations. Two Kasteleyn orientations are called equivalent if one can be obtained from the other by a sequence of moves reversing orientations of all edges adjacent to a vertex. The number of non-equivalent Kasteleyn orientations of a surface graph of genus g is 22g and is equal to the number of non-equivalent spin structures on the surface. The Pfaffian formula expresses the partition function of the dimer model as an alternating sum of Pfaffians of Kasteleyn operators, one for each equivalence class of Kasteleyn orientations. This formula was proved in [6] for the torus, and it was stated in [7] that for other surfaces, the partition function of the dimer model is equal to the sum of 22g Pfaffians. The formal combinatorial proof of this fact and the exact description of coefficients for all oriented surfaces first appeared in [12] and [16] (see also [4]). A

188

D. Cimasoni, N. Reshetikhin

combinatorial proof of such formula for non-orientable surfaces can also be found in [16]. The partition function of free fermions on a Riemann surface of genus g is also a linear combination of 22g Pfaffians of Dirac operators. Each term in this sum corresponds to a spin structure [1]. Assuming that dimer models are discretizations of free fermions on Riemann surfaces, one should expect a relation between Kasteleyn orientations and spin structures and between the Kasteleyn operator for a given Kasteleyn orientation and the Dirac operator in the corresponding spinor bundle. Numerical evidence relating the critical dimer model on a square and triangular lattices in the thermodynamical limit with Dirac operators can be found in [2] for g = 2. An explicit construction relating a spin structure on a surface with a Kasteleyn orientation on a graph with dimer configuration was suggested in [11]. Furthermore, for bipartite graphs with critical weights, the Kasteleyn operator can be naturally identified with a discrete version of the Dirac operator [8]. This gives an interesting relation between dimer models and the theory of discrete meromorphic functions [14]. In this paper we investigate further the relation between Kasteleyn orientations and spin structures, and use this relation to give a geometric proof of the Pfaffian formula for closed surfaces. Below is a brief summary of our main results. Recall some basic notions. A dimer configuration on a graph Γ is a perfect matching on vertices where matched vertices are connected by edges. Given two such configurations D and D
, set ∆(D, D
) = (D ∪ D
) \ (D ∩ D
). A surface graph is a graph Γ embedded into a surface Σ as the 1-squeletton of a CW-decomposition of Σ. A Kasteleyn orientation of a surface graph is an orientation of edges of the graph, such that the product or relative orientations of boundary edges of each face is negative (see Sect. 4). One of our results is that any dimer configuration D on a surface graph Γ ⊂ Σ induces an isomorphism of affine H 1 (Σ; Z2 )-spaces ψ D : (K(Γ )/ ∼) −→ Q(H1 (Σ; Z2 ), ·) , [K ] −→ q DK

(1)

from the set of equivalence classes of Kasteleyn orientations on Γ ⊂ Σ onto the set of quadratic forms on (H1 (Σ; Z2 ), ·), where · denotes the intersection form on Σ. Furthermore, ψ D = ψ D
if and only if D and D
are equivalent dimer configurations (that is, ∆(D, D
) is zero in H1 (Σ; Z2 )). Since the affine space of spin structures on Σ is canonically isomorphic to the affine space of such quadratic forms, this establishes an isomorphism between equivalence classes of Kasteleyn orientations and spin structures. This correspondence implies easily the following identity. Let D0 be a fixed dimer configuration on a graph Γ . Realize Γ as a surface graph Γ ⊂ Σ of genus g. Let K be any Kasteleyn orientation on Γ ⊂ Σ, and let A K be the associated Kasteleyn matrix. Then,
q K (α) Pf(A K ) = ε K (D0 ) (−1) D0 Z α (D0 ), (2) α∈H1 (Σ;Z2 )

where is the quadratic form associated to K and D0 via (1), ε K (D0 ) is some sign depending on K and D0 , and
w(D), Z α (D0 ) = q DK0

D

the sum being on all dimer configurations D such that ∆(D0 , D) is equal to α in H1 (Σ; Z2 ).

Dimers on Surface Graphs and Spin Structures. I

189

It follows that the partition function of a dimer model on Γ is given by Z=

1
Arf(q DK0 )ε K (D0 )Pf(A K ), 2g

(3)

[K ]

where the sum is taken over the 22g equivalence classes of Kasteleyn orientations on Γ ⊂ Σ, and Arf(q DK0 ) = ±1 denotes the Arf invariant of the quadratic form q DK0 . Note that the sign Arf(q DK0 )ε K (D0 ) does not depend on D0 . The paper is organized as follows. In Sect. 2, we introduce the dimer model on a graph Γ and define composition cycles. Section 3 deals with dimers on surface graphs Γ ⊂ Σ and the definition of an equivalence relation for dimer configurations on surface graphs. In Sect. 4, we recall the definition of a Kasteleyn orientation. We then show that a surface graph Γ ⊂ Σ admits such an orientation if and only if the number of vertices of Γ is even. We prove that, in such a case, the set of equivalence classes of Kasteleyn orientations on Γ ⊂ Σ is an affine H 1 (Σ; Z2 )-space. Finally, we give an algorithmic procedure for the construction of the 22g non-equivalent Kasteleyn orientations on a given surface graph Γ ⊂ Σ of genus g. The core of the paper lies in Sect. 5, where we establish the correspondence (1) stated above. This result is used in Sect. 6 to obtain equations (2) and (3). We also give a formula for the local correlation functions of a dimer model. In the appendix, we collect formulae expressing dimer models in terms of Grassman integrals. 2. The Dimer Model 2.1. Dimer configurations and composition cycles on graphs. Let Γ be a finite connected graph. A perfect matching on Γ is a choice of edges of Γ such that each vertex of Γ is adjacent to exactly one of these edges. In statistical mechanics, a perfect matching on Γ is also known as a dimer configuration on Γ . The edges of the perfect matching are called dimers. An example of a dimer configuration on a graph is given in Fig. 1. In order to have a perfect matching, a graph Γ clearly needs to have an even number of vertices. However, there are connected graphs with an even number of vertices but no perfect matching. We refer to [13] for combinatorial aspects of matchings. Throughout the paper and unless otherwise stated, we will only consider finite graphs which admit perfect matchings. In particular, all the graphs will have an even number of vertices. Given two dimer configurations D and D
on a graph Γ , consider the subgraph of Γ given by the symmetric difference (D∪ D
)\(D∩ D
). The connected components of this subgraph are called (D, D
)-composition cycles or simply composition cycles. Clearly,

Fig. 1. A dimer configuration on a graph

190

D. Cimasoni, N. Reshetikhin

Fig. 2. An example of composition cycles: dimers from D are in solid, and dimers from D
are in traced lines. On this example, there is one (D, D
)-composition cycle of length 2, one of length 4, and one of length 6

each composition cycle is a simple closed curve of even length. This is illustrated in Fig. 2, where the two dimer configurations are shown in black and traced lines. 2.2. Edge weight system. Let D(Γ ) denote the set of dimer configurations on a graph Γ . A weight system on D(Γ ) is a positive real-valued function on this set. A weight system w defines a probability distribution on all dimer configurations: Prob(D) = where Z (Γ ; w) =

w(D) , Z (Γ ; w)


w(D)

D

is the partition function. This probability measure is the Gibbs measure for the dimer model on the graph Γ with the weight system w. We shall focus on a particular type of weight system called edge weight system. Assign to each edge e of Γ a positive real number w(e), called the weight of the edge e. The associated edge weight system on D(Γ ) is given by  w(D) = w(e), e∈D

where the product is over all edges occupied by dimers of D. In statistical mechanics, these weights are called Boltzmann weights. Their physical meaning is  E(e)  w(e) = exp − , T where E(e) is the energy of dimer occupying the edge e and T is the temperature. 2.3. Local correlation functions. Let e be an edge of Γ . The characteristic function of e is the function σe on D(Γ ) given by  1 if e ∈ D; σe (D) = 0 otherwise.

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The expectation values of products of characteristic functions are called local correlation functions, or dimer-dimer correlation functions: < σe1 · · · σek >=

Z (e1 , . . . , ek ; Γ ; w) , Z (Γ ; w)

where Z (e1 , . . . , ek ; Γ ; w) =




w(e)

D e∈D

k  i=1

σei (D) =




w(D).

D e1 ,...,ek

Note that < σe1 · · · σek >= 0 if some edges ei = e j share a common vertex. Note also that σe · σe = σe . Therefore, it may be assumed that ei = e j for i = j. For any dimer configuration D,   w(D) , σe = Z (Γ ; w) e∈D

so we can reconstruct the weight system if we know all local correlation functions. In this sense, local correlation functions carry all the information about the Gibbs measure. 3. Dimers on Surface Graphs 3.1. Surface graphs. Let Σ be a connected oriented closed surface. By a surface graph, we mean a graph Γ embedded in Σ as the 1-squeletton of a cellular decomposition X of Σ. We shall assume throughout the paper that the surface Σ is endowed with the counter-clockwise orientation. Any finite connected graph can be realized as a surface graph. Indeed, such a graph Γ always embeds in a closed oriented surface of genus g, for g sufficiently large. If the genus is minimal, one easily checks that Γ induces a cellular decomposition of the surface Σ. In this paper we will focus on graphs embedded into a surface of fixed genus. 3.2. Equivalent dimer configurations. A dimer configuration on a surface graph Γ ⊂ Σ is simply a dimer configuration on the graph Γ . Such a dimer configuration can be regarded as a 1-chain in the cellular chain complex of X with Z2 -coefficients:
cD = e ∈ C1 (X ; Z2 ). 

e∈D

By definition, ∂c D = v∈Γ v ∈ C 0 (X ; Z2 ), the sum being on all vertices v of Γ . Therefore, given any pair of dimer configurations D and D
on Γ ⊂ Σ, c D + c D
is a 1-cycle:
∂(c D + c D
) = ∂c D + ∂c D
= (v + v) = 0 ∈ C0 (X ; Z2 ). v∈Γ

This 1-cycle is nothing but the union of all (D, D
)-composition cycles. Let ∆(D, D
) denote its homology class in H1 (X ; Z2 ) = H1 (Σ; Z2 ). We shall say that two dimer configurations D and D
are equivalent if ∆(D, D
) = 0 in H1 (Σ; Z2 ). Note that these concepts make perfect sense when Γ is the 1-squeletton of any CW-complex, not necessarily the cellular decomposition of an oriented closed surface.

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4. Kasteleyn Orientations on Surface Graphs Let Γ ⊂ Σ be a surface graph. The counter-clockwise orientation of Σ induces an orientation on each 2-cell, or face of X . An orientation K of the edges of Γ is called a Kasteleyn orientation if for each face f of X , 

ε Kf (e) = −1,

(4)

e∈∂ f

where the product is taken over all boundary edges of f , and  ε Kf (e) =

1 if e is oriented by K as the oriented boundary of the face f ; −1 otherwise.

This is illustrated in Fig. 3. Define the operation of orientation changing at a vertex as the one which flips the orientation of all the edges adjacent to this vertex, as illustrated in Fig. 4. It is clear that such an operation brings a Kasteleyn orientation to a Kasteleyn orientation. Let us say that two Kasteleyn orientations are equivalent if they are obtained one from the other by a sequence of orientation changes at vertices.

4.1. Existence of a Kasteleyn orientation. Theorem 1. There exists a Kasteleyn orientation on a surface graph Γ ⊂ Σ if and only if the number of vertices of Γ is even.

Fig. 3. A Kasteleyn orientation on the boundary edges of a face

Fig. 4. Orientation change at a vertex

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Proof. Let ω be any orientation of the edges of Γ , and let cω ∈ C 2 (X ; Z2 ) be defined by the equation  (−1)cω ( f ) = − εωf (e). e∈∂ f

Note that ω is Kasteleyn if and only if cω = 0. Let V , E and F denote the number of vertices, edges and faces in X , respectively. Since Σ is closed, its Euler characteristic is even, and we get the equality mod 2,
0 = χ (Σ) = V + E + F = V + cω ( f ). f ∈F

Here, each edge e contributes to the number cω ( f ), where f is the face whose oriented boundary contains e with the orientation opposite to ω. Therefore, the number of faces f such that cω ( f ) = 1 has the parity of V . Hence, if V is even, then cω ( f ) = 1 for an even number of faces, so cω is a 2-coboundary. In other words, cω = δσ for some σ ∈ C 1 (X ; Z2 ). Let K be the orientation of the edges of Γ which agrees with ω on e if and only if σ (e) = 0. Clearly, c K = 0, so K is a Kasteleyn orientation. Conversely, let us assume that there is a Kasteleyn orientation K . This means that c K ( f ) = 1 for none of the faces. By the argument above, V is even.  A more constructive proof of this result will be given in Sect. 4.3. 4.2. Uniqueness of Kasteleyn orientations. Let V be a vector space. Recall that an affine V -space is a set S endowed with a map S × S → V , (a, b) → a − b such that: i. for every a, b and c in S, we have (a − b) + (b − c) = a − c ; ii. for every b in S, the map S → V given by a → a − b is a bijection. In other words, an affine V -space is a V -torsor: it is a set endowed with a freely transitive action of the abelian group V . Theorem 2. Let Γ ⊂ Σ be a surface graph with an even number of vertices. Then, the set of equivalence classes of Kasteleyn orientations on Γ ⊂ Σ is an affine H 1 (Σ; Z2 )space. Corollary 1. There are exactly 22g equivalence classes of Kasteleyn orientations on Γ ⊂ Σ, where g denotes the genus of Σ. Proof. Let K(Γ ) denote the set of Kasteleyn orientations on Γ ⊂ Σ. By Theorem 1, it is non-empty. Consider the map ϑ : K(Γ ) × K(Γ ) → C 1 (X ; Z2 ) given by ϑ K ,K
(e) = 0 if K and K
agree on the edge e, and ϑ K ,K
(e) = 1 otherwise. Since K and K
are Kasteleyn orientations,   
(−1)ϑ K ,K
(∂ f ) = (−1)ϑ K ,K
(e) = ε Kf (e) · ε Kf (e) = (−1)(−1) = 1 e∈∂ f

e∈∂ f

e∈∂ f

for any face f . Therefore, δϑ K ,K
( f ) = ϑ K ,K
(∂ f ) = 0, that is, ϑ K ,K
is a 1-cocycle. ϑ

Thus, we get a map K(Γ ) × K(Γ ) → H 1 (Σ; Z2 ). Note that ϑ K ,K
+ ϑ K
,K

= ϑ K ,K

for any K , K
, K

∈ K(Γ ). Also, one easily checks that ϑ K ,K
= 0 in H 1 (Σ; Z2 ) if

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and only if there is a sequence of vertices such that K is obtained from K
by reversing the orientation around all these vertices, i.e., if and only if K ∼ K
. It follows that we have a map (K(Γ )/ ∼) × (K(Γ )/ ∼) −→ H 1 (Σ; Z2 ), ([K ], [K
]) → [K ] − [K
] := [ϑ K ,K
] such that for any K
in K(Γ ), the map (K(Γ )/ ∼) → H 1 (Σ; Z2 ) given by [K ] → [K ]−[K
] is injective. Finally, let us check that this map is onto. Fix a class in H 1 (Σ; Z2 ), and represent it by some 1-cycle σ ∈ Z 1 (X ; Z2 ). Let K be the same orientation as K
whenever σ (e) = 0, and the opposite when σ (e) = 1. Obviously, ϑ K ,K
= σ . Furthermore, K is Kasteleyn since K
is and δσ = 0. Indeed, given a face f , 0 = (δσ )( f ) = σ (∂ f ) = ϑ K ,K
(∂ f ), so 1 =



e∈∂ f (−1)

ϑ K ,K
(e)

= (−1)

e∈∂ f

ε Kf (e). This concludes the proof.



4.3. How to construct Kasteleyn orientations. Given a surface graph Γ ⊂ Σ with an even number of vertices, we know that there are exactly 22g non-equivalent Kasteleyn orientations on Γ . However, the proof given above is not really constructive. For this reason, we now give an algorithm for the construction of these Kasteleyn orientations. The successive steps of the algorithm are illustrated in Fig. 5. 0. Let Γ ⊂ Σ be a surface graph with an even number of vertices, and let g denote the genus of Σ. 1. Consider a system α = α1 ∪ · · · ∪ α2g of simple closed curves on Γ such that Σ cut along α is a 2-disc Σ
. (Note that such curves exist since Γ induces a cellular decomposition of Σ.) 2. The surface graph Γ ⊂ Σ induces a graph Γ
⊂ Σ
which is the 1-squeletton of a cellular decomposition of the 2-sphere S 2 . Fix a spanning tree T of the graph dual to the surface graph Γ
⊂ S 2 , rooted at the vertex corresponding to the face S 2 \Σ
. 0.

1.

α2 α1

Γ ⊂Σ 2.

3.

4.

.

T

Γ ⊂ S2

e∗

Fig. 5. An example of the explicit construction of a Kasteleyn orientation

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3. Orient the edges of Γ
which do not intersect T respecting the following condition: whenever two edges of Γ
in ∂Σ
are identified in Σ, their orientations agree. Now, every edge in ∂Σ
is oriented, except one. Let us denote it by e∗ . 4. Orient edges of Γ
such that the Kasteleyn condition (4) holds for the faces corresponding to the leaves of T . Moving down the tree (from the leaves to the root), orient each crossing edge such that the Kasteleyn condition holds for all faces left behind. This gives a Kasteleyn orientation of Γ
⊂ Σ
. By the condition in step 3, it induces a Kasteleyn orientation on Γ ⊂ Σ, provided that the orientation of the last edge e∗ satisfies this condition. It turns out to be the case if and only if the number of vertices of Γ is even. (This is an easy consequence of Theorem 1.) Therefore, we have constructed a Kasteleyn orientation K on Γ ⊂ Σ. To obtain the 22g non-equivalent ones, proceed as follows. 5. Consider a family of simple closed curves β1 , . . . , β2g on Σ avoiding the vertices of Γ , and forming a basis of H1 (Σ; Z2 ). 6. Consider the Kasteleyn orientation K , and some subset I ⊂ {1, . . . , 2g}. For all i ∈ I , change the orientation of all the edges in Γ that intersect βi . The resulting orientation K I is clearly Kasteleyn, as ∂ f · βi is even for every face f and index i. Furthermore, one easily checks that K I and K J are non-equivalent if I = J . Hence, we have constructed the 22g non-equivalent Kasteleyn orientations on Γ ⊂ Σ. 5. Kasteleyn Orientations as Discrete Spin Structures We saw in Corollary 1 that there are exactly 22g non-equivalent Kasteleyn orientations on a surface graph Γ ⊂ Σ, where g denotes the genus of Σ. It is known that this is also the number of non-equivalent spin structures on Σ. This relation between the number of Kasteleyn orientations and the number of spin structures is not accidental. Kasteleyn orientations of surface graphs can be regarded as discrete versions of spin structures. This statement will be made precise in the present section. (See in particular Corollary 3.) 5.1. The quadratic form associated to a Kasteleyn orientation. Let V be a finite dimensional vector space over the field Z2 , and let ϕ : V × V → Z2 be a fixed bilinear form. Recall that a function q : V → Z2 is a quadratic form on (V, ϕ) if q(x + y) = q(x) + q(y) + ϕ(x, y) for all x, y ∈ V . Note that the difference (that is, the sum) of two quadratic forms on (V, ϕ) is a linear form on V . Therefore, one easily checks that the set Q(V, ϕ) of quadratic forms on (V, ϕ) is an affine V ∗ -space, where V ∗ denotes the dual of V . Fix a Kasteleyn orientation K on a surface graph Γ ⊂ Σ. Given an oriented simple closed curve C on Γ , set  ε K (C) = εCK (e), e∈C

where εCK (e) is equal to +1 (resp. −1) if the orientations on the edge e given by C and K agree (resp. do not agree). For a fixed dimer configuration D on Γ , let  D (C) denote the number of vertices v in C whose adjacent dimer of D sticks out to the left of C in Σ.

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Theorem 3. Given a class α ∈ H1 (Σ; Z2 ), represent it by oriented simple closed curves C1 , . . . , Cm in Γ . If K is a Kasteleyn orientation on Γ ⊂ Σ, then the function q DK : H1 (Σ; Z2 ) → Z2 given by K (α) qD

(−1)

= (−1)

 i< j

Ci ·C j

m  (−ε K (Ci ))(−1) D (Ci ) i=1

is a well-defined quadratic form on (H1 (Σ; Z2 ), ·), where · denotes the intersection form. We postpone the proof of this result to the next subsections. Let us first investigate some of its consequences. Proposition 1. (i) Let D be a fixed dimer configuration on Γ . If K and K
are two
Kasteleyn orientations on Γ ⊂ Σ, then q DK − q DK maps to [K ] − [K
] via the canonical isomorphism Hom(H1 (Σ; Z2 ); Z2 ) = H 1 (Σ; Z2 ). (ii) Let K be a fixed Kasteleyn orientation on Γ ⊂ Σ. If D and D
are two dimer configurations on Γ , then q DK − q DK
∈ Hom(H1 (Σ; Z2 ); Z2 ) is given by α → α · ∆(D, D
). Proof. Let C be a simple closed curve in Γ representing a class α in H1 (Σ; Z2 ). By definition, K
)(α)

(−1)(q D −q D K

=



εCK (e)

e∈C

 e∈C




εCK (e) =



(−1)ϑ K ,K
(e) = (−1)ϑ K ,K
(C) .

e∈C

This proves the first point. To check the second one, observe that (−1)(q D −q D
)(α) = (−1) D (C)+ D
(C) . K

K

Clearly,  D (C) +  D
(C) ≡ C · ∆(D, D
) (mod 2), giving the proposition.



Corollary 2. Any dimer configuration D on a surface graph Γ ⊂ Σ induces an isomorphism of affine H 1 (Σ; Z2 )-spaces ψ D : (K(Γ )/ ∼) −→ Q(H1 (Σ; Z2 ), ·) , [K ] −→ q DK from the set of equivalence classes of Kasteleyn orientations on Γ ⊂ Σ onto the set of quadratic forms on (H1 (Σ; Z2 ), ·). Furthermore, ψ D = ψ D
if and only if D and D
are equivalent dimer configurations. Proof. The first part of Proposition 1 exactly states that ψ D is an isomorphism of affine H 1 (Σ; Z2 )-spaces. By the second part, ψ D = ψ D
if and only if the homomorphism H1 (Σ; Z2 ) → Z2 given by the intersection with ∆(D, D
) is zero. By Poincaré duality, this is the case if and only if ∆(D, D
) = 0. 

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5.2. Spin structures on surfaces. Recall that the fundamental group of S O(n) is infinite cyclic if n = 2 and cyclic of order 2 if n ≥ 3. Hence, S O(n) admits a canonical 2-fold cover, denoted by Spin(n) → S O(n). Let M be an oriented n-dimensional Riemannian manifold, and let PS O → M be the principal S O(n)-bundle associated to its tangent bundle. A spin structure on M is a principal Spin(n)-bundle P → M together with a 2-fold covering map P → PS O which restricts to the covering map Spin(n) → S O(n) on each fiber. Equivalently, a spin structure on M is a cohomology class ξ ∈ H 1 (PS O ; Z2 ) whose restriction to each fiber F gives the generator of the cyclic group H 1 (F; Z2 ). It is well-known that such a spin structure exists if and only if the second StiefelWhitney class of M vanishes. In such a case, the set S(M) of spin structures on M is endowed with a natural structure of affine H 1 (M; Z2 )-space. The 2-dimensional case is particularly easy to deal with for several reasons. First of all, any compact orientable surface Σ admits a spin structure, as its second StiefelWhitney class is always zero. Furthermore, spin structures can be constructed using a certain class a vector fields, that we now describe. Let f be a non-vanishing vector field on Σ \ σ , where σ is some finite subset of Σ. Recall that the index of the singularity x ∈ σ of f is defined as the degree of the circle map t → f (γx (t))/| f (γx (t))|, where γx : S 1 → Σ \ σ is a (counter-clockwise) parametrization of a simple closed curve separating x from the other singularities of f . Let us denote by Vev (Σ) the set of vector fields on Σ with only even index singularities. We claim that any such vector field f defines a spin structure ξ f on Σ. Indeed, consider a 1-cycle c in PS O (that is, a closed framed curve in Σ) and let us assume that c avoids the singularities of f . Then, let ξ f (c) ∈ Z2 be the winding number modulo 2 of f along c with respect to the framing of c. Since all the singularities of f have even index, ξ f (c) = 0 if c is a 1-boundary, and ξ f (c) = 1 if c is a small simple closed curve with tangential framing. Therefore, it induces a well-defined cohomology class ξ f ∈ Hom(H1 (PS O ; Z2 ); Z2 ) = H 1 (PS O ; Z2 ) which restricts to the generator of the cohomology of the fibers. So ξ f is a spin structure on Σ, and we have a map Vev (Σ) −→ S(Σ) ,

f −→ ξ f .

We shall need one last result about spin structures on surfaces, due to D. Johnson [5]. Given a spin structure ξ ∈ S(Σ), let qξ : H1 (Σ; Z2 ) → Z2 be the function defined as follows. Represent α ∈ H1 (Σ; Z2 ) by a collection of disjoint regular simple closed curves γ1 , . . . , γm : S 1 → Σ. For all i and all t ∈ S 1 , complete the unit tangent vector γ˙i (t)/|γ˙i (t)| to a positive orthonormal basis of Tγi (t) Σ. This gives disjoint framed closed curves in Σ, that is, a 1-cycle c in PS O . Set qξ (α) = ξ(c) + m. Johnson’s theorem asserts that qξ is a well-defined quadratic form on (H1 (Σ; Z2 ), ·), where · denotes the intersection form. Furthermore, the map S(Σ) −→ Q(H1 (Σ; Z2 ), ·) , ξ −→ qξ is an isomorphism of affine H 1 (Σ; Z2 )-spaces. 5.3. Proof of Theorem 3. Let Γ ⊂ Σ be a surface graph, with Σ counter-clockwise oriented. Given a Kasteleyn orientation K and a dimer configuration D on Γ ⊂ Σ, Kuperberg [11] constructs a vector field f (K , D) ∈ Vev (Σ) as follows. Around each vertex of Γ , make the vectors point to the vertex. At the middle of each edge, make the

198

D. Cimasoni, N. Reshetikhin

Fig. 6. Kuperberg’s construction

vector point 90 degrees clockwise relative to the orientation K of the edge. Extend this continuously to the whole edges, as described in Fig. 6. The Kasteleyn condition (4) ensures that the vector field extends to the faces with one singularity of even index in the interior of each face. However, this vector field f˜(K ) has an odd index singularity at each vertex of Γ . This is where the dimer configuration D enters the game: contract the odd index singularities in pairs along the dimers of D. The resulting vector field f (K , D) has even index singularities: one in the interior of each face of Σ, and one in the middle of each dimer of D. Gathering the results of the previous section and Kuperberg’s construction, we get the following composition of maps: K(Γ ) × D(Γ ) −→ Vev (Σ) −→ S(Σ) −→ Q(H1 (Σ; Z2 ), ·) , (K , D) −→ qξ f (K ,D) . We are left with the proof that, given any Kasteleyn orientation K and dimer configuration D, the resulting quadratic form q = qξ f (K ,D) coincides with the function q DK defined in the statement of Theorem 3. So, given α ∈ H1 (Σ; Z2 ), represent it by a collection of simple closed curves C1 , . . . , Cm in Γ . (This is always possible m as Γ induces a cellular decomposition of Σ.) Since q is a quadratic form and α = i=1 [Ci ], (−1)q(α) = (−1)

 i< j

Ci ·C j

m 

(−1)q([Ci ]) .

i=1

Therefore, we just need to check that if C is an oriented simple closed curve in Γ , then (−1)q([C])+1 = ε K (C)(−1) D (C) . Consider the oriented regular curve γ in Σ which follows C slightly on its left, and goes around the middle of each dimer it meets, except if it meets the same dimer twice. In this case, γ stays close to C, as illustrated in Fig. 7. Clearly, γ is a regular oriented simple closed curve in Σ. Furthermore, it is homologous to C and it avoids all the singularities of f (K , D). We now have to check that the winding number ω of f (K , D) along γ with respect to its tangential framing satisfies (−1)ω = ε K (C)(−1) D (C) . One easily checks that ω is equal (mod 2) to the winding number ω0 of f (K , D) along γ0 , where γ0 is the regular curve which goes around the middle of each dimer it meets, including the ones it meets twice. By construction of f (K , D), ω0 is equal to the winding number ω˜ of f˜(K ) along γ˜ , where γ˜ is the regular curve which avoids all the dimers of D and all the vertices of Γ , as described in Fig. 7. The latter winding number ω˜ can be computed locally by cutting γ˜ into pieces: one piece γ˜e for each edge e of C, and one piece γ˜v for each vertex v of Γ adjacent to a

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199

Fig. 7. The oriented simple closed curve C ⊂ Γ and its associated regular curves γ , γ0 and γ˜ . The curve C is in solid, the dimers are in traced lines

Fig. 8. Computation of the local winding numbers

dimer of D sticking out to the left of C. The theorem now follows from the following case study. 1. Let us assume that e is an edge of C such that εCK (e) = +1. In this case, the vector field f˜(K ) along γ˜e in the tangential framing of γ˜e defines a curve which is homotopically trivial, as illustrated in Fig. 8. Hence, its contribution to ω˜ is null. 2. Consider now the case of an edge e of C such that εCK (e) = −1. This time, f˜(K ) along γ˜e defines a simple close curve around the origin (see Fig. 8). Its contribution to ω˜ is equal to 1 (mod 2). 3. Let v be a vertex of C with a dimer of D sticking out of v to the left of C. Then, the vector field f˜(K ) along γ˜v induces a simple closed curve around the origin, so its contribution to ω˜ is equal to 1 (mod 2). The case illutrated in Fig. 8 is when K orients the dimer from v to its other boundary vertex. The other case is similar. Gathering all the pieces, the winding number of f˜(K ) along γ˜ is equal to e∈C εCK (e) (−1) D (C) . This concludes the proof of Theorem 3.  Using Johnson’s theorem, we have the following immediate consequence of Corollary 2. Corollary 3. Any dimer configuration D on a surface graph Γ ⊂ Σ induces an isomorphism of affine H 1 (Σ; Z2 )-spaces ψ D : (K(Γ )/ ∼) −→ S(Σ)

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D. Cimasoni, N. Reshetikhin

from the set of equivalence classes of Kasteleyn orientations on Γ ⊂ Σ onto the set of spin structures on Σ. Furthermore, ψ D = ψ D
if and only if D and D
are equivalent dimer configurations.

6. Pffafian Formulae for the Partition Function and Correlation Functions 6.1. The Kasteleyn matrix and its Pfaffian. Let K be a Kasteleyn orientation on a surface graph Γ ⊂ Σ with an even number of vertices. Enumerate these vertices by 1, 2, . . . , N = 2n. The Kasteleyn matrix is the 2n × 2n skew-symmetric matrix A K (Γ ; w) = A K whose entry aiKj is the total weight of all edges from i to j minus the total weight of all edges from j to i. More formally, aiKj =




εiKj (e)w(e),

e

where the sum is on all edges e in Γ between the vertices i and j, εiKj (e) = 0 if i = j, and  1 if e is oriented by K from i to j; εiKj (e) = −1 otherwise, if i = j. Consider a dimer configuration D on Γ given by edges e1 , . . . , en matching vertices i  and j for  = 1, . . . , n. It determines an equivalence class of permutations σ : (1, . . . , 2n) → (i 1 , j1 , . . . , i n , jn ) with respect to permutations of pairs (i  , j ) and transpositions (i  , j ) → ( j , i  ). We will write this as σ ∈ D. Given such a permutation σ , define n 

ε K (D) = (−1)σ

=1

εiK j (e ),

where (−1)σ denotes the sign of the permutation σ . Note that this expression does not depend on the choice of σ ∈ D, but only on the dimer configuration D. Theorem 4. Let Γ ⊂ Σ be a surface graph with an even number of vertices. For any dimer configuration D0 on Γ and any Kasteleyn orientation K on Γ ⊂ Σ, ε K (D0 )Pf(A K ) =




K (α) qD

(−1)

0

Z α (D0 ),

α∈H1 (Σ;Z2 )

where q DK0 is the quadratic form on H1 (Σ; Z2 ) associated to K and D0 , and Z α (D0 ) =




w(D),

D

the sum being on all dimer configurations D such that ∆(D0 , D) = α.

(5)

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201

Proof. Recall that the Pfaffian of a skew-symmetric matrix A = (ai j ) of size 2n is given by
Pf(A) = (−1)σ aσ (1)σ (2) · · · aσ (2n−1)σ (2n) , [σ ]∈Π

where the sum is on the set Π of matchings of {1, . . . , 2n}. Therefore,
Pf(A K ) = (−1)σ aσK(1)σ (2) · · · aσK(2n−1)σ (2n) [σ ]∈Π

=




(−1)σ

[σ ]∈Π

=

n  =1

D

=

εσK(1)σ (2) (e1 )w(e1 ) · · ·

e1


(−1)σ








εσK(2n−1)σ (2n) (en )w(en )

en

εσK(2−1)σ (2) (e )w(e )

ε (D)w(D), K

D

where the sum is on all dimer configurations D on Γ . Hence,
ε K (D0 )Pf(A K ) = ε K (D0 )ε K (D)w(D). D

Let us denote by e1 , . . . , en the edges of Γ occupied by dimers of D, and by e10 , . . . , en0 the edges occupied by dimers of D0 . Fix permutations σ and τ representing the dimer configurations D and D0 , respectively, and set ν = τ ◦ σ −1 . By definition, n 

ε K (D0 )ε K (D) = (−1)τ = (−1)ν

=1 n  =1

ετK(2−1)τ (2) (e0 ) · (−1)σ

n  =1

εσK(2−1)σ (2) (e )

K 0 K εν(σ (2−1))ν(σ (2)) (e )εσ (2−1)σ (2) (e ).

Note that the permutation ν depends on the choice of σ ∈ D and τ ∈ D0 , but it always brings the perfect matching D to D0 . Moreover, one can choose representatives σ ∈ D and τ ∈ D0 such that ν is the counter-clockwise rotation by one edge of every (D0 , D)composition cycle C1 , . . . , Cm . For this particular choice of representatives, we have ε K (D0 )ε K (D) = (−1)

m 

m

i=1 (length(Ci )+1)

ε K (Ci ) =

i=1

m 

(−ε K (Ci )).

i=1

Here, we use the fact that the length of a permutation cycle is the length of the corresponding composition cycle, and that the length of each composition cycle is even. Recall the quadratic form q DK0 of Theorem 3. Since the Ci ’s are disjoint (D0 , D)-composition cycles, Ci · C j = 0 and  D0 (Ci ) = 0 for all i, j. Therefore, m  i=1

The theorem follows.



K (∆(D ,D)) qD 0

(−ε K (Ci )) = (−1)

0

.

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D. Cimasoni, N. Reshetikhin

6.2. The partition function. Given a dimer configuration D0 , Theorem 4 provides 22g linear equations (one for each equivalence class of Kasteleyn orientation) with 22g unknowns (the functions Z α (D0 )). We want to use these equations to express the dimer partition function Z = α Z α (D0 ) in terms of Pfaffians. Let V × V → Z2 , (α, β) → α · β be a non-degenerate bilinear form on a Z2 -vector space V . Recall that the Arf invariant of a quadratic form q : V → Z2 on (V, ·) is given by Arf(q) =

1
(−1)q(α) . |V | α∈V

Lemma 1. Let q, q
be two quadratic forms on (V, ·). Then,


Arf(q)Arf(q
) = (−1)q(∆) = (−1)q (∆) , where ∆ ∈ V satisfies (q + q
)(α) = ∆ · α for all α ∈ V . Proof. First note that q + q
is a linear form on V . Since the bilinear form · is nondegenerate, there exists ∆ ∈ V such that (q + q
)(α) = α · ∆ for all α ∈ V . Furthermore,
(q + q
)(∆) = ∆ · ∆ = 0, so (−1)q(∆) = (−1)q (∆) . Let us now compute the product of the Arf invariants: 1
1


(−1)q(α)+q (β) = (−1)q(α)+q (β+∆) . Arf(q)Arf(q
) = |V | |V | α,β∈V

α,β∈V

Using the equality q(α) + q
(β + ∆) = q(α) + q
(β) + q
(∆) + β · ∆ = q(α) + q(β) + q
(∆) = q(α + β) + α · β + q(∆), we obtain Arf(q)Arf(q
) =

(−1)q(∆)
(−1)q(α+β)+α·β |V | α,β

=

(−1)q(∆) 
|V |

α

1+




 (−1)q(α+β)+α·β .

α =β

We are left with the proof that the latter sum is zero:



(−1)q(α+β)+α·β = (−1)q(γ ) (−1)α·β = (−1)q(γ ) (n 0γ − n 1γ ), α =β

γ =0

α+β=γ

γ =0

where n iγ is the cardinality of the set Nγi = {(α, β) ∈ V × V | α + β = γ and α · β = i} for i = 0, 1. Since γ = 0 and (V, ·) is non-degenerate, there exists x ∈ V such that x · γ = 1. Then, the map (α, β) → (α + x, β + x) induces a bijection Nγ0 → Nγ1 . Hence n 0γ = n 1γ for all γ = 0, and the lemma is proved. 

Dimers on Surface Graphs and Spin Structures. I

203

Using this lemma, it is easy to check that if dim(V ) = 2n, then there are exactly 22n−1 + 2n−1 quadratic forms on (V, ·) with Arf invariant 1 and 22n−1 − 2n−1 forms with Arf invariant −1. Theorem 5. The partition function of a dimer model on a surface graph Γ ⊂ Σ for a closed surface of genus g is given by the formula 1
Z= g Arf(q DK0 )ε K (D0 )Pf(A K ), (6) 2 [K ]

where the sum is taken over all equivalence classes of Kasteleyn orientations. Each summand is defined for a Kasteleyn orientation but it depends only on its equivalence class. Furthermore, the sign Arf(q DK0 )ε K (D0 ) does not depend on D0 . Proof. Recall that there is a free, transitive action of H 1 (Σ; Z2 ) on the set of equivalence classes of Kasteleyn orientations on Γ ⊂ Σ (Theorem 2). Let us denote by K φ the result of the action of φ ∈ H 1 (Σ; Z2 ) on a Kasteleyn orientation K . Then, by the first part of Proposition 1, (−1)

Kφ

K +q (q D D )(α) 0

0

= (−1)φ(α) =: χα (φ).

Here χα ’s are characters of irreducible representations of H 1 (Σ; Z2 ). By Eq. (5),


ε K φ (D0 )Pf(A K φ ) =

Kφ

q D (α)

(−1)

Z α (D0 )

0

α∈H1 (Σ;Z) Kφ

q D (β)

for all φ ∈ H 1 (Σ; Z). Multiplying these equations by (−1) over all φ ∈ H 1 (Σ; Z), we get


Kφ

q D (β) K φ 0

(−1)

ε

φ

α

φ




=

K (α)+q K (β) qD D

(−1)

0




0

α

Z α (D0 ) =



φ

and taking the sum

Kφ Kφ

q (α)+q D (β) 0 (−1) D0 Z α (D0 )

(D0 )Pf(A K φ ) =

Using the orthogonality formula

0

χα (φ)χβ (φ)Z α (D0 ).

φ

χα (φ)χβ (φ) = 22g δαβ , we obtain

Kφ 1
q D (α) K φ 0 (−1) ε (D0 )Pf(A K φ ). 22g

φ

Therefore, the partition function Z = Z=

1 2g



α

Z α (D0 ) is given by




σ K φ Pf(A K φ ),

φ∈H 1 (Σ;Z2 )

where σ K φ = ε K φ (D0 )

1 2g


α∈H1 (Σ;Z2 )

Kφ

q D (α)

(−1)

0

K

= ε K φ (D0 )Arf(q D0φ ).

204

D. Cimasoni, N. Reshetikhin

This gives the formula Z=

1 2g




K

Arf(q D0φ )ε K φ (D0 )Pf(A K φ ).

φ∈H 1 (Σ;Z2 )

Since H 1 (Σ, Z2 ) acts transitively and freely on the equivalence classes of Kasteleyn orientations, equality (6) follows.
If K and K
are equivalent Kasteleyn orientations, then q DK0 = q DK0 . Therefore,





Arf(q DK0 ) = Arf(q DK0 ). On the other hand, ε K (D0 ) = (−1)µ ε K (D0 ) and Pf(A K ) = (−1)µ Pf(A K
), where µ is the number of vertices of Γ around which the orientation was flipped. Therefore, the summand Arf(q DK0 )ε K (D0 )Pf(A K ) does not depend on the choice of the representative in the equivalence class [K ]. Let us finally check that the sign Arf(q DK0 )ε K (D0 ) does not depend on D0 . Let D be another dimer configuration on Γ . By Proposition 1, Lemma 1 and the proof of Theorem 4, A(q DK0 )ε K (D0 )A(q DK )ε K (D) = (−1)q D (∆(D0 ,D)) Arf(q DK0 )A(q DK ) = 1. K

This concludes the proof of the theorem.



6.3. Local correlation functions. In order to express local correlation functions < σe1 · · · σek > as combinations of Pfaffians, let us recall several facts of linear algebra. Let A = (ai j ) be a matrix of size 2n. Given an ordered subset I of the ordered set α = (1, . . . , 2n), let A I denote the matrix obtained from A by removing the i th row and the i th column for all i ∈ I . Also, let (−1)σ (I ) denote the signature of the permutation which sends α to the ordered set I (α\I ). If A is skew-symmetric, then for all ordered sets of indices I = (i 1 , j1 , . . . , i k , jk ), ∂ k Pf(A) = (−1)σ (I ) Pf(A I ). ∂ai1 j1 · · · ∂aik jk Furthermore, if A is invertible, then Pf(A) = 0 and (−1)σ (I ) Pf(A I ) = (−1)k Pf(A)Pf((A−1 )α\I ). So, let e1 , . . . , ek be edges of the graph Γ , and let i  , j be the two boundary vertices of e for  = 1, . . . , k. For simplicity, we shall assume that Γ has no multiple edges. Finally, set I = (i 1 , j1 , . . . , i k , jk ). Applying the identities above to the Kasteleyn matrices Aφ = A K φ , Theorem 5 gives k  =1

w(e )

k 1
Kφ  φ ∂k ∂k Z = g σ ai j φ Pf(Aφ ) φ ∂w(e1 ) · · · ∂w(ek ) 2 ∂a · · · ∂a φ =1 i 1 j1 i k jk k 1
Kφ  φ φ = g σ ai j (−1)σ (I ) Pf(A I ). 2 φ

=1

Dimers on Surface Graphs and Spin Structures. I

205

If the Kasteleyn matrix is invertible for any Kasteleyn orientation of Γ , this expression is equal to k (−1)k
K φ  φ σ ai j Pf(Aφ )Pf((Aφ )−1 α\I ). 2g φ

=1

Since < σe1 · · · σek >=

∂k Z w(e1 ) · · · w(ek ) , Z ∂w(e1 ) · · · ∂w(ek )

the correlation functions are given by  < σe1 · · · σek >= (−1)

k

φ

φ σ K φ Pf(Aφ )  ai j Pf((Aφ )−1 α\I ) .  K φ φ Pf(A ) φσ

if the Kasteleyn matrix is invertible for all possible Kasteleyn orientations of Γ . If Γ is a planar graph, then the Kasteleyn matrix is always invertible since its Pfaffian is equal to the partition function. Therefore, < σe1 · · · σek >= (−1)k aiK1 j1 · · · aiKk jk Pf((A K )−1 α\I ), where K is any Kasteleyn orientation on Γ ⊂ S 2 . On the other hand, there are graphs where some Kasteleyn matrix is not invertible. For example, consider a square lattice on a torus. Then, the Kasteleyn matrix corresponding to the spin structure with Arf invariant −1 is not invertible (see [15]). A. Dimers and Grassman Integrals A.1 Grassman integrals and Pfaffians. Let V be an n-dimensional vector space. Its exterior algebra ∧V = ⊕nk=0 ∧k V is called the Grassman algebra of V . The choice of a linear basis in V induces an isomorphism between ∧V and the algebra generated by elements φ1 , . . . , φn with defining relations φi φ j = −φ j φi . The isomorphism identifies the linear basis in V with the generators φi . Choose an orientation on V . Together with the basis in V , this defines a basis in the top exterior power of V . The integral over the Grassman algebra of V of an element a ∈ ∧V is the coordinate of a in the top exterior power of V with respect to this basis. It is denoted by

a dφ. In physics, elements φ are called Fermionic fields. More precisely, they are called neutral fermionic fields (or neutral fermions). They are called charged fermions if there is an action of U (1) on V . Recall that the Pfaffian of a skew symmetric matrix A of even size n is given by Pf(A) =


1 (−1)σ aσ (1)σ (2) · · · aσ (n−1)σ (n) , n 2 2 ! σ ∈Sn n 2

206

D. Cimasoni, N. Reshetikhin

where the sum is over all permutations σ ∈ Sn . (This formula is easily seen to be equivalent to the one stated in Sect. 6.) Expanding the exponent into a power series, one gets the following identity in ∧V : 

π exp

n 1


2

ai j φi ∧ φ j



= Pf(A) φ1 ∧ · · · ∧ φn ,

i, j=1

where π : ∧V → ∧n V is the projection to the top exterior power. In terms of Grassmann integral, this can be expressed as

n  1
(7) φi ai j φ j dφ = Pf(A). exp 2 i, j=1

Let us now assume that the space V is polarized, i.e. that V = W ⊕ W ∗ with W some vector space and W ∗ its dual. Then, the Grassman algebra of V is isomorphic to the tensor product of Grassman algebras for W and for W ∗ in the category of super-vector spaces. That is, if the dimension of W is k, the Grassman algebra of V is isomorphic to the algebra generated by ψi , ψi∗ , i = 1, . . . k with defining relations ψi ψ j = −ψ j ψi , ψi ψ ∗j = −ψ ∗j ψi , and ψi∗ ψ ∗j = −ψ ∗j ψi∗ . Such an isomorphism is specified by the choice of a linear basis in W . Note that such a choice induces a basis in V (take the dual basis in W ∗ ) and an orientation on V given by the ordering ψ1 , . . . , ψn , ψ1∗ , . . . , ψn∗ . The Grassmann integral of an element a ∈ ∧V with respect to this choice of basis in W is denoted by

a dψdψ ∗ . Expanding the exponent, we obtain

exp

k 


 k(k−1) ψi ai j ψ ∗j dψdψ ∗ = (−1) 2 Det(A).

(8)

i, j=1

Comparing this formula with (7), we obtain the following well known identity 

k(k−1) 0 A = (−1) 2 Det(A). Pf −At 0 Now consider the space U = V ⊕ V ∗ . Let φ1 , . . . , φn be a basis in V and φ1∗ , .√. . , φn∗ √ φ + −1φ ∗

be the dual basis in V ∗ . Using the change of variables χi = i √ i , χi∗ = 2 together with the equalities (7) and (8), we obtain

n n  1
1
2 Pf(A) = exp ai j φi φ j + ai j φi∗ φ ∗j dφdφ ∗ 2 2 i, j=1

= (−1)

n(n−1) 2



exp

i, j=1

n 


 χi ai j χ ∗j dχ dχ ∗

i, j=1

= Det(A). One can easily derive other Pfaffian identites in a similar way .

φi − −1φi∗ √ , 2

Dimers on Surface Graphs and Spin Structures. I

207

A.2 Dimer models on graphs and Grassmann integrals. Let Γ ⊂ Σ be a surface graph, and let A K = (aiKj ) be the Kasteleyn matrix associated with a Kasteleyn orientation K on Γ ⊂ Σ. By Theorem 5 and identity (7), the partition function for dimers on Γ is given by

 1
1
Z= g Arf(q DK0 )ε K (D0 ) exp φi aiKj φ j dφ, 2 2 [K ]

i, j∈V (Γ )

where V (Γ ) denotes the set of vertices of Γ . Recall that ε K (D0 ) = (−1)σ n=1 εiK j , where the dimer configuration D0 matches vertices i  and j for  = 1, . . . , n, and σ denotes the permutation (1, . . . , 2n) → (i 1 , j1 , . . . , i n , jn ). Taking into account the identity (−1)σ

n  =1

εiK j dφ1 . . . dφ2n =

n  =1

εiK j dφi dφ j ,

the formula for the partition function can be written as

 1
1
Z= g φi aiKj φ j D K φ, exp 2 2 [K ]

i, j∈V (Γ )

where D K φ = Arf(q DK0 )ε K (D0 ) dφ = Arf(q DK0 )

n  =1

εiK j dφi dφ j .

Similarly, local correlation functions can be written as < σe1 · · · σek

k  1
1
K >= g φi ai j φ j aiKl jl φil φ jl D K φ. exp 2 Z 2 [K ]

i, j ∈I /

l=1

Here il and jl are the boundary vertices of the edge el , and I = (i 1 , j1 , . . . , i k , jk ). Acknowledgements. We are grateful to Peter Teichner and Andrei Okounkov for discussions, and to Richard Kenyon and Greg Kuperberg for useful remarks and comments. We also thank Faye Yaeger who was kind to type a large part of this paper. Finally, we thankfully acknowledge the hospitality of the Department of Mathematics of the University of Aarhus, via the Niels Bohr initiative. The work of D.C. was supported by the Swiss National Science Foundation. The work of N.R. was supported by the NSF grant DMS-0307599, by the CRDF grant RUM1-2622, and by the Humboldt Foundation.

References 1. Álvarez-Gaumé, L., Bost, J.-B., Moore, G., Nelson, P., Vafa, C.: Bosonization on higher genus Riemann surfaces. Commun. Math. Phys. 112, 503–552 (1987) 2. Costa-Santos, R., McCoy, B.: Dimers and the critical Ising model on lattices of genus > 1. Nucl. Phys. B 623, 439–473 (2002) 3. Cohn, H., Kenyon, R., Propp, J.: A variational principle for domino tilings. J. Amer. Math. Soc. 14, 297–346 (2001) 4. Dolbilin, N., Zinovyev, Yu., Mishchenko, A., Shtanko, M., Shtogrin, M.: Homological properties of twodimensional coverings of lattices on surfaces. (Russian) Funktl. Anal. i Pril. 30, 19–33 (1996); translation in Funct. Anal. Appl. 30, 163–173 (1996)

208

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5. 6. 7. 8.

Johnson, D.: Spin structures and quadratic forms on surfaces. J. London Math. Soc. (2) 22, 365–373 (1980) Kasteleyn, W.: Dimer statistics and phase transitions. J. Math. Phys. 4, 287–293 (1963) Kasteleyn, W.: Graph Theory and Theoretical Physics. London: Academic Press, 1967, pp. 43–110 Kenyon, R.: The Laplacian and Dirac operators on critical planar graphs. Invent. Math. 150, 409–439 (2002) Kenyon, R., Okounkov, A.: Planar dimers and Harnack curves. Duke Math. J. 131, 499–524 (2006) Kenyon, R., Okounkov, A., Sheffield, S.: Dimers and amoebae. Ann. of Math. (2) 163, 1019–1056 (2006) Kuperberg, G.: An exploration of the permanent-determinant method. Electron. J. Combin. 5, Research Paper 46, (1998) 34 pp. (electronic) Galluccio, A., Loebl, M.: On the theory of Pfaffian orientations. I. Perfect matchings and permanents. Electron. J. Combin. 6, Research Paper 6 (1999), 18 pp. (electronic) Lovasz, L., Plummer, M.D.: Matching theory North-Holland Mathematics Studies, 121, Annals of Discrete Mathematics, 29. Amsterdam: North-Holland Publishing Co., 1986 Mercat, C.: Discrete Riemann surfaces and the Ising model. Comm. Math. Phys. 218, 177–216 (2001) McCoy, B., Wu, T.T.: The two-dimensional Ising model. Cambridge MA: Harvard University Press, 1973 Tesler, G.: Matchings in graphs on non-orientable surfaces. J. Combin. Theory Ser. B 78, 198–231 (2000)

9. 10. 11. 12. 13. 14. 15. 16.

Communicated by L. Takhtajan

Commun. Math. Phys. 275, 209–254 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0258-7

Communications in

Mathematical Physics

An Index for 4 Dimensional Super Conformal Theories Justin Kinney1 , Juan Maldacena2 , Shiraz Minwalla3,4 , Suvrat Raju4 1 2 3 4

Department of Physics, Princeton University, Princeton, NJ 08544, USA Institute for Advanced Study, Princeton, NJ 08540, USA Tata Institute of Fundamental Research, Mumbai 400005, India Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA. E-mail: [email protected]

Received: 10 October 2006 / Accepted: 25 December 2006 Published online: 6 June 2007 – © Springer-Verlag 2007

Abstract: We present a trace formula for an index over the spectrum of four dimensional superconformal field theories on S 3 × time. Our index receives contributions from states invariant under at least one supercharge and captures all information – that may be obtained purely from group theory – about protected short representations in 4 dimensional superconformal field theories. In the case of the N = 4 theory our index is a function of four continuous variables. We compute it at weak coupling using gauge theory and at strong coupling by summing over the spectrum of free massless particles in Ad S5 × S 5 and find perfect agreement at large N and small charges. Our index does not reproduce the entropy of supersymmetric black holes in Ad S5 , but this is not a contradiction, as it differs qualitatively from the partition function over supersymmetric states of the N = 4 theory. We note that entropy for some small supersymmetric Ad S5 black holes may be reproduced via a D-brane counting involving giant gravitons. For big black holes we find a qualitative (but not exact) agreement with the naive counting of BPS states in the free Yang Mills theory. In this paper we also evaluate and study the partition function over the chiral ring in the N = 4 Yang Mills theory. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. 4 Dimensional Superconformal Algebras and their Unitary Representations 2.1 The 4 dimensional conformal algebra . . . . . . . . . . . . . . . . . . 2.2 Unitary representations of the conformal group . . . . . . . . . . . . 2.3 Unitary representations of d = 4 superconformal algebras . . . . . . . 2.4 The null vectors in short representations . . . . . . . . . . . . . . . . 2.5 Indices for four dimensional super conformal algebras . . . . . . . . . 3. A Trace Formula for the Indices of Superconformal Algebra . . . . . . . . 3.1 The commuting subalgebra . . . . . . . . . . . . . . . . . . . . . . . 3.2 I W L expanded in sub-algebra characters with I L as coefficients . . .

. . . . . . . . . .

210 212 212 213 215 217 218 219 220 220

210

J. Kinney, J. Maldacena, S. Minwalla, S. Raju

3.3 The Witten index I W R . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Computation of the Index in N = 4 Yang Mills on S 3 . . . . . . . . . . . . 4.1 Weak coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Strong coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. The Partition Function over BPS States . . . . . . . . . . . . . . . . . . . . 5.1 Partition function at
= 0 in free Yang Mills . . . . . . . . . . . . . . 5.2 Cohomology at strong coupling: Low energies . . . . . . . . . . . . . . 5.3 Cohomology at strong coupling: High energies . . . . . . . . . . . . . 5.4 Cohomology at intermediate energies: Giant gravitons and small black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . th th 6. Partition Functions over 21 , 41 and 18 BPS States . . . . . . . . . . . . . .

221 223 223 226 227 227 229 231

6.1 Enumeration of 18 , quarter and half BPS cohomology . . . . . . . . . 6.2 Protected double trace operator in the 20 . . . . . . . . . . . . . . . . . 6.3 Large N limits and phase transitions . . . . . . . . . . . . . . . . . . . 7. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. The d = 4 Superconformal Algebra . . . . . . . . . . . . . . . . . A.1 The commutation relations . . . . . . . . . . . . . . . . . . . . . . . . A.2 An oscillator construction of the algebra . . . . . . . . . . . . . . . . . Appendix B. Algebraic Details Concerned with the Index . . . . . . . . . . . . . B.1 Superconformal indices from joining rules . . . . . . . . . . . . . . . . B.2 The index I W L as a sum over characters . . . . . . . . . . . . . . . . . B.3 Representation theory of the subalgebra SU (2, 1|m − 1) . . . . . . . . Appendix C. Conventions and Computations for the N = 4 Index . . . . . . . . C.1 Weights of the supercharges . . . . . . . . . . . . . . . . . . . . . . . . C.2 Racah Speiser algorithm . . . . . . . . . . . . . . . . . . . . . . . . . C.3 State content of ‘graviton’ representations . . . . . . . . . . . . . . . . C.4 Character of SU (3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.5 Translation between bases . . . . . . . . . . . . . . . . . . . . . . . . C.6 Index on the Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix D. Comparison of the Cohomological Partition Function and the Index

236 238 240 242 242 242 243 244 244 245 246 247 247 248 248 249 249 250 251

th

233 235

1. Introduction Supersymmetry is a powerful tool for extracting exact information about quantum field theories. Supersymmetry algebras that contain R-charges in the right–hand side have special BPS multiplets. These multiplets occur at special values of energies or conformal dimensions determined by their charge, and have fewer states than the generic representation. An infinitesimal change in the energy of a special multiplet turns it into a generic multiplet with a discontinuously larger number of states. One might be tempted to use this observation to conclude that the number of short representations cannot change under variation of any continuous parameter of the field theory; however there is a caveat. It is sometimes possible for two or more BPS representations to combine into a generic representation. For this reason only states that cannot combine with other multiplets to form a long representation are guaranteed to be protected. In this paper we construct some quantities, called indices, that receive contributions only from those BPS states that cannot combine into long representations. The indices that we construct are defined for 4 dimensional superconformal field theories (with arbitrary number of supersymmetries) on S 3 × time. They take the form I W = T r [(−1) F eµi qi ],

(1.1)

Index for 4 Dimensional Super Conformal Theories

211

where qi are charges that commute with a particular supercharge. Our indices closely resemble the Witten index [1], and are invariant under all continuous deformations of the theory that preserve superconformal invariance1 . We demonstrate that our indices I W contain all the information about protected states that can be obtained by group theory alone, and so should be useful in the study of general super-conformal field theories. The indices I W are functions of 2, 3 and 4 continuous variables for N = 1, 2, 4 superconformal field theories respectively. In the case of the N = 4 Yang Mills theory we explicitly compute this index IYWM in the free limit. Upon taking the large N limit the index receives contributions only from states with energies of order one at all chemical potentials of order one. In other words IYWM does not undergo the deconfinement phase transition described in [2, 3]. Moreover we find that IYWM agrees perfectly with the index evaluated over the spectrum of free ten dimensional massless fields propagating on Ad S5 × S 5 . This agreement provides a check on the AdS/CFT conjecture in the BPS sector, which ends up containing the same information as the matching of chiral primary operators in [4, 5]. Related to the fact that the index never ‘deconfines’, in the limit of very small chemical potential, with charges growing like q ∼ N 2 we find that the index IYWM grows rather slowly. In particular, it does not grow fast enough to account for the entropy of the BPS black holes in Ad S5 × S 5 found in [6–8]. This is not a contradiction with Ad S/C F T ; the entropy of a black hole counts all supersymmetric states with a positive sign whereas our index counts the same states up to sign. It is possible for cancellations to ensure that the index is much smaller than the partition function evaluated over supersymmetric states of the theory. This is certainly what happens in the free N = 4 theory, where both quantities (the index and the partition function) may explicitly be computed, and is presumably also the case at strong coupling. It may well be possible to provide a weak coupling microscopic counting of the entropy of BPS black holes [6–8] in Ad S5 × S 5 ; however such an accounting must incorporate some dynamical information about N = 4 super Yang Mills beyond the information contained in the superconformal algebra. In this paper we make some small steps towards understanding the entropy of these black holes. In particular we provide a counting of the entropy for small black holes in terms of D-branes and giant gravitons in the interior. The counting is rather similar to the one performed for the D1D5p black holes [9]. We also note that, for large (large compared to the AdS radius) black holes a naive computation of the simple partition function of BPS states in the free theory gives a formula which has similar features to the black hole answer. The indices (1.1) do not exhaust all interesting calculable information about supersymmetric states in all superconformal field theory; in specific examples it is possible to extract more refined information about supersymmetric states by adding extra input. An explicit example where dynamical information allows us to make more progress is the computation of the chiral ring [5, 10]. In the case of N = 4 Yang Mills theory, we write down explicit counting formulas for 1/2, 1/4 and 1/8 BPS states. The counting can be done in terms of N particles in harmonic oscillator potentials. For very large charges the entropy in these states grows linearly in N . By taking the large N limit of these partition functions we show that they display a second order phase transition which corresponds to the formation of Bose-Einstein condensate. The structure of this paper is as follows. In Sect. 2 we review the unitary representations of the conformal and superconformal algebra, and list the linear combinations 1 More generally they are invariant under all deformations of the theory that preserve the corresponding supercharge.

212

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of (numbers of) short representations that form indices in these algebras. In Sect. 3 we define the Witten Indices that are the main topic of this paper, and explain how these are related to the indices of Sect. 2. This comparison allows us to argue that our indices capture all group theoretically protected information about short representations in superconformal field theories. In Sect. 4 we turn to the N = 4 Super Yang Mills theory. We compute our index in this theory in free Yang Mills on S 3 and show that it agrees perfectly, in the large N limit, with the same index computed over supergravitons in Ad S5 × S 5 . In Sect. 5 we continue our study of supersymmetric states in the N = 4 1 th supersymmetric Yang Mills theory on S 3 . We compute the partition function over 16 states in free Yang Mills using perturbation theory and in strongly coupled Yang Mills using gravity, and compare the two results. In Sect. 6 we present exact formulas for the th th partition function over 41 and 18 BPS states in N = 4 Yang Mills. For large charges we find that the free energy of these states scales linearly in N . This free energy displays a second order transition which is associated to the formation of a Bose-Einstein condensate. This paper contains two related but distinct streams. Sections 2 and 3 below concern themselves with the detailed nature of unitary representations of the superconformal algebra. Sections 4, 5 and 6 study the supersymmetric states of the N = 4 Yang Mills theory on S 3 . The link between these two streams is the Witten Index, defined in Sect. 3.1. The reader who is interested only in the definition of the index and the results for N = 4 Yang Mills could proceed directly to Sect. 3.1 where the index is defined, then to Sects. 4, 5 and 6 for computations in the N = 4 Yang Mills theory. On the other hand, the reader who is interested principally in the algebraic aspects of this index, including the demonstration that the Witten Index captures all protected information about superconformal field theories in four dimensions, could focus on Sections two and three. In this revised version of the paper we have added Sect. 6.2 which shows, using the index, that a particular double trace operator in the 20 of S O(6) is protected. While this paper was being completed we saw [11] which overlaps with parts of Sect. 3. 2. 4 Dimensional Superconformal Algebras and their Unitary Representations In this section we study the structure of representations of conformal and superconformal algebras. Our goal is to understand which representations, or combinations of representations, are protected. This will allow us to show that all protected information that can be obtained by using group theory alone is captured by the index we will define in Sect. 3.1. The reader who is willing to accept this fact (and is otherwise uninterested in the structure of unitary representations of the superconformal algebra), can just jump to Sect. 3.1 and from there to Sect. 4. We start this section with a discussion of the conformal algebra and then we discuss the superconformal algebra. 2.1. The 4 dimensional conformal algebra. The set of killing vectors Mµν = −i(xµ ∂ν − xν ∂µ ), Pµ = −i∂µ , K µ = i(2xµ x.∂ − x 2 ∂µ ) and H = x.∂ form a basis for infinitesimal conformal diffeomorphisms of R 4 . These killing vectors generate the algebra [H, Pµ ] = Pµ , [H, K µ ] = −K µ , [K µ , Pν ] = 2(δµν H − i Mµν ),

Index for 4 Dimensional Super Conformal Theories

213

[Mµν , Pρ ] = i(δµρ Pν − δνρ Pµ ), [Mµν , K ρ ] = i(δ 
µρ K ν − δνρ K µ ), [Mµν , Mρσ ] = i δµρ Mνσ + δνσ Mµρ − δµσ Mνρ − δνρ Mµσ .

(2.1)

Consider a 4 dimensional Euclidean quantum field theory. It is sometimes possible to combine the conformal killing symmetries of the previous paragraph with suitable action on fields to generate a symmetry of the theory. In such cases the theory in question is called a conformal field theory (CFT). The Euclidean path integral of a CFT may be given a useful Hilbert space interpretation via a radial quantization. Wave functions (kets) are identified with the path integral, with appropriate operator insertions, over the unit 3 ball surrounding the origin. Dual wave functions (bras) are obtained by acting µ on kets with the conformal symmetry corresponding to inversions x µ = xx 2 2 . Under an inversion, the killing vectors of the previous paragraph transform as Mµν → Mµν , H → −H , Pµ → K µ . As a consequence, the operators Mµν , Pµ , K ν are represented on the CFT Hilbert space (2.1) with the hermiticity conditions † Mµν = Mµν ,

Pµ = K µ† .

(2.2)

Radial quantization of the CFT on R 4 is equivalent to studying the field theory on S 3 × time. The operators Mµν generate the S O(4) rotational symmetries of S 3 , and H is the Hamiltonian. From this point of view the conjugate generators Pµ and K µ are less familiar; they act as ladder operators, respectively raising and lowering energy by a single unit. The Hilbert space of a CFT on S 3 × time may be decomposed into a sum of irreducible unitary representations of the conformal group. The theory of these representations was studied in detail by [12]. We present a brief review below, as a warm up for the superconformal algebra (see [13] and references therein for a recent discussion).

2.2. Unitary Representations of the Conformal Group. Any irreducible representation i of the conformal group can be written as some direct sum of representations, Rcompact , of the compact subgroup S O(4) × S O(2):  i R S O(4,2) = Rcompact . (2.3) i

The states within a given S O(4)× S O(2) representation all have the same energy. As the energy spectrum of any sensible quantum field theory is bounded from below, the representations of interest to us all possess a particular set of states with minimum energy. λ We will call these states (which we will take to transform as Rcompact ) the lowest weight λ because the K µ states. The K µ operators necessarily annihilate all the states in Rcompact have negative energy. We can now act on these lowest weight states with an arbitrary number of Pµ (‘raising’) operators to generate the remaining states in the representation. We will use the charges of the lowest weight state |λ ≡ |E, j1 , j2  to label this representation. We use the fact that S O(4) = SU (2) × SU (2); j1 and j2 are standard representation labels of these SU (2)s. 2 As a consequence, a bra may be thought of as being generated by a path integral, performed with appropriate insertions, on R 4 minus the unit 3 ball. The scalar product between a bra and a ket is the path integral with insertions both inside and outside the unit sphere - over all of space.

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It is important that not all values of E, j1 , j2 yield unitary representations of the conformal group. For a representation to be unitary, it is necessary for all states to have positive norm. Acting on the lowest energy states with Pµ , we obtain (via a Clebsh Gordan decomposition) states that transform in the representations (E +1, j1 ±1/2, j2 ±1/2). The norm of these states may be calculated using the commutation relations (2.1) [14]. The states with lowest norm turn out to have quantum numbers (E + 1, j1 − 21 , j2 − 21 ), and this norm is given by  2 = E − j1 − 1 + δ j1 0 − j2 − 1 + δ j2 0 . 2

(2.4)

Unitarity then requires that (i) E ≥ j1 + j2 + 2 j1 = 0 j2 = 0, (ii) E ≥ j1 + j2 + 1 j1 j2 = 0.

(2.5)

The special case j1 = j2 = 0 has an additional complication. In this case the norm of the level 2 state P 2 |ψ is proportional [14] to E(E − 1) and so is negative for 0 < E < 1. The representation with E = 0 is annihilated by all momentum operators and represents the vacuum state. The representation at E = 1 is short and it obeys the equation P 2 |E = 0 so it is a free field in the conformal field theory. Unitary representations exist even when this bound is strictly saturated. The zero norm states, and all their descendants, are simply set to zero in these representations3 making them shorter than generic. Now consider a one parameter (fixed line) of conformal field theories. An infinitesimal variation of the parameter that labels the theory will, generically, result in an infinitesimal variation in the energy of all the long representations of the theory. However a short representation can change its energy only if it turns into a long representation. In order for this to happen without a discontinuous jump in the spectrum of the CFT (i.e. a phase transition), it must pair up with some other representation, to make up the states of a long representation with energy at just above the unitarity threshold. Groups of short representations that can pair up in this manner are not protected; the numbers of such representations can jump discontinuously under infinitesimal variations of a theory. However consider an index I that is defined as a sum of the form I = α[i]n[i],

(2.6)

where i runs over the various short representations of the theory, n[i]s are the number of representations of the i th variety, and α[i] are fixed numbers chosen so that I evaluates to zero on any collection of short representations that can pair up into long representations. By definition such an index is unaffected by groups of short representations pairing up and leaving, as it evaluates to zero anyway on any set of representations that can. It follows that an index cannot change under continuous deformations of the theory. We will now argue that the conformal algebra does not admit any nontrivial indices. In order to do this we first list how a long representation decomposes into a sum of other representations (at least one of which is short) when its energy is decreased so that it hits the unitarity bound. Let us denote the representations as follows. A E, j1 , j2 denotes the generic long representation, C j1 , j2 denotes the short representations with j1 and j2 3 The consistency of this procedure relies on the fact that, at the unitarity bound, zero norm states are orthogonal to all states in the representation. As a consequence the inner product on the representation modded out by zero norm states is well defined and positive definite.

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215

both not equal to zero, B Lj1 denotes the short representations with j2 = 0, B jR2 the short ones with j1 = 0. Finally we denote the special short representation at E = 1 and j1 = j2 = 0 by B. As the energy is decreased to approach the unitarity bound we find lim χ [A j1 + j2 +2+ , j1 , j2 ] = χ [C j1 , j2 ] + χ [A j1 + j2 +3, j1 − 1 , j2 − 1 ],

→0

2

lim χ [A j1 +1+ , j1 ,0 ] =

→0

2

χ [B Lj1 ] + χ [C j1 − 1 , 1 ], 2 2

lim χ [A j2 +1+ ,0, j2 ] = χ [B jR2 ] + χ [C 1 , j2 − 1 ],

→0

2

2

lim χ [A1+ ,0,0 ] = χ [B] + χ [A3,0,0 ].

(2.7)

→0

In (2.7) and throughout this paper, the symbol χ denotes the super-character on a representation.4  It follows from (2.7) that i αi n i is an index only if αC j1 , j2 = 0, α B L + αC j1

j1 − 21 , 21

= 0, α B R + αC 1

1 2 , j2 − 2

j2

= 0, α B = 0.

(2.8)

The only solution to these equations has all α to zero; consequently the conformal algebra admits no nontrivial indices. The superconformal algebra will turn out to be more interesting in this respect. 2.3. Unitary Representations of d = 4 Superconformal Algebras. In the next two subsections we review the unitary representations of the d = 4 superconformal algebras [15] that were studied in [16–19, 14, 20, 21]. A supersymmetric field theory that is also conformally invariant, actually enjoys superconformal symmetry, a symmetry algebra that is larger than the union of conformal and super symmetry algebras. The bosonic subalgebra of the N = m superconformal algebra consists of the conformal algebra times U (m), except in the special case m = 4, where the R symmetry algebra is SU (4). The fermionic generators of this algebra consist of the 4m supersymmetry generators Q αi j and Q¯ iα˙ , together with the super conformal generators Sαi , S¯α˙ . The generators transform under S O(4) × U (m) as indicated by their index structure (an upper i index indicates a U (m) fundamental, while a lower i index is a U (m) anti-fundamental). The commutation relations of the algebra are listed in detail in Appendix A.1. In particular, {Sαi , Q β j } = δi (J1 )βα + δαβ Ri + δi δαβ ( j

j

j

4−m H +r ), 2 4m

β

(2.9)

j

where (J1 )α are the SU (2) generators in spinor notation, Ri are the SU (m) generators and r is the U (1) generator. As in the previous subsection, radial quantization endows these generators with hermiticity properties; specifically (Q αi )† = Sαi , ( Q¯ iα˙ )† = S¯αi˙ .

(2.10)

The theory of unitary representations of the superconformal algebra is similar to that of the conformal algebra. Irreducible representations are labeled by the energy E and the SU (2) × SU (2) and U (m) representations of their lowest weight states. We label 4 I.e. T r (−1) F G, where R is an arbitrary representation, G is an arbitrary group element, and F is the R Fermion number, which plays no role in the representation theory of the conformal group, but will be important when we turn to superconformal groups below.

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U (m) representations by their U (1) charge r (normalized such that Q αi has charge +1 and Q¯ iα˙ has charge -1) and the integers Rk (k = 1 . . . m − 1), the number of columns of height k in the Young Tableaux for the representation.5 Lowest weight states are annihilated by the S but, generically, not by the Q operators. Q αi have E = 21 and transform in the SU (2) × SU (2) × U (m) representation with quantum numbers j1 = 21 , j2 = 0, r = 1, R1 = 1, Ri = 0 (i > 1). Let |ψa  be the set of lowest weight states of this algebra that transforms in the representation (E, j1 , j2 , r, Ri ). The states Q αi |ψa  transform in all the Clebsh Gordan product representations; the lowest norm among these states occurs for those that have quantum numbers (E + 21 , j1 − 21 , j2 , r + 1, R1 + 1, R j ); the value of the norm of these states is given by [14] 2χ1 2 = E + 2δ j1 ,0 − E 1 ( j1 , r, Ri ) , m−1 r (4 − m) p=1 (m − p)R p E 1 ≡ 2 + 2 j1 + 2 + . m 2m

(2.11)

In a similar fashion, of the states of the form Q¯ αi ˙ |ψ the lowest norm occurs for those that transform in (E + 21 , j1 , j2 − 21 , r − 1, Rk , Rm−1 + 1), and the norm of these states is equal to [14] 2χ2 2 = E + 2δ j2 ,0 − E 2 ( j1 , j2 , r, Ri )  2 m−1 r (4 − m) p=1 p R p − . E 2 ≡ 2 + 2 j2 + m 2m

(2.12)

Clearly unitarity demands that χ1 2 ≥ 0 and χ2 2 ≥ 0. As for the conformal group, representations with either χ1 2 = 0 or χ2 2 = 0 or both zero are allowed. In such representations the zero norm states and all their descendants are simply set to zero, yielding short representations. In the special case j1 = 0 the positivity of the norm at level 2 yields more information. Of states of the form Q αi Q β j |ψa  (where |ψa  are the lowest weight states), those that have the smallest norm transform in the representation (E + 1, 0, j2 , r + 2, R1 + 2, R j ). The norm of these states turns out to be proportional to (E − E 1 )(E − E 1 + 2), where E 1 is defined in (2.11). Thus unitarity disallows representations in the window E 1 − 2 < E < E 1 . Representations at E = E 1 − 2 and E = E 1 are both short and both allowed. Representations at E = E 1 − 2 are special because they are separated from long representations (with the same value of all other charges) by an energy gap of two units. All these remarks also apply to the special case j2 = 0, upon replacing Q αi with Q¯ iα˙ and E 1 with E 2 . In [20], Dolan and Osborn, performed a comprehensive tabulation of short representations of the d = 4 superconformal algebras. We will adopt a notation that is slightly different from theirs. Representations are denoted by xL xR E, j1 , j2 ,r,Ri , where  xL =

a if E > E 1 c if E = E 1 and j1 ≥ 0 , b if E = E 1 − 2 and j1 = 0

(2.13)

5 R may also be thought of as the eigenvalues of the highest weight vectors under the diagonal generator k Rk whose k th diagonal entry is unity, (k +1)th entry is −1, and all other are zero, in the defining representation.

Index for 4 Dimensional Super Conformal Theories

217

and  x = R

a if E > E 2 c if E = E 2 and j2 ≥ 0 . b if E = E 2 − 2 and j2 = 0

(2.14)

Further, we will usually omit to specify the first (energy) subscript on all short representations as this energy is determined by the other charges. Thus representations denoted by aa are long; all other representations are short.

2.4. The null vectors in short representations. We now study the nature of the null vectors in short representations in more detail. Consider a representation of the type cx, with j1 > 0, where x is either a, c or b. Such a representation has χ1 2 = 0. The descendants of the null-state form another representation of the superconformal algebra. This representation also has null states6 characterized by their own value of (χ1 2 , χ2 2 ). A short calculation7 shows that χ1 2 , χ2 2 )/ = (0, χ2 2 ). It follows that the Q null states of a representation of type cx are generically also of the type cx. The exception to this rule occurs when j1 = 0, in which case the null states are of type bx. Of course analogous statements are also true for Q¯ null states. All of this may be summarized in a set of decomposition formulae, for the supercharacters,   χ [R] = Tr R (−1)2(J1 +J2 ) G , (2.15) where G is an arbitrary element of the superconformal group. These formulae describe how a long representation decomposes into a set of short representation when its energy hits the unitarity bound: lim χ [aa E 1 + , j1 , j2 ,r,Ri ] = χ [ ca j1 , j2 ,r,Ri ] + χ [ ca j1 − 1 , j2 ,r +1,R1 +1,R j ], E 1 > E 2 ,

→0

2

lim χ [aa E 2 + , j1 , j2 ,r,Ri ] = χ [a c j1 , j2 ,r,Ri ] + χ [a c j1 , j2 − 1 ,r −1,Rk ,Rm−1 +1 ], E 2 > E 1 ,

→0

2

c c j1 , j2 ,r,Ri ] + χ [ c c j1 − 1 , j2 ,r +1,R1 +1,R j ] lim χ [aa E 2 + , j1 , j2 ,r,Ri ] = χ [

→0

2

+ χ [ c c j1 , j2 − 1 ,r −1,Rk ,Rm−1 +1 ] 2

+ χ [ c c j1 − 1 , j2 − 1 ,r,R1 +1,Rl ,Rm−1 +1 ], E 1 = E 2 , 2

2

(2.16) where, in this equation and, as far as possible, in the rest of the paper, we use the index convention i = 1 . . . m − 1,

j = 2 . . . m − 1, k = 1 . . . m − 2, l = 2 . . . m − 2. (2.17)

On the right-hand side of (2.16) we have used the notation given in Table 1. 6 When we say that a short representation has ‘null states’ of a particular type we mean the following. When we lower the energy of a long representation down to its unitarity bound (E 1 or E 2 ), the long representation splits into a positive norm short representation m, plus a set of null representations m  . We describe this situation by the words ‘the short representation m has null representations m  . As is clear from this definition, it is meaningless to talk of the null state content of representations of the sort bx or xb, as these representations are separated by a gap from long representations. 7 (χ  2 , χ  2 ) = ( 1 + 1 − 2(m − 1)/m − (4 − m)/2m, χ 2 − 1 + 2/m + (4 − m)/2m) = (0, χ 2 ). 2 2 1 2 2 2

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Table 1. Short Representations Symbol

Denotation

ca j1 , j2 ,r,Ri

ca j1 , j2 ,r,Ri , j1 ≥ 0 ba0, j2 ,r +1,R1 +1,R j , j1 = − 21 ac j1 , j2 ,r,Ri , j2 ≥ 0 ab j1 ,0,r −1,Rk ,Rm−1 +1 , j2 = − 21 cc j1 , j2 ,r,Ri , j1 ≥ 0, j2 ≥ 0 cb j1 ,0,r −1,Rk ,Rm−1 +1 , j2 = −1 2 , j1 ≥ 0 bc0, j2 ,r +1,R1 +1,R j , j1 = − 21 , j2 ≥ 0

a c j1 , j2 ,r,Ri

c c j1 , j2 ,r,Ri

bb0,0,r,R1 +1,Rl ,Rm−1 +1 , j1 = j2 = − 21

2.5. Indices for four dimensional super conformal algebras. We now turn to a study of the indices for these algebras. Using (2.16) it is not difficult to convince oneself that the set of indices for the our dimensional superconformal field theories is a vector space that is spanned by 1. The number of representations of the sort bx with R1 = 0 or R1 = 1 plus the number of representations of the sort xb with Rm−1 = 0 or Rm−1 = 1. 2. The indices I jL2 , r ,M,R j =

M 

(−1) p+1 n[ cx p , j2 , r − p,M− p,R j ] 2

(2.18)

p=−1

for all values of

r and non-negative integral values of j2 , M, R j . In the case m = 1 we do not have the indices M or R j and the sum runs from p = −1 to infinity. In the m = 4 case, simply ignore the r and

r subindices. 3. The indices I jR1 ,r  ,Rk ,N =

M 

(−1) p+1 n[x c j1 , p ,r  + p,Rk ,N − p ] 2

(2.19)

p=−1

for all values of r  and non-negative integral values of of j1 , Rk , N , with the same remarks for m = 1, 4. In the special case that representations that contribute to the sum in (2.18) and (2.19) have quantum numbers on which E 1 = E 2 8 , the indices (2.18) and (2.19) are subject to the additional constraints N  p=−1

(−1) p I Lp ,r  + p,M,R ,N − p = 2

M 

l

p=−1

(−1) p I Rp ,r  − p,M− p,R ,N (E 1 = E 2 ) (2.20) 2

l

for all values of r  = −∞ . . . ∞, and non-negative integral values of M, N , Rl . These results are explained in more detail in Appendix B.1, where we also present a detailed listing of the absolutely protected multiplets, for the N = 1, 2, 4 superconformal algebras. 8 If this relation is true for any term that contributes to the sum, it is automatically true on all other terms as well.

Index for 4 Dimensional Super Conformal Theories

219

3. A Trace Formula for the Indices of Superconformal Algebra The supercharges Q αi transform in the fundamental or (1, 0, . . . , 0) representation of 1 SU (m). Let Q ≡ Q − 2 ,1 , the supercharge whose SU (2) × SU (2) quantum numbers are ( j13 , j23 ) = (− 21 , 0), that has r = 1, and whose SU (m) quantum numbers are (1, 0, . . . 0). Let S ≡ Q † . Then (see (2.9)) 2{S, Q} = H − 2J1 − 2

m−1  k=1

m−k (4 − m)r Rk − = E − (E 1 − 2) ≡
. (3.1) m 2m

It follows from (3.1) that every state in a unitary representation of the superconformal group has
≥ 0. Note that the Jacobi identity implies that Q and S commute with
. Consider a unitary representation R of the superconformal group that is not necessarily irreducible. Let R
0 denote the linear vector space of states with
=
0 > 0. It follows from (3.1) that if |ψ is in R
0 then |ψ = Q

S Q |ψ + S |ψ.
0
0

(3.2)

Q S the Let R
denote the subspace of R
0 consisting of states annihilated by Q and R
0 0 set of states in R
0 that are annihilated by S. It follows immediately from (3.2) (and the Q S and that S|ψ = |ψ   is a one to unitarity of the representation) that R
0 = R
0 + R
0 Q S (Q/
provides the inverse map). to R
one map from R
0 0 0 Now consider the Witten index

  I W L = T r R (−1) F exp(−β
+ M) ,

(3.3)

where M is any element of the subalgebra of the superconformal algebra that commutes with Q and S. We discuss this subalgebra in detail in the next subsection. It follows Q that the states in R
0 do not contribute to this index, the contribution of R
0 cancels S . Consequently, I W L receives contributions only from states with against that of R
0
= 0, i.e. those states that are annihilated by both Q and S. Thus, despite appearances, (3.3) is independent of β. As no long representation contains states with
= 0, such representations do not contribute to I W L . It also follows from continuity that I W L evaluates to zero on groups of short representations that a long representation breaks up into when it hits unitarity threshold. As a consequence I W L is an index; it cannot change under continuous variations of the superconformal theory, and must depend linearly on the indices, I L and I R , listed in the previous section. We will explain the relationship between I W L and I L in more detail in Subsect. 3.2 and 3.3 below. The main result of the following subsections is to show that (3.3) (and its I W R version) completely capture the information contained in the indices defined in the previous section, which is all the information about protected representations that can be obtained without invoking any dynamical assumption. In Appendix B.2 we derive most of the results of Sect. 2 in a way that uses a smaller amount of group theory.

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3.1. The commuting subalgebra. In this subsection we briefly describe the subalgebra of the superconformal algebra that commutes with the SU (1|1) algebra spanned by Q, S,
. The N = m, d = 4 superconformal algebra is the super-matrix algebra SU (2, 2|m)9 . Supersymmetry generators transform as bifundamentals under the bosonic subgroups SU (2, 2) and SU (m). It is not difficult to convince oneself that the commuting subalgebra we are interested in is SU (2, 1|m − 1)10 . The generators of SU (2, 1|m − 1) are related to those of SU (2, 2|m) via the obvious reduction. In more detail, the bosonic subgroup of SU (2, 1|m − 1) is SU (2, 1) × U (m − 1). The U (m − 1) factor sits inside the superconformal U (m) setting all elements in the first row and first column to zero, except for the 11 element which is set to one. The Cartan elements (E  , j2 , r  , Ri ) of the subalgebra are given in terms of those for the full algebra by E  = E + j1 ,

j2 = j2 , r  =

m−p (m − 1)r − R p , Rk = Rk+1 , (3.4) m m m−1 p=1

where Rk are the Cartan elements of U (m − 1) and r  is the U (1) charge in U (m − 1). We will think of (3.4) as defining a (many to one) map from (E, j1 , j2 , r, Ri ) to (E  , j2 , r  , Ri ). We will be interested in the representations of the subalgebra, SU (2, 1|m − 1), that are obtained by restricting a representation of the full algebra, SU (2, 2|m), to states with
= 0. Null vectors, if any, of SU (2, 1|m − 1) are inherited from those of SU (2, 2|m). Is is possible to show that SU (2, 1|m − 1) is short only when SU (2, 2|m) is one of the representations cb, cc or if R is bx with R1 = 0. See Appendix B.3 for further discussion. 3.2. I W L expanded in sub-algebra characters with I L as coefficients. In this subsection we present a formula for Index I W L as a sum over super characters of the commuting subalgebra, SU (2, 1|m − 1), more details may be found in Appendix B.2. It is not difficult to convince oneself (see Appendix B.2) that on any short irreducible representation R of the superconformal algebra SU (2, 2|m), I W L evaluates to the supercharacter of a single irreducible representation R  of the subalgebra SU (2, 1|m−1). More specifically we find

I W L [bx0, j2 ,r,Ri ] = χsub [b], c], I W L [cx j1 , j2 ,r,Ri ] = (−1)2 j1 +1 χsub [

(3.5)

where χsub is the supercharacter   χs [R  ] = Tr R  (−1)2J2 G  ,

(3.6)

where G  is an element of the Cartan subgroup. The vectors b and c specify the highest weight of the representation of the subalgebra in the Cartan basis [E  , j2 , r  , Rk ] defined 9 For m = 4 we have P SU (2, 2|4). 10 Or P SU (2, 1|3) for m = 4.

Index for 4 Dimensional Super Conformal Theories

221

in (3.4), 3 b = [ r − 2r  , j2 , r  , R j ], 2 c = [3 + 3( j1 + r/2) − 2r  , j2 , r  , R j ],

(3.7)

where r  is the function defined in (3.4); we emphasize the fact that it depends on r and R1 only through the combination r − R1 . Notice that the functions that specify the character of the subalgebra, (3.7), are not one to one. In fact, it follows from (3.5), (3.7), that I W L evaluates to the same subalgebra character for each representation R that appears in the sum in (2.18), for fixed values of j2 , r  , M, Ri . Notice that by formally setting j1 = −1/2 in the second line of (3.7) we get the Cartan values for the subalgebra that we expect for the representation b according to the definition of c in Table 1. This implies that we can replace c in (3.5) by c. More specifically I W L [ c p , j2 , r − p,M− p,R j ] = (−1) p I W L [ c0, j2 , r ,M,R j ]. 2

(3.8)

It follows immediately from (3.5), (3.8), that I W L , evaluated on any (in general reducible representation) A of the superconformal algebra evaluates to   n[bx0, j2 ,r,0,Ri ]χsub [b 0 ] + n[bx0, j2 ,r,1,Ri ]χsub [b 1 ] I W L [A] = j2 ,r,Ri

+



j2 ,r  ,M,Ri

I jL2 , r ,M,Ri χsub [

c0 ],

(3.9)

where b 0,1 are given by (3.7) with R1 = 0, 1 respectively and c 0 is given by (3.7) with j1 = 0, r =

r , R1 = M. The quantities n[xx j1 , j2 ,r,Ri ] in (3.9) are the number of copies of the irreducible representation, with listed quantum numbers, that appear in A, and I jL2 , r ,M,Ri are the indices (2.18) made out of these numbers. Notice that most of the discussion in this section goes through unchanged if we were to consider the supergroup SU (2|4) (or SU (2|m)). The representation theory of this group was studied in [22, 23] and the index was used in [24] to analyze various field theories with this symmetry. The index for the plane wave matrix model is given by an expression like (4.3) below but without the denominators (this is then inserted into (4.1)). Notice that the fact that the index for N = 4 Yang Mills and the index for the plane wave matrix model are different implies that we cannot continuously interpolate between N = 4 super Yang Mills and the plane wave matrix model while preserving the SU (2|4) symmetry. In [25] BPS representations and an index for SU (1|4) were considered. 3.3. The Witten Index I W R . As in Sect. 2, we may define a second index I W R . The − 21 theory for this index is almost identical. We focus on the supercharge, Q¯ m−1 which has SU (2) × SU (2) quantum numbers, ( j13 , j23 ) = (0, − 21 ), r = −1 and SU (4) quantum numbers (0, 0, . . . , 1). Let S¯ = Q¯ † . It is then easy to verify (see Appendix A) that ¯ Q} ¯ = H − 2J2 − 2 2{ S,

m−1  k=1

k (4 − m)r ¯ Rk + = E − (E 2 − 2) ≡
. m 2m

(3.10)

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It follows from (3.10) that every state in a unitary representation of the superconformal ¯ ≥ 0. group has
Following (3.3) we define   ¯ , ¯ + M) I W R = T r R (−1) F exp(−β


(3.11)

¯ where M¯ is the part of the superconformal algebra that commutes with Q¯ and S. The Cartan elements of this subalgebra are given in terms of those of the algebra by E  = E + j2 ,

j1 = j1 , r  =

(m − 1)(r + Rm−1 )  p + R p , Rk = Rk . (3.12) m m m−2 p=1

Note that r  depends on r and Rm−1 only through the combination r + Rm−1 . We then find that the index (3.11) is zero on long representations and for c, b representations it is equal to

¯ I W R [bx j1 ,0,r,Ri ] = χsub [b],

¯ I W R [cx j1 , j2 ,r,Ri ] = (−1)2 j2 +1 χsub [c],

(3.13)

¯ c ¯ in the basis where the representation of the subalgebra is specified by the vectors b, [E  , j1 , r  , Rk ] specified by (3.12), 3 b ¯ = [− r + 2r  , j1 , r  (r + Rm−1 , Rk ), Rk ], 2 c ¯ = [3 + 3( j2 − r/2) + 2r  , j1 , r  (r + Rm−1 , Rk ), Rk ],

(3.14)

where r  is the function in (3.12). We find that on a general representation (not necessarily irreducible) of the superconformal algebra, I W R evaluates to I W R [R] =

  n[xb j1 ,0,r,Rk ,0 ]χsub [b ¯ 0 ] + n[xb j1 ,0,r,Rk ,1 ]χsub [b ¯ 1 ] j1 ,r,Ri



+ j1

,r  ,R

k ,N

I jR1 ,r  ,Rk ,N χsub [c ¯0 ],

(3.15)

¯ are given by (3.14) with R where b0,1 ¯0 is given by (3.14) m−1 = 0, 1 respectively and c

with j2 = 0, r = r  , Rm−1 = N . The quantities n[xx j1 , j2 ,r,Ri ] in (3.9) are the number of copies of the irreducible representation, with listed quantum numbers, that appear in R, and I jR1 ,r  ,Rk ,N are the indices (2.19) made out of these numbers. The main lesson we should extract from (3.9), (3.15) is that each of the indices defined in Sect. Two are multiplied by different SU (1, 2|m − 1) (or SU (2, 1|m − 1)) characters in (3.9), (3.15). This shows that the Witten indices (3.3), (3.11) capture all the protected information that follows from the supersymmetry algebra alone.

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223

Table 2. Letters with
= 0 Letter X, Y, Z

(−1)F [E; j1 , j2 ] [1, 0, 0]

[q1 , q2 , q3 ] [1, 0, 0] + cyclic

[R1 , R2 , R3 ] [0, 1, 0] + [1, −1, 1] + [1, 0, −1]

ψ+,0;−++ + cyc

−[ 23 , 21 , 0] [2, 1, 0] [ 25 , 21 , 0]

[− 21 , 21 , 21 ] + cyc [ 21 , 21 , 21 ] [0, 0, 0] [ 21 , 21 , 21 ]

[1, −1, 0], [0, 1, −1], [0, 0, 1]

ψ0,±,+++ F++ ∂++ ψ0,−;+++ + ∂+− ψ0,+;+++ = 0 ∂+±

−[ 23 , 0, ± 21 ]

[1, 21 , ± 21 ]

[0, 0, 0]

[0, 0, 0]

[1, 0, 0] [0, 0, 0] [1, 0, 0]

4. Computation of the Index in N = 4 Yang Mills on S3 4.1. Weak coupling. We will now evaluate the index (3.3) for free N = 4 Yang Mills on S 3 . In the free theory this index may be evaluated either by simply counting all gauge invariant states with
= 0 and specified values for other charges [2, 3] or by evaluating a path integral [3]. The two methods give the same answer. We will give a very brief description of the path integral method, referring the reader to [3] for all details. One evaluates the path integral over the
= 0 modes of all the fields of the N = 4 theory on S 3 × S 1 with periodic boundary conditions for the fermions around S 1 (to deal with the (−1) F insertion) and twisted boundary conditions on all charged fields (to insert the appropriate chemical potentials). While the path integral over all other modes may be evaluated in the one loop approximation, the path integral over the zero mode of A0 on this manifold must be dealt with exactly (as the integrand lacks a quadratic term for this mode, the integral over it is always strongly coupled at every nonzero coupling no matter how weak). Upon carefully setting up the problem one finds that the integral over A0 is really an integral over the holonomy matrix U , and the index I W L evaluates to

IY M =

[dU ] exp

  1 f (t m , y m , u m , w m )tr(U † )m trU m , m

(4.1)

where f (t, y, u, w) is the index I W L evaluated on the space of ‘letters’ or ‘gluons’ of the N = 4 Yang Mills theory. As a consequence, in order to complete our evaluation of the index (4.1) we must merely evaluate the single letter partition function f . f may be evaluated in many ways. Group theoretically, we note that the letters of Yang Mills theory transform in the ‘fundamental’ representation of the superconformal group (the representation whose quantum lowest weight state has quantum numbers E = 1, j1 = j2 = 0, R1 = R3 = 0 and R2 = 1). f is simply the supertrace over this representation, which we have evaluated using group theoretic techniques in Appendix C. It is useful, however, to re-derive this result in a more physical manner. The full set of single particle
= 0 operators in Yang Mills theory is given by the fields listed in Table 2 below, acted on by an arbitrary number of the two derivatives ∂+± (see the last row of Table 2) modulo the single equation of motion listed in the second to last row of Table 2. In Table 2 we have listed both the SU (4) Cartan charges R1 , R2 , R3 used earlier in this paper, as well as the S O(6) Cartan charges, q1 , q2 , q3 (the eigenvalues in each of the 3 planes of the embedding R 6 ) of all fields.

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To find f we evaluate (3.3) by summing over the letters f = =



(−1) F t 2(E+ j1 ) y 2 j2 v R2 w R3

letters t 2 (v + w1

+

w 3 v ) − t (y

+ 1y ) − t 4 (w +

(1 − t 3 y)(1 −

1 v

+

v 6 w ) + 2t

t3 y)

.

(4.2)

Remarkably the expression for 1 − f factorizes 1− f =

(1 − t 2 /w)(1 − t 2 w/v)(1 − t 2 v) . (1 − t 3 y)(1 − t 3 /y)

(4.3)

The expression for IYWML is well defined (convergent) only if t, y, v, w have values such that every contributing letter has a weight of modulus < 1; applying this criterion to the three scalars and the two retained derivatives yields the restriction t 2 v < 1, t 2 /w < 1, t 2 v/w < 1, t 3 y < 1, t 3 /y > 1. It follows from (4.3) that f < 1 for all legal values of chemical potentials. We will now proceed to evaluate the integral in (4.1), using saddle point techniques, in the large N limit (note, however, that (4.1) is the exact formula valid for all N ). To process this formula, we convert the integral over U to an integral over its N 2 eigenvalues eiθ j . We can conveniently summarize this information in a density distribution ρ(θ ) with:

π dθ ρ(θ ) = 1. (4.4) −π

This generates an effective action for the eigenvalues given by [3]

2 S[ρ(θ )] = N dθ1 dθ2 ρ(θ1 )ρ(θ2 )V (θ1 − θ2 ) =

∞ N2  |ρn |2 Vn (T ), 2π

(4.5)

n=1

with Vn =

2π (1 − f (t n , y n , u n , w n )), n

ρn =

dθρ(θ )einθ .

(4.6)

As (1 − f ) is always positive for all allowed values of the chemical potential, it is clear that the action (4.5) is minimized by ρn = 0, n > 0; ρ0 = 1. The classical value of the action vanishes on this saddle point, and the index is given by the gaussian integral of the fluctuations of ρn around zero. This allows us to write   IYWML 

N =∞

=

∞  n=1

1 1−

f (t n , y n , v n , w n )

.

(4.7)

Index for 4 Dimensional Super Conformal Theories

225

If we think about the ’t Hooft limit of the theory it is also interesting to compute the index over single trace operators. This is given by Z s. t. = − =

∞  ϕ(r ) r =1 t 2 /w

r

1 − t 2 /w

  log 1 − f (t r , y r , vr , wr )

+

t 2 w/v vt 2 t 3 /y t3y + − − , 2 2 3 1 − vt 1 − t w/v 1 − t /y 1 − t3y

(4.8)

 ) −x r where ϕ is the Euler Phi function and we used that r ϕ(r r log(1 − x ) = 1−x . The result (4.7) is simply the multiparticle contribution that we get from (4.8). Note that the action (4.5) vanished on its saddle point; as a consequence (4.7) is independent of N in the large N limit. This behavior, which is is reminiscent of the partition function of a large N gauge theory in its confined phase, is true of (4.7) at all finite values of the chemical potential. In this respect the index I Y M behaves in a qualitatively different manner from the free Yang Mills partition function over supersymmetric states (see the next section). This partition function displays confined behavior at large chemical potentials (analogous to low temperatures) but deconfined behavior (i.e. is of 2 order e N ) at small chemical potentials (analogous to high temperature). It undergoes a sharp phase transition between these two behaviors at chemical potentials of order unity. Several recent studies of Yang Mills theory on compact manifolds have studied such phase transitions, and related them to the nucleation of black holes in bulkduals [2, 3, 26–31]. The index IYWML does not undergo this phase transition, and is always in the ‘confined’ phase. We interpret this to mean that it never ‘sees’ the dual supersymmetric black hole phase. At first sight we might think that this is a contradiction, since the black holes give a large entropy. On the other hand we are unaware of a clear argument which says that black holes should contribute to the index. For example, it is unclear whether the Euclidean black hole geometry should contribute to the path integral that computes the index. While the Lorentzian geometry of the black hole is completely smooth, if we compactify the Euclidean time direction with periodic boundary conditions for the spinors, then the corresponding circle shrinks to zero size at the horizon, which would represent a kind of singularity. See the next section and Appendix D for a mechanism for how this phenomenon (the excision of the black hole saddle point) might work in Lorentzian space. We now present the expression for the index in a new set of variables that are more symmetric, and for some purposes more convenient, in the study of Yang Mills theory. We will use these variables in the next section. Let us choose to parameterize charges in the subalgebra by J2 , L 1 = E + q1 − q2 − q3 , L 2 = E + q2 − q1 − q3 , L 3 = E + q3 − q1 − q2 . (4.9) Note that L i are positive for all Yang Mills letters. A simple change of basis, (see Appendix C) yields (1 − e−2γ1 )(1 − e−2γ2 )(1 − e−2γ3 ) 
, 1− f =
1 − e−ζ −γ1 −γ2 −γ3 1 − e+ζ −γ1 −γ2 −γ3

(4.10)

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where 

f =

(−1) F eγ1 L 1 +γ2 L 2 +γ3 L 3 +2ζ j2 .

(4.11)

letters

In Sect. Six we will write an explicit exact formula for the index (4.1) for γ3 = ∞. Further studies on the spectrum of free Yang Mills can be found in [32–35].

4.2. Strong coupling. According to the AdS/CFT correspondence, N = 4 Yang Mills theory on S 3 at large N and large λ has a dual description as a weakly coupled IIB theory on the large radius Ad S5 × S 5 . At fixed energies in the large N limit, the spectrum of the bulk dual is a gas of free gravitons, plus superpartners, on Ad S5 × S 5 . In this subsection section we will compute the index IYWML over this gas of masssless particles, and find perfect agreement with (4.7). Note that states with energies of order one do not always dominate the partition function at chemical potentials of unit order. At small values of the chemical potential, the usual partition function of strongly coupled Yang Mills theory is dominated by black holes. However, as we have explained in the previous subsection, we do not see an argument for the black hole saddle point to contribute to the index, and apparently it does not. We now turn to the computation. When the spectrum of (single particle) supergravitons of Type I I B supergravity compactified on Ad S5 × S 5 is organized into representations of the superconformal group, we obtain representations that are built on a lowest weight state that is a SU (2) × SU (2) in the (n, 0, 0) S O(6) = (0, n, 0) SU (4) representation of the R-symmetry group [36]. The representation with n = 1 is the Yang-Mills multiplet. The representation with n = 2 is called the ‘supergraviton’ representation. These representations preserve 8 of 16 supersymmetries. In the language of Sect. 2, they are of the form bb. When restricted to
= 0, they yield a representation of the subalgebra that we shall call Sn . Sn has lowest weights E  = n, j2 = 0, R2 = n, R3 = 0. The states of Sn are tabulated explicitly in Appendix C. The state content of n = 1 is somewhat different and is tabulated separately. This can also be found by looking at the list of Kaluza Klein modes in [36]. The index on single-particle states may now be calculated in a straightforward manner. The supercharacter of Sn may be read off from the appendix and is given by SU (3)

χ Sn =

(t 2n χn,0

SU (3)

(v, w) − t 2n+1 χn−1,0 (v, w)(y + 1/y) + . . .) (1 − t 3 y)(1 − t 3 /y)

.

(4.12)

The SU (3) character that occurs above is described by the Weyl character formula described in Appendix C. To obtain the index, we simply need to calculate

Isp =

∞  n=2

χ Sn + χ S1 .

(4.13)

Index for 4 Dimensional Super Conformal Theories

227

The sums in (4.13) are all geometric and are easily performed, yielding the single particle contribution Isp =

vt 2 t 3 /y t3y t 2 /w t 2 w/v + − − . + 1 − t 2 /w 1 − vt 2 1 − t 2 w/v 1 − t 3 /y 1 − t3y

(4.14)

This matches precisely (4.8). To obtain the index for the Fock-space of gravitons we use the formula, justified in Appendix C, that relates the index of one particle to the index of the Fock space.   1 WL n n n n Igrav = exp Isp [t , v , w , y ] n n =

∞  n=1

(1 − t 3n /y n )(1 − t 3n y n ) (1 − t 2n /w n )(1 − v n t 2n )(1 − t 2n w n /v n )

(4.15)

in perfect agreement with (4.7). Finally, let us point out that the value of the index is the same in N = 1 marginal deformations of N = 4.11 5. The Partition Function over BPS States In this section we will compute the partition function over BPS states that are annihilated by Q and S in N = 4 Yang Mills at zero coupling and strong coupling. We perform the first computation using the free Yang Mills action, and the second computation using gravity and the AdS/CFT correspondence, together with a certain plausible assumption. Specifically, we assume that the supersymmetric density of states at large charges is dominated by the supersymmetric black holes of [6–8]. At small values of chemical potentials (when these supersymmetric partition functions are dominated by charges that are large in units of N 2 ) we find that these partition functions are qualitatively similar at weak and strong coupling but differ in detail, in these two limits. Moreover, each of these partition functions differs qualitatively from index computed in the previous section. Before turning to the computation, it may be useful to give a more formal description of the BPS states annihilated by Q and S. Q may formally be thought of as an exterior derivative d, its Hermitian conjugate S is then d ∗ and
is the Laplacian dd ∗ + d ∗ d. States with
= 0 are harmonic forms that, according to standard arguments (see [37], those arguments may all be reworded in the language of Q and S and Hilbert spaces) are in one to one correspondence with the cohomology of Q. I W L , the (−1)degr ee weighted partition function over this cohomology is simply the (weighted) Euler character over this cohomology. 5.1. Partition function at
= 0 in free Yang Mills. Let   Z f r ee = T r
=0 x 2H eµ1 q1 +µ2 q2 +µ3 q3 +2ζ J2 ,

(5.1)

11 These theories have the superpotential T r [eβ φ φ φ − e−β φ φ φ + c(φ 3 + φ 3 + φ 3 )]. If c is nonzero, 1 2 3 1 3 2 1 2 3 then we should set all chemical potentials γi to be equal in the original N = 4 result, since we lose two of the U (1) symmetries.

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J. Kinney, J. Maldacena, S. Minwalla, S. Raju −β

where x = e 2 , and q1 , q2 , q3 correspond to the S O(6) Cartan charges (related to R1 , R2 , R3 by the formulas in Appendix C). In Free Yang Mills theory this partition function is easily computed along the lines described in Subsect. 4.1; the final answer is given by the formula [2, 3]  

T rU n T rU −n  n n+1 n f B (x , nµi , nζ )+(−1) f F (x , nµi , nζ ) Z = DU exp , n n (5.2) where U is a unitary matrix and the relevant ‘letter partition functions’ are given by fB = fF =

(eµ1 + eµ2 + eµ3 )x 2 + x 4 (1 − x 2 eζ )(1 − x 2 e−ζ ) x 3 (2 cosh ζ e

µ1 +µ2+µ3 2

µ1+µ2−µ3

µ1−µ2+µ3

−µ1 +µ2 +µ3 2

+e 2 +e 2 +e (1 − x 2 eζ )(1 − x 2 e−ζ )

)−x 5 e

µ1 +µ2 +µ3 2

.

(5.3)

As explained in the previous section, (5.2) and (5.3) describe a partition function that undergoes a phase transition at finite values of chemical potentials. For chemical potentials such that f B + f F < 1, the integral in (5.2) is dominated by a saddle point on which |T rU n | = 0 for all n. In this phase the partition function is obtained from the one loop integral about the saddle point (as in Sect. 4.1) and is independent of N in the large N limit. The density of states√in this phase grows exponentially with energy, ρ(E) ∝ eβ H E , where β H = − ln( 7−32 5 ) = 1.925 and the system undergoes a phase transition when the effective inverse temperature becomes smaller than β H (e.g., on −β H

setting all other chemical potentials to zero, this happens at x = e 2 ). At smaller values of chemical potentials (5.2) is dominated by a new saddle point. In particular, in the limit ζ  1 and β  1, the integral over U in (5.1) is dominated by a saddle point on which T rU n T rU −n = N 2 for all n, the partition function reduces to  1 f B (x n , nµi , nζ ) + (−1)n+1 f F (x n , nµi , nζ ) . ln Z = N 2 (5.4) n n In the rest of this subsection we will, for simplicity, set µ1 = µ2 = µ3 = µ and thereby focus on that part of cohomology with q1 = q2 = q3 ≡ q. The relevant letter partition functions reduce to fB = fF =

3eµ x 2 + x 4 , (1 − x 2 eζ )(1 − x 2 e−ζ )  3µ µ e 2 (2 cosh ζ − x 2 ) + 3e 2 x 3 (1 − x 2 eζ )(1 − x 2 e−ζ )

.

(5.5)

In the limit β  1, ζ  1 (5.4) reduces to ln Z = N 2 where

1 f (µ), (β 2 − ζ 2 )

 µ 3µ f (µ) = ζ (3) + 3Pl(3, eµ ) − 3Pl(3, −e 2 ) − Pl(3, −e 2 )

(5.6)

(5.7)

Index for 4 Dimensional Super Conformal Theories

229

and the PolyLog function is defined by Pl(m, x) =

∞  xn . nm

(5.8)

n=1

This partition function describes a system with energy E, angular momentum j2 , S O(6) charge (q, q, q) and entropy S given by12 2 j1 E ∼ 2 2 N N 2 j2 N2 q N2 S N2 where f  (µ) = g(µ) = 3



β f (µ) , (β 2 − ζ 2 )2 ζ f (µ) =2 2 , (β − ζ 2 )2 g(µ) = 2 , β − ζ2 3 f (µ) − µg(µ) = , β2 − ζ 2 =2

 µ 3µ 1 1 2 2 Pl(2, e ) − Pl(2, −e ) − Pl(2, −e ) . 2 2 µ

(5.9)

(5.10)

We see that for high temperatures, this partition function looks like a gas of massless particles in 2+1 dimensions. Note that in this limit E ∼ 2 j1  q. We will sometimes be interested in the partition function with only those chemical potentials turned on that couple to charges that commute with Q and S. This is achieved if we choose µ = β3 . In the limit β  1, ζ  1 we have µ  1 and the partition function and charges are given by (5.6) and (5.9) with µ ∼ 0; note that f (0) = 7ζ (3) 2 and g(0) = π4 . Note that, although the index IYWML and the cohomological partition function Z free both traces over Q cohomology, the final results for these two quantities in Free Yang Mills theory are rather different. For instance, at finite but small values of chemical potentials, ln Z free is proportional to N 2 (see (5.6)) while IYWML is independent of N (see (4.7)). Clearly cancellations stemming from the fluctuating sign in the definition of IYWML cause the index to see a smaller effective number of states. In Appendix D we explain, in more detail, how this might come about. 5.2. Cohomology at strong coupling: Low energies. We now turn to the study of Q cohomology at strong coupling and low energies. In this limit the cohomology is simply that of the free gas of supergravitons in Ad S5 × S 5 , and may be evaluated following the method of Subsect. 4.2. We will calculate the quantity   (5.11) Z = Tr x 2H z 2J1 y 2J2 v R2 w R3 12 Physically, the equations below describe Free Yang Mills theory at fixed values of charges in the limit T → 0 (T is the temperature). In the free theory this limit retains only supersymmetric states at all values of charges. On the other hand the black holes in [6–8] are supersymmetric in the same limit only for a subfamily of charges. See the next section for more discussion on this puzzling difference.

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over the supergraviton representations restricted to states of
= 0. We recall that the single particle states form an infinite series of short reps of the N = 4 superconformal algebra where the primary is a lorentz scalar with energy n with R-charges [0, n, 0]. The trace over single particle states may be easily calculated. The answer is13 numbos + numfer , den den = (1 − x 2 /w)(1 − x 2 v)(1 − x 2 w/v)(1 − x 2 z/y)(1 − x 2 zy), Z sp =

numfer = x 3 /y + x 3 y + x 3 z/v + vx 3 z/w + wx 3 z − 2x 5 z +vx 7 z + x 7 z/w + wx 7 z/v + x 7 z 2 /y + x 7 z 2 y, numbos = vx 2 + x 2 /w + wx 2 /v − x 4 /v − vx 4 /w − wx 4 +2x 6 + x 6 z/(yv) + vx 6 z/(wy) + wx 6 z/y − x 8 z/y +x 6 zy/v + vx 6 zy/w + wx 6 zy − x 8 zy + x 4 z 2 + x 10 z 2 .

(5.13)

The full (multi-particle) partition function over supersymmetric states may be obtained by applying the formulas of Bose and Fermi statistics to (5.13). Special limits of (5.13) will be of interest in the next section. For instance, the limit z → 0 focus on states with
= 0 and j1 = 0, i.e. (1/8) BPS states. In this limit (5.13) becomes 1/8

1 − (1 − x 2 /w)(1 − vx 2 )(1 − wx 2 /v) + x 6 , (1 − x 2 /w)(1 − vx 2 )(1 − wx 2 ) x 3 (y + 1/y) . = 2 (1 − x /w)(1 − vx 2 )(1 − wx 2 /v)

Z bos−sp = 1/8

Z fer−sp

(5.14)

In terms of the γi variables introduced at the end of Subsect. 4.1,

1/8

1 − (1 − e−2γ1 )(1 − e−2γ2 )(1 − e−2γ3 ) + e−2(γ1 +γ2 +γ3 ) , (1 − e−2γ1 )(1 − e−2γ2 )(1 − e−2γ3 )   e−γ1 −γ2 −γ3 eζ + e−ζ . = (1 − e−2γ1 )(1 − e−2γ2 )(1 − e−2γ3 )

Z bos−sp = 1/8

Z fer−sp

(5.15)

13 In the notation of the previous subsection, with y = eζ , res res    res = T r x 2H y 2J2 u 2 qi = numbos + numfer , Z sp res den den = (1 − x 2 u 2 )3 (1 − x 2 /y)(1 − x 2 y),

numfer = 3ux 3 − 2u 3 x 5 + 3u 5 x 7 + (u 3 x 3 )/y + (u 3 x 7 )/y +u 3 x 3 y + u 3 x 7 y, numbos = 3u 2 x 2 + x 4 − 3u 4 x 4 + 2u 6 x 6 + u 6 x 10 +(3u 4 x 6 )/y − (u 6 x 8 )/y + 3u 4 x 6 y − u 6 x 8 y.

(5.12)

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Applying the formulas for Bose and Fermi statistics, it is now easy to see that the partition function over the Fock space, including multi-particle states, is given by Z 1/8 (ζ, γ1 , γ2 , γ3 )  ∞ sζ −(2n+1)γ1 −(2m+1)γ2 −(2r +1)γ3 )  s=±1 (1 + e e = . (1 − e−2nγ1 −2mγ2 −2r γ3 )(1 − e−(2n+2)γ1 −(2m+2)γ2 −(2m+2)γ3 ) n,m,r =0

(5.16) Finally, in order to compute the rate of growth of the cohomological density of states with respect to energy, we set z, y, v, w → 1. This gives the “blind” single particle partition function which is x 2 (3 − 2x 2 + 8x 4 − 2x 6 + x 8 ) , (1 − x 2 )5 x 3 (5 − 2x 2 + 5x 4 ) = . (1 − x 2 )5

bl Z bos−sp = bl Z fer−sp

(5.17)

The full partition function is given by Z

bl

= exp

  bl bl  Z bos−sp (x n ) + (−1)n+1 Z fer−sp (x n ) n

n

.

(5.18)

Let x =e

−β 2

.

(5.19)

At small β this partition function is approximately given by ln Z =

63ζ (6) . 4β 5

(5.20)

It follows that the entropy as a function of energy is given by S(E) = h log n(E) ∼

6 5



315ζ (6) 4

1 6

E 5/6 ≈ 2.49E 5/6 .

(5.21)

Note that this is slower than the exponential growth of the same quantity at zero coupling.

5.3. Cohomology at strong coupling: High energies. Gutowski and Reall [6, 7], and Chong, Cvetic, Lu and Pope [8] have found a set of supersymmetric black holes in global Ad S5 × S 5 , that are annihilated by the supercharges Q and S. These black holes presumably dominate the supersymmetric cohomology at energies of order N 2 or larger. In this subsection we will translate the thermodynamics of these supersymmetric black holes to gauge theory language, and compare the results with the free cohomology of Subsect. 5.1.

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Restricting to black holes with q1 = q2 = q3 = q these solutions constitute a two parameter set of solutions, with thermodynamic charges, translated to Yang Mills language via the AdS/CFT dictionary14 , E N2 j1 + j2 N2 j1 − j2 N2 q N2

[(1 − a)(1 − b) + (1 + a)(1 + b)(2 − a − b)] , 2(1 − a)2 (1 − b)2 (a + b)(2a + b + ab) , 2(1 − a)2 (1 − b) (a + b)(a + 2b + ab) , 2(1 − a)(1 − b)2 (a + b) , 2(1 − a)(1 − b) √ π(a + b) a + b + ab . (1 − a)(1 − b)

= (a + b) = = =

S = N2

(5.22)

Setting a = 1 − (β  + ζ  ) and b = 1 − (β  − ζ  ), and assuming β   1, ζ   1, (5.22) reduces to 2 j1 E 8β  ∼ ∼ ,   N2 N2 (β 2 − ζ 2 )2 2 j2 −8ζ  ∼ ,   N2 (β 2 − ζ 2 )2 q 1 ∼ 2  , N2 β −ζ 2 √ S 2 3π ∼ 2  . N2 β −ζ 2

(5.23)

Equations (5.22) and (5.9) are have some clear similarities15 in form, but also have one important qualitative difference. Equation (5.9) has one additional parameter absent in (5.22). After setting the three S O(6) charges equal the Q cohomology is parametrized by 3 charges, whereas only a two parameter set of supersymmetric black hole solutions are available. We should emphasize that in the generic, non-BPS situation black hole solutions are available for all values of the 4 parameters q, j2 , j1 and E [8]. It is thus possible to continuously lower the black hole energy while keeping q, j2 and j1 fixed at arbitrary values. The temperature of the black hole decreases as we lower its energy, until it eventually goes to zero at a minimum energy. However the extremal black hole thus obtained is supersymmetric (its mass saturates the supersymmetric bound) only on a 2 dimensional surface in the 3 dimensional space of charges parameterized by q, j2 and j1 . For every other combination of charges the zero temperature black holes are not supersymmetric (their mass is larger than the BPS bound). We are unsure how this should be 3 14 We have set g = 1 in [8] and set E C F T = E Chong et al /G 5 , where G 5 = G N 5 /R Ad S is the value of Newton’s constant in units where the Ad S5 radius is set to one. Sher e = SChong et al /G 5 . For N = 4 Yang Mills we have G 5 = π 2 . To convert formulas in [6, 7] simply set this value for the five dimensional Newton 2N constant in their expressions. 15 This observation has also been made by H. Reall and R. Roiban.

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233

interpreted16 . It is possible that, for other combinations of charges, the cohomology is captured by as yet undiscovered supersymmetric black solutions. In order to compare the cohomologies in (5.9) and (5.22) in more detail, we choose µ in (5.9) so that the equations for E/N 2 and q/N 2 in (5.9) and (5.22) become identical (after a rescaling of β  and ζ  ). This is achieved provided that f (µc )2 = 16g(µc )3 .

(5.24)

This equation is easy to solve numerically. We find µc = −0.50366 ± .00001 and that f (µc ) = 5.7765, g(µc ) = 1.2776. Plugging in µ = µc into the entropy formula in (5.9) we then find SField SBlack−Hole

=

f (µ) −µ 3 g(µ) = 1.2927. √ 2 3π

(5.25)

Another way to compare (5.9) and (5.22) is the following. First notice that the charge q is much smaller than the energy in this limit, q  E. Let us set µ = β/3 which is the value that we have in the index (though we do not insert the (−1) F we have in the index). Since we are taking the limit where β is small we can evaluate f in (5.9) at zero, f (0) = 7ζ (3). By comparing the energies and entropies in (5.9) and (5.22) and writing the free energy as E = N 2 cβ −3 , where c is a “central charge” that measures the number of degrees of freedom, then we can compute cgravity cfree−field−theory

=

π3 ∼ 0.35458 . . . . 14ζ (3)33/2

(5.26)

It is comforting that this value is lower than one since we expect that interactions would remove BPS states rather than adding new ones. A similar qualitative agreement between the weak and strong coupling was observed between the high temperature limit of uncharged black holes and the free Yang Mills theory [38], where the ratio (5.26) is 3/4. Note that for µ = β/3 we can approximate g in the expression for the charge in (5.9) by g(0) = 0. This agrees qualitatively with the expression coming from black holes. 5.4. Cohomology at intermediate energies: Giant gravitons and small black holes. Let us set j2 = 0 or a = b in (5.22). We then expand the resulting expression for low values of a, E q ∼ 3a ∼ 3 2 , N2 N j1 ∼ 3a 2 , N2 √ S ∼ 2π 2a 3/2 . 2 N

(5.27)

It is possible to count the entropy of these black holes using D-branes in Ad S. This is not the same problem as counting them in the field theory, but perhaps these results might be a good hint for the kind of states that we should look at in the field theory. 16 Note that our index I W L , when specialized to states with q = q = q , also depends on two rather than 1 2 3 YM 3 parameters.

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In the small charge limit the black holes look very similar to the black holes that appear in toroidal compactifications of type IIB on T 5 . Let us recall first how the entropy of these black holes is counted [9]. We view the black holes as arising from two sets of intersecting D3 branes n 3 and n 3 which intersect along a circle which is one of the . One can then add momentum L 0 along this circle. Their entropy is given circles of T 5 by S = 2π n 3 n 3 L 0 . This entropy arises as follows. Let us focus on the T 4 that is orthogonal to the common circle. The D3 branes can form any homomorphic surface on this T 4 . The number of complex moduli of these surfaces goes as n 3 n 3 . There is an equal number of Wilson lines and there are 4n 3 n 3 fermions. This gives central charge c = 6n 3 n 3 and then using the Cardy formula we get the entropy. We will now repeat the same counting for small black holes in our context. First we recall that the theory contains giant graviton D3 branes which can carry some of the charge. Let us recall the description in [39] for giant gravitons on the 5-sphere. We 3 take an arbitrary holomorphic 2-complex dimensional surface in5 C and we intersect it with |z i | = 1. This gives a 3-real dimensional surface on S which will be a giant graviton. Let us focus first on surfaces that are invariant under ψ translations, where ψ is an angle that rotates all z i → eiψ z i . The holomorphic surface in C 3 is specified by a homogeneous polynomial of degree n,  n 1 +n 2 +n 3 =n

Cn 1 ,n 2 ,n 3 z 1n 1 z 2n 2 z 3n 3 = 0.

(5.28)

Think of S 5 as an S 1 fibration on C P 2 . Then (5.28) defines a holomorphic surface in C P 2 and the resulting giant graviton on S 5 consists of this surface plus the S 1 fiber which is parametrized by the angle ψ. For example, the maximum size giant graviton that wraps an S 3 in S 5 [40] corresponds to the equation z 1 = 0. The number of complex parameters characterizing the curve (5.28) goes as d ∼ n 2 /2.

(5.29)

In order to compute the charge under the U (1) gauge field that performs translations in ψ we need to know how many times this curve wraps the C P 1 inside C P 2 [41, 42]. It is easy to see that this number is n. The amount of wrapping of this curve over the C P 1 in C P 2 is n. So the total charge under the generator J that rotates all the angles is

q=



qi = N n.

(5.30)

i

We define the overall U (1) charge

q to be the change in phase when we shift ψ → ψ +2π . q ψ is the shift in phase for a state of charge

q. So we have that ei

Our strategy is as follows. The total charge that we have at our disposal is

q = 3q. We split it as

q = (3q − n N ) + n N . The second term will be realized by n giant gravitons and the first by momentum along ψ. In other words, the n giant gravitons are D3 branes that are intersecting at points on the C P 2 (and form a smooth surface (5.28)) and are coincident along the fiber parametrized by ψ. We have many moduli of this configurations counted by (5.29). The momentum L 0 ≡ (3q − n N ) will be carried by these oscillations. In other words the D3 branes wrapped along ψ give us an effective string with central charge c = 6d = 3n 2 .

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235

Then the entropy is 



S = 2π cL 0 /6 = 2π

1 2 n (3q − n N ). 2

(5.31)

Note that we have not said anything about the value of n. We now maximize the entropy with respect to n; we get n = 2q/N ,

(5.32)

√ q 3/2 S = 2π 2 N

(5.33)

and substituting in (5.31) we obtain

in agreement with (5.27). Notice, however that there is an important difference between the computation that is done here and the usual computation for the D1D5 p system. In the latter case it is possible to vary the parameters of the compactification to go to a regime where the amount of energy contained in momentum is much smaller than the energy of the D-branes, which is a necessary condition for being able to view the momentum as producing small oscillations on the D-branes. In the discussion of this section it is not possible to satisfy this condition. Equation (5.32) implies that the energy contained in oscillations of the branes is comparable to the brane tensions, and there is no obvious parameter that we can adjust to change this fact17 . As a consequence the discussion of this section falls short of qualifying as a completely satisfactory derivation of (5.33) (note, nonetheless, that all factors work bang on). This point of view lets us also heuristically understand why we need to have angular momentum j1 . This arises as follows. The system we had above is very similar to the D1D5 p system in the NS sector, since the fermions are anti-periodic in the ψ direction. Recall that the D1D5 p black hole has j1 = 0 [43](though j2 can be non-zero18 ) and this black hole naturally arises in the Ramond sector. When we perform a spectral flow to c the NS sector we get j1 = 12 . In our case, we cannot choose the fermions to be periodic along ψ due to the way the circle is fibered over C P 2 . However, writing down the same 2 2 c = n4 = Nq 2 . formula as we had for the D1D5 p in the NS sector we would get j1 = 12 2

On the other hand we get j1 = 3 Nq 2 from (5.27), which has a different numerical coefficient. It would be nice to compute j1 properly and see whether it agrees with the black hole answer. 6. Partition Functions over 21 ,

1 th 4

and

1 th 8

BPS States th

In this section we will study the partition function over 18 , a quarter and half BPS supersymmetric states in N = 4 Yang Mills. We will compute these partition functions in free Yang Mills, at weak coupling using naive classical formulas, and at strong couth pling using the AdS/CFT correspondence. In the case of quarter and 18 BPS states, our 17 One would like to increase the radius of the ψ circle without changing anything else, but this would not be a solution to the gravity equations. 18 It can be seen that for small black holes the formulas in [8] allow j to be non-zero with a bound similar 2 to the one in [43].

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free and weak coupling partition functions are discontinuously different. However the weak coupling and strong coupling partition functions agree with each other (see [44] for an explanation). 1 th It is possible that something similar will turn out to be true for the 16 cohomology (see [1] for a possible mechanism). This makes the enumeration of the weakly coupled Q cohomology an important problem. We hope to return to this problem in the near future. th

6.1. Enumeration of 18 , quarter and half BPS cohomology. In this subsection we will enumerate operators in the anti-chiral ring, i.e. operators that are annihilated by Q α1 , with α = ± 21 , and their Hermitian conjugates (these are the charges we called Q and Q  in previous sections19 ). All such states have
= 0 and j1 = 0. It is not possible to isolate the contribution of these states to IY M (note the index lacks a chemical potential that couples to j1 ); nonetheless we will be able to use dynamical information to count these states below. This enumeration is easily performed in the free theory. Only the letters X, Y, Z , ψ0,±,+++ (see Table 2) and no derivatives contribute in this limit. We will denote these ¯ i (i = 1 . . . 3) and W¯ α˙ (α˙ = ±) below. Note that these letters all have j1 = 0 letters by  and E = q1 + q2 + q3 . The partition function    µi qi + 2ζ j2 Z cr − f r ee = T r exp (6.1) i

is given by the expression on the RHS of (5.2) with fB =

3 

eµi ,

f F = 2 cosh ζ e

µ1 +µ2 +µ3 2

.

(6.2)

i=1

Note that 1 − f B − f F is positive at small µi but negative at large µi . We conclude that the partition function (6.1) undergoes the phase transition described in [2, 3] at finite values of the chemical potentials, and that its logarithm evaluates to an expression of order N 2 at small µi . We now turn to the weakly interacting theory. As explained in [5, 44], at any nonzero coupling no matter how small, the set of supersymmetric operators is given by all ¯ i , W¯ α˙ modulo relations [ ¯ i,  ¯ j ] = [ ¯ i , W¯ α˙ ] = 0 gauge invariant anti-chiral fields  and {W¯ α˙ , W¯ β˙ } = 0 (the first of these follows from ∂¯ i W¯ = 0 where W¯ is the superpotential). In general there can be corrections to these relations (see [44] ). We assume that such corrections do not change the number of elements in the ring. In fact, if we go to the Coulomb branch of N = 4 we get a U (1) N theory with no corrections at the level of the two derivative action. The chiral primary operators at a generic point of this moduli space are the same as all the operators that we are going to count. It is now easy to enumerate the states in the chiral ring. The relations in the previous paragraph ensure that all the basic letters commute or anticommute, and so may be simultaneously diagonalized, so we must enumerate all distinct polynomials of traces of diagonal letters. This is the same thing as enumerating all polynomials of the 3N bosonic and 2N fermionic eigenvalues that are invariant under the permutation group S N . We 19 If we had chosen states annihilated by Q ¯ α˙ we would have obtained the chiral ring. 1

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237

f f may now formally substitute the eigenvalues φ¯ i and W¯ α˙ ( f = 1 . . . N ) with bosonic f f and fermionic creation operators ai and wα˙ ; upon acting on the vacuum these produce states in the Hilbert space of N particles, each of which propagates in the potential of a 3 dimensional bosonic and a 2 dimensional fermionic oscillator. The permutation symmetry ensures that the particles are identical bosons or fermions depending on how many fermionic oscillators are excited. As a consequence we conclude that the cohomological partition function is given by the coefficient of p N in

Z 1/8 ( p, γ1 , γ2 , γ3 , ζ ) ∞  p N Z N (γ1 , γ2 , γ3 , ζ ) = =

N =0 ∞ 



n,m,r =0

(1 −

sζ −(2n+1)γ1 −(2m+1)γ2 −(2r +1)γ3 ) s=±1 (1 + p e e . p e−n2γ1 −m2γ2 −r 2γ3 )(1 − p e−(2n+2)γ1 −(2m+2)γ2 −(2m+2)γ3 )

(6.3)

Further discussion on these 1/8 BPS states can be found in [10]. We may now specialize both the free and the interacting cohomologies listed above to 1 th 4 BPS cohomology by taking the limit γ3 → ∞. The only letters that contribute in this ¯ 1 and  ¯ 2 (X, Y of Table 2). The final result for the interacting cohomology limit are  may be written as Z 1/4 ( p, γ1 , γ2 ) =

∞  N =0

1 . (1 − p e−n2γ1 −m2γ2 ) n,m=0

p N Z N (γ1 , γ2 ) = ∞

(6.4)

For a more explicit construction of 1/4 BPS operators see [45] and references therein. It is instructive to compare the γ3 → ∞ limit (6.4) of (6.3) to the same limit of the partition function over Q cohomology of the previous section that also simplifies dramatically in this limit.The only letters that contribute in this limit are X, Y, +,++− (where the indices refer to j1 , q1 , q2 , q3 charges). Further, it is easy to verify that Q+,++− ∝ [X, Y ]. As a consequence the matrices X and Y should commute and may be diagonalized; furthermore the matrix ψ must also be diagonal (so that Q annihilates it). The cohomology in this limit is thus given by the partition function of N particles in a 2 bosonic and one fermionic dimensional harmonic oscillator. Z=



p N T r [y 2J1 e−γi L i ] =

 (1 + pye−2(n+1)γ1 −2(m+1)γ2 ) . (1 − pe−2nγ1 −2mγ2 )

(6.5)

n,m≥0

N

The index I W L over this cohomology is then computed by setting y = −1. At this special value, terms in the numerator with values m, n cancel against terms in the denominator with m + 1, n + 1 leaving only   IYWML = IYWML (N ) = p N T r N [(−1) F e−γi L i ] N

=

(1 − p)

N

1 . −n2γ1 )(1 − pe−n2γ2 ) (1 − pe n=1

∞

(6.6)

This is an exact formula for the γ3 → ∞ limit of the index IYWML . Multiplying it with (1 − p) and setting p to unity, we recover the large N result (4.7) (at γ3 = ∞).

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It is also possible to further specialize (6.4) to the half BPS cohomology (of states annihilated by supercharges with q1 = 21 ) by taking the further limit γ2 → ∞ to obtain Z 1/2 ( p, γ1 ) =

∞  N =0

1 . −n2γ1 ) n=0 (1 − p e

p N Z N (γ1 ) = ∞

(6.7)

Note that the free half BPS cohomology, interacting half BPS cohomology and the γ2 , γ3 → ∞ limit of IYWML all coincide. On the other hand the free quarter BPS cohomology sees many more states than the interacting quarter BPS cohomology which, in turn, sees a larger effective number of states than the γ3 → ∞ limit of the index. The ¯ 1,  ¯ 2 and ψ+,0;++− , which are last quantity, the index, receives contributions from  all the states in Table Two which have L 3 = 0. This index sees a smaller number of states as a consequence of cancellations involving the presence of the fermion ψ+,0;++− . th Again, the 18 BPS free cohomology sees more states than the interacting cohomology, which in turn sees more states than the index, with no restrictions on chemical potentials. More explicitly, we can see that for very large charges, or very small chemical potentials the entropy of (6.7) is that of N harmonic oscillators, which correspond basically to the eigenvalues. Similarly, (6.4), and (6.3) give the entropy of 2N and 3N harmonic oscillators respectively. All these entropies are basically linear in N in the large N limit. The intuitive reason is that the matrices commute, and so do not take advantage of the full non-abelian structure of the theory. 6.2. Protected double trace operator in the 20. As an example of a practical application of the exact partition function over the chiral ring (derived in the previous subsection) and the index IY M (defined and computed in Sect. 3 and 4), in this subsection we will demonstrate that the scaling dimension of a particular double trace Yang Mills operator is protected.20 Consider SU (N ) N = 4 Yang Mills theory. Let us first study the set of operators with quantum numbers (q1 , q2 , q3 ) = (3, 1, 1) and j1 = j2 = 0. Using
≥ 0 we conclude that such operators have E ≥ 5; we will be interested in operators that saturate this equality. Let us first consider free Yang Mills theory. The set of all such operators is easy to list; we find T r [Wα W α ]T r [X 2 ] , T r [Wα X ]T r [W α X ], T r [X 2 ]T r [X Y Z ], T r [X 2 ]T r [X Z Y ] , T r [X Y ]T r [X 2 Z ] , T r [Y Z ]T r [X 3 ].

T r [X Z ]T r [X 2 Y ], (6.8)

Turning now to the interacting theory, we note that all but one of these operators belongs to the 1/8 BPS chiral ring, and so has protected scaling dimension21 . Indeed it is not difficult to check that the appropriate coefficient in (6.3) (after subtracting the U (1) part and the single trace contribution) is 6 implying that 6 of the 7 operators in (6.8) have protected dimensions. The unprotected operator in (6.8) is simply O  = tr [X 2 ]tr [X [Y, Z ]]. Note that the operators studied in the previous paragraph have
= 0, L 1 = 6, L 2 = 2, L 3 = 2, J2 = 0. Notice that states with quantum numbers (q1 , q2 , q3 ) = (5/2, 1/2, 1/2), j1 = 21 , j2 = 0 share these values for
, L i , j2 ; (and, moreover, are 20 This subsection was added in the revised version to answer a question raised by M. Bianchi. 21 In general the interacting operator with good scaling dimension will have a complicated form, admitting

admixtures with single trace operators.

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239

unique in this regard in the double trace sector of the SU (N ) theory). As a consequence we will now list all double trace operators in the free theory with these quantum numbers. They are tr [ψ+,−++ X ]tr [X 2 ], tr [ψ+,+−+ Y ]tr [X 2 ], tr [ψ+,++− Z ]tr [X 2 ], tr [ψ+,+−+ X ]tr [X Y ], tr [ψ+,++− X ]tr [X Z ].

(6.9)

It follows from the discussion above that contribution of double trace operators with L 1 = 6, L 2 = L 3 = 4, J2 = 0 to the index IY M is (7 − 5)e−6γ1 −2γ2 −2γ3 = 2e−6γ1 −2γ2 −2γ3 . As IY M is not renormalized, it must be that 4 out of the 5 operators listed in (6.9) are exactly protected. The single non-protected operator is easily identified; at infinite N it is the operator
 O = tr [ψ+,−++ X ] + tr [ψ+,+−+ Y ] + tr [ψ+,++− Z ] tr [X 2 ] = Q 1 tr [X X¯ + Y Y¯ + Z Z¯ ]tr X 2 . + 2 ,1

(6.10)

Note that Q − 1 ,1 O ∝ O  ; we see that the two non-protected operators O and O  are 2 married together in the same long multiplet. We have concluded that four double trace operators of the form (6.9) are exactly protected. At the end of this subsection we will demonstrate that while three of these four operators are SU (1, 2|3) descendants, a fourth is and SU (1, 2|3) primary. As we have explained in Sect. 3, the decomposition of the index IY M into characters of SU (1, 2|3) yields information about linear combinations of the number of short representations of the Yang Mills theory. In the case at hand, the existence of precisely one protected primary with these quantum numbers implies the existence of exactly one double trace cc type representation with quantum numbers q1 = 2, q2 = q3 = 0, (or R1 = R3 = 0, R2 = 2), j1 = j2 = 0 (such a representation has E = 4). This is an operator of the schematic form O I J Q J K − 16 δ I K O L J Q J L , where O I J is the single trace operator in the 20 of S O(6). (This form is schematic because this operator will mix with single trace operators, see for example [46].) Indeed this operator was studied in [47–49] and a proof that it is protected was was given in [50–52], based on the non-renormalization theorem in [53]. The arguments of this subsection may be regarded as an alternate proof of this non-renormalization. To complete this subsection we will now demonstrate that 3 of the operators in (6.9) are SU (1, 2|3) descendants. In fact the operators in question will turn out to be descendants of 1/2 and 1/4 BPS states in bb representations. In other words, some of them result from the action of SU (1, 2|3) generators on conformal primaries which have lower conformal weight. So let us understand the protected bb representations with E = 4. One of them arises from the 1/2 BPS chiral primary operator tr [X 2 ]2 . We can now consider the SU (1, 2|3) descendants of it. By analyzing in more detail the action of the supercharges we find that two of the states in (6.9) are SU (1, 2|3) descendants of tr [X 2 ]2 . Another operator that we should consider is the 1/4 BPS double trace operator that is in the 84 (the protected nature of this operator follows from the partition function of chiral primary operators (6.3), or (6.4)-in other words, it gives rise to operators in the chiral ring). This operator has SU (4) Dynkin labels R = (2, 0, 2). It turns out that there is one SU (1, 2|3) descendant of the 84 with the quantum numbers appearing in (6.9), completing our demonstration.

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6.3. Large N limits and phase transitions. In this subsection we will study the large N limit of the partition functions (6.3), (6.4), (6.7). We will first briefly consider the limit N → ∞ at fixed values of the chemical potentials, and show that in this limit these partition functions reproduce the supergravity answers (5.15). We will then turn to large N limits in which the chemical potentials scale with N . We find that the formulas for 1/4 and 1/8 BPS states lead to large N phase transitions. This phase transition is the Bose-Einstein condensation of the lowest mode, the ground state of the harmonic oscillators we had in the previous subsection. In the N → ∞ and fixed chemical potential the partition functions (6.7), (6.4), (6.3), become independent of N . This limit is most easily evaluated by multiplying the partition functions by (1 − p)22 and setting p = 1. The entropy then grows as a gas of massless particles in one, two and three dimensions respectively. For half BPS states we have [54] 1 . (1 − e−n2γ1 ) n=0

Z 1/2 (γ1 ) = ∞

(6.11)

Clearly, in the large N limit, (6.11) may be thought of as the multiparticle partition function for a system of bosons with Z 1/2−sp =

∞  n=1

e−2nγ1 =

1 − 1; 1 − e−2γ1

(6.12)

note that (6.12) is the same as the supergravity result (5.15) in the limits γ2 → ∞, γ1 → ∞. Similarly the large N limit of (6.4) is the multiparticle partition function for a system of bosons whose single particle partition function is  1 Z 1/4−sp = − 1, (6.13) e−2nγ1 −2mγ2 = −2γ1 )(1 − e−2γ2 ) (1 − e n,m which is the same as (5.15) in the limit γ3 → ∞. In a similar fashion, in the large N limit of (6.3) is precisely the multiparticle partition function (5.16), a system of bosons and fermions, whose single particle partition functions, are given by (5.15). We now turn to large N limits of these partition functions in which we will allow the chemical potentials to scale with N . Let us start with the 1/2 BPS case, and set γ1 = γ . This case does not have a phase transition. We write

∞  1 −2nγ log(1 − p e )∼− d x log(1 − p e−x ). (6.14) log Z (γ , p) = − 2γ 0 n We can first solve for p by writing

∞ 1 p e−x 1 log(1 − p). dx =− N = p ∂ p log Z = 2γ 0 1 − p e−x 2γ

(6.15)

≡ N 2γ . Then (6.15) is independent of N and it has a solution for We can now write β all values of β . We can then write the partition function as  ∞  1 −x −β −β log Z N (γ ) = N d x log[1 − (1 − e )e ] − log(1 − e ) . (6.16) 0 β 22 This step cancels the divergent contribution of the ground state of the harmonic oscillator in this limit. We will have a lot more to say about this below.

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241

. For large We see that this formula is of order N . There is no transition as a function of β  1, it turns out that (6.16) is independent of N when expressed in terms of values of β γ . This can be most easily seen by setting p = 1 in (6.14). As expected the change in behavior happens at a temperature (2γ )−1 ∼ N which is when the trace relations start we find that (6.16) becomes log Z N ∼ N [− log β +1], being important. For very small β which captures the large temperature behavior of N harmonic oscillators plus an 1/N ! statistical factor. Let us now consider 1/4 BPS states. Let us set γ1 = γ2 = γ . For sufficiently large temperatures we approximate the partition function as

∞  1 −(n 1 +n 2 )2γ log Z (β, p) = − log(1 − pe )∼ d x x[− log(1 − pe−x )]. 2 (2γ ) 0 n ,n 1

2

(6.17) Now we find a new feature when we compute 1 N= (2γ )2

dxx

1 pe−x = Pl[2, p], 1 − pe−x (2γ )2

(6.18)

where Pl[2, p] is the PolyLog function. Now we see that for the lowest value of the chemical potential, p = 1, we get Nmax =

1 π2 . (2γ )2 6

(6.19)

√ ≡ 2γ N we see that there is a critical temperature, β c2 = π 2 , at which there Defining β 6 is a phase transition obtained by setting Nmax = N in (6.19). At temperatures smaller than this temperature we have a Bose-Einstein condensation of the ground state of the harmonic oscillator. In this regime the free energy Z N (β) is given by (6.17) with p = 1. For higher temperatures we are supposed to solve for p using (6.18) and then insert it in (6.17) to compute the free energy. We get  ∞  1  ) = N )e−x )] − log p(β ) , log Z N (β d x x[− log(1 − p( β (6.20) 2 0 β ) is the solution to (6.18). Then for large temperatures we have log Z N ∼ where p(β 2 + 1] which captures the entropy of N 2-dimensional harmonic oscillators N [− log β c we have a second order plus the 1/N ! statistical factor. It is possible to see that at β phase transition. One can find similar results for the 1/8 BPS states. We set γi = γ . In this case  = 2γ N 1/3 . The results are similar. For low the rescaled temperature is given by β temperatures the answer is independent of N and for high temperatures we have a free  . Again there energy which is linear in N and is a function of the rescaled temperature β is a second order phase transition corresponding to the Bose-Einstein condensation of the ground state of the harmonic oscillator. If we think of these harmonic oscillators as arising from D3 branes wrapping the S 3 , then we could think of this condensation as responsible for the fact that the S 3 is contractible, in the spirit of the transition in [55]. It would be nice to see if this can be made more precise.

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7. Discussion In this paper we have considered an index that counts protected multiplets for general four dimensional superconformal field theories. This quantity captures all the information about protected multiplets that can be obtained purely from group theory. We have focused on the applications of this index to the N = 4 Yang Mills theory. It is possible that (and it would be interesting if) our index turns out to be a useful tool in the study of N = 1 and N = 2 superconformal theories as well. Indices of the form that we have constructed have obvious counterparts in superconformal theories in d = 3, 5, and 6. It is possible that some of these indices (whose theory we have not worked out in detail) could have interesting applications. The later half of this paper was devoted to a study of the supersymmetric states of N = 4 Yang Mills on S 3 . We computed this index for this theory and found that it precisely agrees with the free supergravity spectrum when we take the large N limit. The index, however, does not reflect the large entropy of BPS black holes in Ad S5 . This is not a contradiction because there is no clear argument from the supergravity point of view which says that the black holes should contribute to the index. A satisfactory Yang Mills accounting of the entropy of the BPS black holes of [6–8] remains an important outstanding problem. We have not even awareness of a field theoretic understanding of a rather gross feature of these black holes; the fact that supersymmetric solutions are known only when a certain special relation between the charges is obeyed. We think it should be possible to use weakly coupled Yang Mills theory to count the entropy of BPS black holes in Ad S5 × S 5 . In such a counting one will have to put in some extra information about the dynamics of the theory (over and above the superconformal algebra), see Sect. 6. In this connection it is encouraging that the counting of BPS states in the free theory (without the (−1) F ) has some qualitative agreement with the black hole results. Acknowledgements. We would like to thank O. Aharony, D. Berenstein, M. Bianchi, R. Gopakumar, L. Grant, S. Lahiri, J. Marsano, K. Papadodimas, H. Reall, R. Roiban, G. Rossi, N. Seiberg, A. Strominger and M. Van Raamsdonk for useful discussions. The work of JM was supported in part by DOE grant #DE-FG0290ER40542. The work of SM and SR was supported in part by an NSF Career Grant PHY-0239626, DOE grant DE-FG01-91ER40654, and a Sloan Fellowship.

Appendix A. The d = 4 Superconformal Algebra A.1. The Commutation Relations. γ

γ

γ

γ˙

γ˙

γ˙

˙

γ

[(J1 )αβ , (J1 )δ ] = δβ (J1 )αδ − δδα (J1 )β , [(J2 )αβ˙˙ , (J2 )δ˙ ] = δβ˙ (J2 )αδ˙˙ − δδα˙˙ (J2 )β˙ , ˙

˙

[(J1 )αβ , P γ δ ] = δβ P α δ − (1/2)δβα P γ δ ,

[(J1 )αβ , K γ δ˙ ] = δγα K β δ˙ − (1/2)δβα K γ δ˙ , ˙

˙

˙

˙ − (1/2)δβα˙˙ P δγ , [(J2 )αβ˙˙ , P δγ ] = δβδ˙ P αγ α˙ α˙ [(J2 )αβ˙˙ , K δγ ˙ ] = δδ˙ K βγ ˙ − (1/2)δβ˙ K δγ ˙ , ˙

˙

[H, P α β ] = P α β ,

Index for 4 Dimensional Super Conformal Theories ˙

˙

[H, K α β ] = −K α β , ˙

243

˙

˙

˙

[K α β˙ , P γ δ ] = δβδ˙ (J1 )γα + δαγ (J2 )δβ˙ + δβδ˙ δαγ H, γ

[(J1 )αβ , Q γ n ] = δβ Q αn − (1/2)δβα Q γ n , γ

γ

γ

[(J2 )αβ , Q¯ n ] = δβ Q¯ αn − (1/2)δβα Q¯ n , [K α β˙ , Q γ n ] = δαγ S¯βn˙ , γ

[Pα β˙ , Q¯ n ] = δαγ Sn β˙ , 1 γn 1 1 1 Q , [H, Q¯ αn˙ ] = Q¯ αn˙ , [H, Sαn ] = − Sαn , [H, S¯αn˙ ] = − S¯αn˙ , 2 2 2 2 [r, Q γ n ] = Q γ n , [r, Q¯ αn˙ ] = − Q¯ αn˙ , [r, Sαi ] = −Sαi , [r, S¯αi˙ ] = Sαi˙ , H 4−m j j j {Sαi , Q β j } = δi (J1 )βα + δαβ Ri + δi δαβ ( + r ), 2 4m n , {Q αm , Q¯ αm˙ } = P α α˙ δm m n ¯ {Sαm , Sα˙ } = K α α˙ δm , [H, Q γ n ] =

[Ri , Q αp ] = δi Q α j − (1/m)δi Q αp , j

p

p

j

p

p

[Rnm , Rq ] = δqm Rn − δn Rqm .

(A.1)

The Cartan generators are H,J1 = (J1 )22 = −(J1 )11 , J2 = (J2 )22 = −(J2 )11 , Rn = n+1 . While we have used script letters here, the Cartan generators above are the − Rn+1 same as those used in the rest of the paper. The eigenvalue under H is the energy E, the eigenvalues under J1 , J2 are the angular momenta j1 , j2 and the eigenvalues under Ri are the R-charges ri . Notice that in the way that we have defined the generators the commutation relations of the J s and the Rs differ by a sign. For this reason, in the case of m = 2, the structure of BPS states and null vectors is not symmetric under the exchange of the J and the R quantum numbers. Rnn

A.2. An oscillator construction of the algebra. It is possible to find an explicit oscillator construction of this algebra following [56]. We introduce two sets of bosonic oscillators a α , bα˙ , α, α˙ = 1, 2 with adjoints aα , bα˙ . In addition, we introduce fermionic oscillators α n with adjoints αn , n = 1..4. As expected the a and b oscillators will transform as Lorentz spinors whereas the fermionic oscillators will transform in the fundamental representation of SU (4). The generators of the superconformal group are now defined as below: 1 H = (a α aα + bα˙ bα˙ ) + 1, 2 1 α (J1 )β = a α aβ − δβα a γ aγ , 2 1 α˙ γ˙ α˙ α˙ (J2 )β˙ = b bβ˙ − δβ˙ b bγ˙ , 2 P α α˙ = a α bα˙ , K α α˙ = aα bα˙ , 1 (A.2) Rsn = αs α n − δsn αt α t , m

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Q αn Q¯ αn˙ Sαn S¯αn˙ r C

= aα αn , = bα˙ αn , = aα αn , = bα˙ α n , = αn α n , = bα˙ bα˙ − a α aα − αn α n .

While C appears in the oscillator construction it commutes with all the generators of the algebra and is not really part of it. When we construct representations of the algebra using oscillators we fix the total value of C. Appendix B. Algebraic Details Concerned with the Index B.1. Superconformal indices from joining rules. As we have explained in Sect. 2, any index is given by the sum (2.6) where the sum runs over representations of the algebra, and the coefficients α[i] are chosen such that I evaluates to zero on every combination of short representations that can pair into a long representation. The simplest indices for the superconformal algebra are given by α[i] = 0 for all i = i 0 for some specific i o ; this choice of α[i] defines an index only when the representation i 0 never makes an appearance on the right-hand side of (2.16). An inspection of (2.16) and Table 2 shows that this is true of the representations of the form bx with R1 = 0 or R1 = 1 and representations of the form xb with any of Rm−1 = 0 or Rm−1 = 1. The number of all such representations constitutes an index. We briefly pause to list these special representations in the most physically relevant cases m = 1, 2, 4. Protected representations do not exist in the N = 1 algebra (m = 1). In the N = 2 algebra (m = 2) they consist of SU (2) R singlets with j1 = 0 and E = 2 j2 + r/2, SU (2) R doublets with j1 = 0 and E = 2 j2 + r/2 + 2 and chirality flips ( j1 ↔ j2 , r ↔ −r ) of these. In the N = 4 (m = 4) algebra they are representations with R1 = 0 = j1 and E = 2 j2 + R2 + R3 /2, with R1 = 1, j1 = 0 and E = 2 j2 + 3/2 + R2 + R3 /2, and the chirality flips ( j1 ↔ j2 , Ri ↔ R4−i ) of these. Note that this includes representations with j1 = j2 = R1 = R3 = 0 and E = R2 ; these are the famous chiral primaries (gravitons) of the N = 4 theory. Let us now turn to indices that have support on representations that do appear on the RHS of (2.16). We first consider indices built out of representations of the form ca. It follows from the first equation in (2.16) that, on an index α[ ca j1 , j2 ,r,R1 ,R j ] + α[ ca j1 − 1 , j2 ,r +1,R1 +1,R j ] = 0. 2

(B.1)

To begin with let us assume m > 1. Notice that the two representations that appear in have equal values of j2 , r  ≡ r − R1 , M ≡ R1 + 2 j1 , R j ( j = 2 . . . m − 1).

(B.2)

The number of representations with given values for these conserved quantum numbers is M + 2 (recall j1 varies in half integer units from − 21 up to M/2; see Table 2). The α coefficients for these representations are constrained by M + 1 equations. We conclude that there is exactly one index for any given set of charges (B.2) that obeys E 1 > E 2 ; this index is given by (2.18).

Index for 4 Dimensional Super Conformal Theories

245

In a very similar fashion, we conclude that (2.19) also defines an index provided E2 > E1. If E 1 = E 2 the last equation in (2.16) applies. We have a total of (M + 2)(N + 2) representations with given values for M, N , r  ≡ r + 2 j1 − 2 j2 and Rl . The α  s corresponding to these representations are constrained by (M + 1)(N + 1) equations (from the last equation in (2.16)). This leaves us with an M + 2 + N + 2 − 1 dimensional linear vector space of indices. A convenient basis for these indices is given by N + 2 indices (2.18) (see the LHS of the equation below) (plus the M + 2 indices of the form (2.19), (see the RHS of the equation below) subject to the single additional constraint (2.20). Finally we turn to the special case m = 1 (the N = 1 algebra). As we have remarked above, no representations are absolutely protected in this case. The two indices (2.18) and (2.19), formally continue to be protected; however the expressions for these indices I jL2 ,r  = I jR1 ,r  =

∞ 

(−1)( p+1) n[ cx p , j2 ,r  + p ], 2

p=−1 ∞ 

(−1) p n[x c j1 , p ,r  − p ]

(B.3)

2

p=−1

(where r  = r + 2 j1 and r  = r − 2 j2 ) now involve a sum over an infinite number of representations, and so could diverge. 1

B.2. The index I W L as a sum over characters. We define Q  ≡ Q 2 ,1 , the SU (2) partner 1 of Q ≡ Q − 2 ,1 . Q  has charge
= −2. All other supercharges either have
= 0 (Q itself and all the supercharges in the SU (1, 2|m − 1) subalgebra) or
= 2 (all other supercharges). There are also negative
bosonic generators. For example the SU (2) spin operator J1+ has
= −2 and it appears in the anticommutation relation between S and Q  . We also get negative
states among the raising operators of SU (m). Notice that we can rederive some of the results in Sect. Two as follows. We start with the anticommutation relation (3.1). From this we derive that all states should obey
≥ 0. Now suppose that we start with the highest weight state with Cartan charges |ψ0  = |E, j1 , j2 , r, Ri . This state has the lowest value of
among all the level zero states, which is
0 = E − (E 1 − 2), where (E 1 − 2) is the combination of charges appearing in the right-hand side of (3.1). Notice that we cannot lower the value of
by acting with bosonic generators since this is a highest weight state under the SU (2) × SU (2) × SU (m) subalgebra. We now start acting with the supercharges Q α,i on this state. When we act with Q  we lower the value of
. If Q  does not annihilate the state we conclude that
0 −2 ≥ 0. If it is strictly bigger, then we have the representations of the generic type, which we called a in Sect. Two. When it becomes equal to zero, then we get representations of the type c, which obey E = E 1 . The final possibility is that Q  annihilates the level zero state. Using the anticommutator of Q  and S  we notice that this can happen only if
0 + 4 j1 = 0. Since,
0 ≥ 0, this implies that j1 = 0 and E = E 1 − 2. So we have a representation called b in Sect. Two. Using some of these ideas it is possible also to understand the structure of the null vectors. For that it is convenient to consider level one states of the form J − Q  |ψ0  and Q|ψ0 . Using the anticommutation relations it is possible to show that the determinant of the 2 × 2 matrix of inner products among these states is proportional to 2 j1 (
0 − 2). So we find that in the case that
0 = 2 we have a null state. It is also easy to show

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that the state Q  |ψ0  has positive norm (if
0 ≥ 2). This latter state transforms in the representation with highest weights (E + 21 , j1 + 21 , j2 , r + 1, R1 + 1, R j ). Thus the zero norm state we just mentioned transforms in the representation of the form (E + 21 , j1 − 21 , j2 , r + 1, R1 + 1, R j ). This follows from the fact that we have one state with these quantum numbers plus the fact that if we had any other states with higher weights then we would be able to decrease the value of
for this state below 2 and we could change it in a continuous fashion (as we increase the energy of the original state away from the value that makes it a c type representation), but this is not possible. So we have recovered the statements in Sect Two about the structure of the null vectors of the c type representations, at least for the case j1 > 0. We can similarly continue the analysis for the structure of the null vectors of the c representations with j1 = 0. We can use some of these facts also to learn about the structure of the states that contribute to the index. For this purpose we should note that we get states with
= 0 by applying Q  to |ψ0 . This state has Cartan charges 1 1 (E + , j1 + , j2 , r + 1, R1 + 1, Ri ). 2 2

(B.4)

It is also easy to see that it transforms in the SU (1, 2|m − 1) representation with charges (E  , j2 , r  , R j ) given in terms of the map (3.4) applied to (B.4). In this way we obtain formula (3.5), for type c BPS representations. The factor (−1)2 j1 +1 comes from the statistics factor associated to (B.4) which will not be included when we consider the character in the subalgebra, whose sign depends only on the j2 quantum numbers. For type b BPS representations the highest weight state itself has
= 0, so we find that (E  , j2 , r  , R j ) are directly given by the formula in (3.4) in terms of the Cartan eigenvalues of |ψ0  and we do not get any overall minus sign. This is summarized in (3.7). The character on the SU (1, 2|m − 1) character in (3.5) manifestly depends on the quantum numbers j1 , r, R1 of a c representation only through the combination j1 + r/2, r − R1 . This leads to (3.8) and (3.9) and (3.7) for c representations. B.3. Representation theory of the subalgebra SU (2, 1|m − 1) . The representation theory of the subalgebra is easily worked out, and closely mimics the pattern presented in the previous section. Briefly, representations are labeled by the quantum numbers (E  , j2 , r  , Ri ) that specify the U (1) × SU (2) × U (m − 1) (i = 2 . . . m − 1) quantum numbers of the lowest weight state. Acting on this lowest weight state with the supersymmetries charged under J2 , we find a set of level one states; the lowest norm among  + 1) these states occurs for those that transform in (E  + 21 , j2 − 21 , r  − 1, R j , Rm−2 ( j = 2 . . . m − 2); this norm is given by  2R,sub = E  − 3 j2 + 3δ j2 0 − 3 − 3 +

m−2

k Rk m−1 k=1

r  (4 − m)  ≡ E  + 3δ j2 ,0 − E 2sub ( j2 , r  , Ri ). (m − 1)

(B.5)

Acting on the lowest weight states with supersymmetries uncharged under j2 , we find a set of states; the lowest norm occurs for those states that transform in (E  + 1, j2 ,

Index for 4 Dimensional Super Conformal Theories

247

r  + 1, R1 + 1, R j ) ( j = 2 . . . m − 2). The norm of these states is given by 

2sub,L

3 =E − 2 

m−2 k=1

(m − 1 − k)Rk r  (4 − m)  (r  , Ri ). − ≡ E − E 1sub m−1 2(m − 1)

(B.6)

  and E  ≥ E 2sub . When these inequaliUnitary representations occur when E  ≥ E 1sub ties are strictly satisfied, the representations are long and are denoted by (aa)sub  [E  , j2 , r  , Ri ]. Representations with E  = E 2sub are short, and are denoted by (xc)sub     [ j2 , r , Ri ]. When j2 = 0 the null states of this representation occur at level 2. In addition,  at j2 = 0 we have a short representation at E  = E 2sub −3, denoted by (xb)sub [0, r  , Ri ].   Representations with E = E 1sub are denoted by (bx)sub [ j2 , r  , Ri ]. Now consider a representation R of the full algebra that is of the form cx. The highest 1 weight state of R has
= 2. Acting on this with Q  = Q + 2 ,1 , we obtain a state with charges (E + 21 , J1 + 21 , J2 , r + 1, R1 + 1, Ri ). This state has
= 0 and serves as the  highest weight of the representation R of the subalgebra. If R is of the form bx then its  highest weight state has
= 0 and also serves as the highest weight of R . If R is of the form ax, then it has no states with
= 0. Let us investigate if the representation R  so obtained satisfies the unitarity bounds from (B.5). First, consider the case where R is cx. Then highest weight of R  is specified by the charges given by c in (3.7). Substituting these values of the charges into Eqs. (B.5), (B.6) we find that

3 j  22 = 3δ02 + (E 1 − E 2 ), 2 3 2  1 = 3 j1 + 3 + R1 . 2

(B.7)

So, R  is long unless (E 1 = E 2 ). In this case, R ∼ cc and R  is short. If j2 = 0 it is possible to have E 2 = E 1 + 2, and then R ∼ cb and R  is short. Now, let R = bx. Then 3 j2  22 = 3δ0 − 3 + (E 1 − E 2 ), 2 3 2  1 = R1 . 2

(B.8)

If x is a or c, we have E 1 − E 2 ≥ 2. If this inequality is saturated, R ∼ bc and R  is short. R  may also be short if R1 = 0. Finally, when j2 = 0 and E 1 = E 2 , R ∼ bb and R  is short. Using all of this, the decomposition of long representations as they hit unitarity bound follows immediately; we will not explicitly list the character formulae. Appendix C. Conventions and Computations for the N = 4 Index C.1. Weights of the supercharges. In this subsection we list the weights of the supercharges under the Cartan elements (E, J13 , J23 , R1 , R2 , R3 ). 1 1 Q 1± → { , ± , 0, 1, 0, 0}, 2 2

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1 1 Q 2± → { , ± , 0, −1, 1, 0}, 2 2 1 1 Q 3± → { , ± , 0, 0, −1, 1}, 2 2 1 1 Q 4± → { , ± , 0, 0, 0, −1}, 2 2 1 1 Q¯ 1,± → { , 0, ± , −1, 0, 0}, 2 2 1 1 Q¯ 2,± → { , 0, ± , 1, −1, 0}, 2 2 1 1 Q¯ 3,± → { , 0, ± , 0, 1, −1}, 2 2 1 1 Q¯ 4,± → { , 0, ± , 0, 0, 1}.[3 pt] 2 2

(C.1)

C.2. Racah Speiser algorithm. The Racah Speiser algorithm is an efficient way to calculate tensor products. Consider a highest weight state | > and the complete set of states in another representation |λi >. We denote the half sum of positive roots by ρ. The Racah-Speiser algorithm tells us that to obtain the representations in the tensor product, we need to perform the following steps: 1. First count all representations | + λi >, where  + λi is in the fundamental Weyl Chamber[All weights are non-negative]. 2. If  + λi + ρ is on the boundary of the Weyl chamber, i.e. at least one weight is zero, then throw away this representation. (ρ is the half-sum of the positive roots.) 3. If  + λi is not on the boundary of the Weyl Chamber, there exists a unique Weyl reflection σ such that σ ( + λi + ρ) − ρ is in the Weyl Chamber. Count this representation with a plus or a minus sign depending on the sign of σ . We use this algorithm to obtain the state contents tabulated below. It is interesting that for the Yang Mills multiplet, using the Racah Speiser algorithm automatically gives us the representations corresponding to the equations of motion with negative signs.

C.3. State content of ‘graviton’ representations. As explained in Sect. 4.2 the spectrum of Type I I B supergravity compactified on Ad S5 × S 5 organizes into representations of the superconformal algebra that are built on a lowest weight state that is a scalar in the (n, 0, 0) S O(6) = (0, n, 0) SU (4) representation of the R-symmetry group. When restricted to
= 0, they yield a short representation of the subalgebra that we shall call Sn . Sn has lowest weight [E = n, j2 = 0, R2 = n, R3 = 0]. We may explicitly compute the SU (2, 1) × SU (3) content of Sn by starting with the lowest weight state, repeatedly acting on it with the Q and Q¯ operators, and deleting states of zero norm. This process is expedited by using the Racah Speiser algorithm explained in C.2. In the table below we explicitly list the SU (2, 1) × SU (3) content of Sn using the notation [E  , j2 , R1 , R2 ], where [E  , j2 ] specify the weight of the lowest weight state under the compact U (1) × SU (2) subgroup of SU (2, 1) and [R2 , R3 ] are Dynkin labels for SU (3). This can also be found by looking at the list of Kaluza Klein modes in [36].

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249

Table 3. Content of Sn (−1)F E

J2

R1

R2

n −(n + 21 ) n+1 −(n + 1) n + 23 −(n + 2) n+2 −(n + 25 ) n+3

0

n n−1 n−2 n−1 n−2 n−3 n−1 n−2 n−3

0 0 0 1 1 1 0 0 0

1 2

0 0

1 2

0 0

1 2

0

For n = 2 we just drop the lines containing n − 3. On the other hand, for n = 1 we have further shortening and we find Table 4. Content of S1 (−1)F E

J2

R1

R2

1

0

1 0 0 0 0

0 0 1 0 023

−3 2

−2 3 3

1 2

0 0 0

C.4. Character of SU (3). We wish to compute the quantity χR (θ1 , θ2 ) = Tr R exp i(R1 θ1 + R2 θ2 ),

(C.2)

where R1 and R2 form the Cartan subalgebra of SU (3). We denote the eigenvalues of the highest weight state of a representation, under the operators Ri , by ri . Furthermore, we define v1 = exp −iθ2 , v2 = exp iθ1 , v3 = exp i(θ2 − θ1 ).24 Then, the character (C.2) is given by the Weyl Character Formula [57],   R +1  v 1 v2R1 +1 v3R1 +1   1  −R2 −1 −R2 −1 −R2 −1    v1 v2 v3    1 1 1   1 1 1  . (C.3) χ R1 ,R2 =  v v v   1 2 3   −1 −1 −1   v1 v2 v3     1 1 1  C.5. Translation between bases. S O(6) → SU (4). First, we show how to translate between S O(6) and SU (4) notation. Denote the S O(6) Dynkin labels by q1 , q2 , q3 and 23 This term comes from the fermionic equation of motion, hence it counts with a positive sign. 24 These are the weights of the fundamental representation of SU (3).

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the SU (4) Dynkin labels by R1 , R2 , R3 , R1 R3 + R2 + , 2 2 R1 R3 + , q2 = 2 2 R1 R3 − . q3 = 2 2 q1 =

(C.4)

H  , Ri → L i . Next, we show how to translate between the basis formed by the L i , J2 and the Cartan generators of the subalgebra H + J1 , J2 , R1 , R2 defined above. Note that R1 = R2 , R2 = R3 . Moreover, recall that the L i are specified by (4.9) which we recapitulate here: L 1 = E + q1 − q2 − q3 , L 2 = E + q2 − q1 − q3 , L 3 = E + q3 − q1 − q2 . (C.5) Under the condition
= 0 we find (denoting H + J1 = H  ) 2  H + 3 2 L2 = H  + 3 2  L3 = H + 3 L1 =

2 (2R1 + R2 ), 3 2 (−R1 + R2 ), 3 2 (−R1 − 2R2 ). 3

(C.6)

In turn this implies a relationship between the chemical potentials. If θ H  H  + θ1 R1 + θ2 R2 = γ1 L 1 + γ2 L 2 + γ3 L 3 then, 2 (γ1 + γ2 + γ3 ), 3 2 θ1 = (2γ1 − γ2 − γ3 ), 3 2 θ2 = (γ1 + γ2 − 2γ3 ). 3

θH  =

(C.7)

C.6. Index on the Fock space. Let us say that we have the single particle index Z sp =



xiB −

i



xiF ,

(C.8)

i

where the index i runs over all the bosons and all the fermions. Then the index for a multiparticle system is given by Z Fock =

 (1 − x F ) i

So we find (4.15).

(1 −

i xiB )

=e



1 n n

Z sp (x n )

.

(C.9)

Index for 4 Dimensional Super Conformal Theories

251

Appendix D. Comparison of the Cohomological Partition Function and the Index Let the number of states with charges J1 , J2 , L i be given by e S(J1 ,J2 ,L i ) . Then Z free =





exp S(J1 , J2 , L i ) −

J1 ,J2 ,L i

IYWML

=





 γi L i − ......2ζ J2 ,

i

 exp S(J1 , J2 , L i ) −

J1 ,J2 ,L i



 γi L i − 2ζ J2 (−1)2(J1 +J2 ) ,

(D.1)

i

where we have set all chemical potentials that couple to charges outside SU (2, 1|3) to zero in Z free . Let 



exp N Seff ( j1 , γi ) = 2



 exp S(J1 , J2 , L i ) −

J2 ,L i



 γi L i − 2ζ J2 ,

(D.2)

i

where j1 ≡ J1 /N 2  1 and γi  1. Let us assume that Seff is independent of N in the large N limit. We certainly have this property in the free theory, and we expect it in the interacting N = 4 theory, but it does not have to hold for every theory. We can then rewrite (D.1) as Z free =

 J1

IYWML

=



  exp N 2 Seff ( j1 , ζ, γi ) ,    exp N 2 Seff ( j1 , ζ + πi, γi ) + 2iπ j1 .

(D.3)

J1

j = a(θ, γi ) and that Let us assume that at fixed values of ζ, γi has a maximum at Seff (a + δ, ζ, γi ) ≈ S0 − 2b2 δ 2 , S0 = Seff (a, ζ, γi ).

(D.4)

The contribution of this saddle point to the partition function in the first line of (D.3) is easily estimated25 by  Z free ≈

  2π 2 exp N S 0 . b2 N 2

(D.5)

An estimation of the index in the second line of (D.3) is a more delicate task as the summand changes by large values over integer spacings. To proceed we will assume that Seff ( j1 , ζ, γi ) is a continuous function; i.e. that it does not evaluate to discontinuously different answers for integral and half integral values of J1 . This is a nontrivial assumption, which we believe to be true for free Yang Mills theory, but will not always 25 For instance one could convert the sum into an integral using the Euler McLaurin formula [58] and approximate the integral using saddle points. A more careful estimate may be obtained by Poisson resumming, see the next paragraph.

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be true in every theory. Under this assumption we will now estimate the contribution of the saddle point at j1 = a to the index by  2 2  b m exp − + πim 2N 2 m=−∞   2 ∞   2π N (2π )2 1 2 N 2 S0 ) exp (k − =e b2 N 2 2b2 2 k=−∞    2π π2 ≈ 2 2 2 exp N 2 (S0 − 2 ) , b N 2b

IYWML = e N

2S 0

∞ 

(D.6)

where we have used the Poisson resummation formula to go from the first to the second line of (D.6). Note that the contribution of the saddle point at j1 = a to the index is supressed compared to its contribution to the partition function. Moreover, if S0 < π 2 /2b2 , the 2 contribution of this saddle point is formally of order e−a N ; which means that the neighborhood of the saddle point does not contribute significantly to the index in the large N limit; the index receives its dominant contributions from other regions of the summation domain. An estimation from formulas of (5.6), (5.9) puts us in this regime As a toy example of the suppression described in the last two paragraphs, consider the two identities Z = (2 + 1) N =



2k

N! , k!(N − k)!

2k

N! (−1) N −k . k!(N − k)!

k

I = (2 − 1) N =

 k

(D.7)

The summation over k in the first of (D.7) may be approximated by the integral

1 x N ln x 2 1−x x (1−x) e , (D.8) x=0

which localizes around the saddle point value x s =

2 3

at large N , yielding Z = 3 N . The

contribution to I from this saddle point, on the other hand, is proportional to e N (ln 3− and so is utterly negligible, consistent with the fact that I evaluates to unity.26

π2 3 )

,

References 1. Witten, E.: Constraints On Supersymmetry Breaking. Nucl. Phys. B 202, 253 (1982) 2. Sundborg, B.: The Hagedorn transition, deconfinement and N = 4 SYM theory. Nucl. Phys. B 573, 349 (2000) 26 Actually, a computation very similar to this toy example explains why the index grows more slowly than exponentially with energy in the ‘low temperature phase’ (while the cohomological partition function displays exponential growth in the same phase). The number of states that contribute at energy E to the index is given by the coefficient of x E in (4.7). This is given by a multinomial expansion. When we weight the sum with (−1) F , the multinomial sum stops growing exponentially just like (D.7) above. Hence, the index never goes through a Hagedorn-like transition.

Index for 4 Dimensional Super Conformal Theories

253

3. Aharony, O., Marsano, J., Minwalla, S., Papadodimas, K., Van Raamsdonk, M.: The Hagedorn/deconfinement phase transition in weakly coupled large N gauge theories. Adv. Theor. Math. Phys. 8, 603 (2004) 4. Witten, E.: Anti-de Sitter space and holography. Adv. Theor. Math. Phys. 2, 253 (1998) 5. Aharony, O.: Unpublished 6. Gutowski, J.B., Reall, H.S.: Supersymmetric AdS(5) black holes. JHEP 0402, 006 (2004) 7. Gutowski, J.B., Reall, H.S.: General supersymmetric AdS(5) black holes. JHEP 0404, 048 (2004) 8. Chong, Z.W., Cvetic, M., Lu, H., Pope, C.N.: General non-extremal rotating black holes in minimal five-dimensional gauged supergravity. http://arxiv.org/list/hep-th/0506029, 2005 9. Strominger, A., Vafa, C.: Microscopic Origin of the Bekenstein-Hawking Entropy. Phys. Lett. B 379, 99 (1996) 10. Berenstein, D.: Large N BPS states and emergent quantum gravity. http://arxiv.org/list/hep-th/0507203, 2005 11. Romelsberger, C.: An index to count chiral primaries in N=1 d=4 superconformal field theories. http://arxiv.org/list/hep-th/0510060, 2005 12. Mack, G.: All Unitary Ray Representations Of The Conformal Group SU(2,2) With Positive Energy. Commun. Math. Phys. 55, 1 (1977) 13. Barabanschikov, A., Grant, L., Huang, L.L., Raju, S.: The spectrum of Yang Mills on a sphere. http://arxiv.org/list/hep-th/0501063, 2005 14. Minwalla, S.: Restrictions imposed by superconformal invariance on quantum field theories. Adv. Theor. Math. Phys. 2, 781 (1998) 15. Kac, V.G.: Lie Superalgebras. Adv. Math. 26, 8 (1977) 16. Dobrev, V.K., Petkova, V.B.: All Positive Energy Unitary Irreducible Representations Of Extended Conformal Supersymmetry. Phys. Lett. B 162, 127 (1985) 17. Dobrev, V.K., Petkova, V.B.: On The Group Theoretical Approach To Extended Conformal Supersymmetry: Classification Of Multiplets. Lett. Math. Phys. 9, 287 (1985) 18. Dobrev, V.K., Petkova, V.B.: All positive energy unitary irreducible representations of the extended conformal superalgebra. Talk at the Symposium on Conformal Groups and Structures (Clausthal, 1985), In: Proceedings, eds. A.O. Barut, H.D. Doebner, Lecture Notes in Physics, Vol. 261, (Berlin:Springer-Verlag, 1986) pp. 300–308 19. Dobrev, V.K., Petkova, V.B.: Group Theoretical Approach To Extended Conformal Supersymmetry: Function Space Realizations And Invariant Differential Operators. Fortsch. Phys. 35, 537 (1987) 20. Dolan, F.A., Osborn, H.: On short and semi-short representations for four dimensional superconformal symmetry. Ann. Phys. 307, 41 (2003) 21. Dobrev, V.K.: Characters of the positive energy UIRs of D = 4 conformal supersymmetry. http:// arxiv.org/list/hep-th/0406154, 2004 22. Dasgupta, K., Sheikh-Jabbari, M.M., Van Raamsdonk, M.: Protected multiplets of M-theory on a plane wave. JHEP 0209, 021 (2002) 23. Kim, N., Park, J.H.: Superalgebra for M-theory on a pp-wave. Phys. Rev. D 66, 106007 (2002) 24. Lin, H., Maldacena, J.: Fivebranes from gauge theory. http://arxiv.org/list/hep-th/0509235, 2005 25. Motl, L., Neitzke, A., Sheikh-Jabbari, M.M.: Heterotic plane wave matrix models and giant gluons. JHEP 0306, 058 (2003) 26. Liu, H.: Fine structure of Hagedorn transitions. http://arxiv.org/list/hep-th/0408001, 2004 27. Alvarez-Gaumé, L., Gomez, C., Liu, H., Wadia, S.: Finite temperature effective action, AdS(5) black holes, and 1/N expansion. Phys. Rev. D 71, 124023 (2005) 28. Aharony, O., Marsano, J., Minwalla, S., Wiseman, T.: Black hole - black string phase transitions in thermal 1+1 dimensional supersymmetric Yang-Mills theory on a circle. Class. Quant. Grav. 21, 5169 (2004) 29. Aharony, O., Marsano, J., Minwalla, S., Papadodimas, K., Van Raamsdonk, M.: A first order deconfinement transition in large N Yang-Mills theory on a small S**3. Phys. Rev. D 71, 125018 (2005) 30. Basu, P., Wadia, S.R.: R-charged AdS(5) black holes and large N unitary matrix models. http://arxiv.org/list/hep-th/0506203, 2005 31. Aharony, O., Marsano, J., Minwalla, S., Papadodimas, K., Van Raamsdonk, M., Wiseman, T.: The phase structure of low dimensional large N gauge theories on tori. http://arxiv.org/list/hep-th/0508077, 2005 32. Morales, J.F., Samtleben, H.: Higher spin holography for SYM in d dimensions. Phys. Lett. B 607, 286 (2005) 33. Bianchi, M., Morales, J.F., Samtleben, H.: On stringy AdS(5) × S**5 and higher spin holography. JHEP 0307, 062 (2003) 34. Beisert, N., Bianchi, M., Morales, J.F., Samtleben, H.: On the spectrum of AdS/CFT beyond supergravity. JHEP 0402, 001 (2004) 35. Beisert, N., Bianchi, M., Morales, J.F., Samtleben, H.: Higher spin symmetry and N = 4 SYM. JHEP 0407, 058 (2004) 36. Gunaydin, M., Marcus, N.: The Spectrum Of The S**5 Compactification Of The Chiral N=2, D = 10 Supergravity And The Unitary Supermultiplets Of U(2, 2/4). Class. Quant. Grav. 2, L11 (1985)

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37. Green, M.B., Schwarz, J.H., Witten, E.: Superstring Theory. Vol. 2: Loop Amplitudes, Anomalies And Phenomenology. Chapter 12. Cambridge, Cambridge University Press, (1987) 38. Gubser, S.S., Klebanov, I.R., Peet, A.W.: Entropy and Temperature of Black 3-Branes. Phys. Rev. D 54, 3915 (1996) 39. Mikhailov, A.: Giant gravitons from holomorphic surfaces. JHEP 0011, 027 (2000) 40. McGreevy, J., Susskind, L., Toumbas, N.: Invasion of the giant gravitons from anti-de Sitter space. JHEP 0006, 008 (2000) 41. Berenstein, D., Herzog, C.P., Klebanov, I.R.: Baryon spectra and AdS/CFT correspondence. JHEP 0206, 047 (2002) 42. Barnes, E., Gorbatov, E., Intriligator, K., Wright, J.: Current correlators and AdS/CFT geometry. http://arxiv.org/list/hep-th/0507146, 2005 43. Breckenridge, J.C., Myers, R.C., Peet, A.W., Vafa, C.: D-branes and spinning black holes. Phys. Lett. B 391, 93 (1997) 44. Cachazo, F., Douglas, M.R., Seiberg, N., Witten, E.: Chiral rings and anomalies in supersymmetric gauge theory. JHEP 0212, 071 (2002) 45. D’Hoker, E., Heslop, P., Howe, P., Ryzhov, A.V.: Systematics of quarter BPS operators in N = 4 SYM. JHEP 0304, 038 (2003) 46. Bianchi, M., Eden, B., Rossi, G., Stanev, Y.S.: On operator mixing in N = 4 SYM. Nucl. Phys. B 646, 69 (2002) Bianchi, M., Rossi, G., Stanev, Y.S.; Surprises from the resolution of operator mixing in N = 4 SYM. Nucl. Phys. B 685, 65 (2004) 47. Arutyunov, G., Frolov, S., Petkou, A.: Perturbative and instanton corrections to the OPE of CPOs in N = 4 SYM(4). Nucl. Phys. B 602, 238 (2001); Erratum-ibid. B 609, 540 (2001) 48. Bianchi, M., Kovacs, S., Rossi, G., Stanev, Y.S.: Properties of the Konishi multiplet in N = 4 SYM theory. JHEP 0105, 042 (2001) 49. Eden, B., Jarczak, C., Sokatchev, E., Stanev, Y.S.: Operator mixing in N = 4 SYM: The Konishi anomaly revisited. Nucl. Phys. B 722, 119 (2005) 50. Eden, B., Petkou, A.C., Schubert, C., Sokatchev, E.: Partial non-renormalisation of the stress-tensor four-point function in N = 4 SYM and AdS/CFT. Nucl. Phys. B 607, 191 (2001) 51. Arutyunov, G., Eden, B., Petkou, A.C., Sokatchev, E.: Exceptional non-renormalization properties and OPE analysis of chiral four-point functions in N = 4 SYM(4). Nucl. Phys. B 620, 380 (2002) 52. Arutyunov, G., Frolov, S., Petkou, A.C.: Operator product expansion of the lowest weight CPOs in N = 4 SYM(4) at strong coupling. Nucl. Phys. B 586, 547 (2000); Erratum-ibid. B 609, 539 (2001) 53. Eden, B., Petkou, A.C., Schubert, C., Sokatchev, E.: Partial non-renormalisation of the stress-tensor four-point function in N = 4 SYM and AdS/CFT. Nucl. Phys. B 607, 191 (2001) 54. Corley, S., Jevicki, A., Ramgoolam, S.: Exact correlators of giant gravitons from dual N = 4 SYM theory. Adv. Theor. Math. Phys. 5, 809 (2002) 55. Greene, B.R., Morrison, D.R., Strominger, A.: Black hole condensation and the unification of string vacua. Nucl. Phys. B 451, 109 (1995) 56. Bars, I., Gunaydin, M.: Unitary Representations Of Noncompact Supergroups. Commun. Math. Phys. 91, 31 (1983) 57. Fuchs, J., Schweigert, C.: Symmetries, Lie algebras and representations: A graduate course for physicists. Cambridge: Cambridge University Press, 1997 58. Arfken, G.B., Weber, H.J.: Mathematical methods for physicists. NewYork: Academic Press, 2001 Communicated by N.A. Nekrasov

Commun. Math. Phys. 275, 255–269 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0306-3

Communications in

Mathematical Physics

Navier-Stokes Equation and Diffusions on the Group of Homeomorphisms of the Torus F. Cipriano1 , A. B. Cruzeiro2 1 GFM e Dep. de Matemática FCT-UNL, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal.

E-mail: [email protected]

2 GFM e Dep. de Matemática IST-TUL, Av. Rovisco Pais, 1049-001 Lisboa, Portugal.

E-mail: [email protected] Received: 17 October 2006 / Accepted: 15 January 2007 Published online: 25 July 2007 – © Springer-Verlag 2007

Abstract: A stochastic variational principle for the (two dimensional) Navier-Stokes equation is established. The velocity field can be considered as a generalized velocity of a diffusion process with values on the volume preserving diffeomorphism group of the underlying manifold. Navier-Stokes equation is reinterpreted as a perturbed equation of geodesics for the L 2 norm. The method described here should hold as well in higher dimensions.

1. Introduction V. Arnold (cf. [A] and [A-K]) discovered a beautiful relation between the Euler equation in hydrodynamics and the geometry of diffeomorphisms in L 2 (M) preserving the volume measure of the underlying manifold M. This equation coincides with the geodesic equation for the L 2 metric. In particular geometric properties like curvature reflect the dynamics of the Eulerian fluid. This discovery led in particular to new ideas in the study of geometry and topology of infinite dimensional manifolds. On the other hand it is natural to think of the Navier-Stokes equation as a “perturbation” of the Euler one, corresponding to a similar description, but of an underlying stochastic nature, the stochasticity being encoded in the Laplacian. This work explores this direction. We formulate solutions of Navier-Stokes as critical points of some regularized functional and show how they can be regarded as stochastically perturbed geodesics of Arnold’s model. Our idea finds its roots in the works [N-Y-Z, Y]. Other different stochastic variational formulations for Navier-Stokes can be found, for example in [I-F] or, more recently, in [Go] (cf also [C1]). One could also benefit from a comparison with the approach of [G] as well as the one of [C2]. Our approach relies upon the construction of diffusions on the (infinite dimensional) group of measure preserving homeomorphisms in the torus, a line of work which has recently been developed by [M, F1] and others.

256

F. Cipriano, A. B. Cruzeiro

When M is a compact n-dimensional Riemannian manifold without boundary, we denote by (·, ·) the Riemannian metric and by µ the associated volume element. Let G s , s ≥ 0 be the infinite dimensional group of homeomorphisms on M which belong to H s (the Sobolev space of order s), namely g ∈ G s if g : M → M is a bijection and g, g −1 ∈ H s . For s > n2 + 1 the group G s is a C ∞ infinite dimensional Hilbert manifold (see [S]). For each g ∈ G s , G s is locally diffeomorphic to the Hilbert space Hgs (T M) = {X ∈ H s (M, T M) : π ◦ X = g}, where π : T M → M is the canonical projection. For g ∈ G s denote by L g and Rg respectively the right and left transformations on the group, L g : G s → G s (h → g ◦ h), Rg : G s → G s (h → h ◦ g). The adjoint transformation Ad(g) is usually defined as follows: Ad(g) : G s → G s h → L g Rg−1 (h).

(1.1)

For an arbitrary Lie group G, the Lie algebra G is the space of left invariant vector fields on G, which can be identified with the tangent space at the identity. In the case of the group G s , the Lie algebra G s can be identified with the space of vector fields on M with Sobolev s-regularity. We consider the volume preserving homeomorphism subgroup G sV = {g ∈ G s : g∗ µ = µ}. For g ∈ G sV , Hgs (T M) = {X ∈ H s (M, T M) : π ◦ X = g and ∇ · X = 0}. The Lie algebra GVs of the subgroup G sV corresponds to the space of divergence free vector fields on M which are in H s . We shall deal with the L 2 inner product defined on the Lie algebra GVs by
(X g (x), Yg (x))g(x) dµ(x). (1.2) (X g , Yg ) L 2 = M

Since this inner product is right invariant it defines a metric on each tangent space Tg (G sV ). Therefore, G sV remains endowed with a Riemannian structure. Arnold ([A]) has given a variational formulation for the Hydrodynamic Euler equation. More precisely, the motion of an ideal fluid (i.e., non viscous and incompressible) on M corresponds to a flow on G 0V which is critical for the energy functional
1 T 2 g(t) ˙ dt. (1.3) S[g] = L2 2 0 ∂u + u · ∇u = ∇ p, ∇ · u = 0. Its velocity satisfies the Euler equation ∂t In this work we consider the Navier-Stokes equation on the two dimensional torus T2 during the time interval [0, T ], ∂u + u · ∇u − νu = ∇ p, ∂t ∇ · u = 0.

(1.4)

Navier-Stokes Equation and Diffusions of Homeomorphisms

257

This equation describes the motion of an incompressible fluid on T2 with viscosity ν > 0. The vector field u(t, ·) represents the velocity of the fluid and the function p the pressure. Since the incompressibility condition ∇ · u = 0 is intrinsically associated with the space GVs , and the Brownian motion is closely related with the Laplacian operator, a stochastic approach of the above framework using G sV -valued processes seems to be natural when the viscosity is strictly positive. We introduce a concept of the solution of (the deterministic) Navier-Stokes equation as the mean velocity of some stochastic flow. First we define a generalization of the energy functional (1.3) for stochastic flows with values in G 0V . Let (, Ft , P) be a probability space endowed with an increasing filtration Ft . Given a stochastic flow gω (t) with values in G 0V and adapted to the filtration, we generalize the above energy functional by considering:
T  1 1 Ft  E gω (t + t) − gω (t) 2L 2 dt, S[g] = E  lim (1.5) t→0 t 2 0 where the increment gω (t + t) − gω (t) is understood as the difference in the respective local coordinates and E Ft is the conditional expectation with respect to the filtration. Notice that formally, when ν → 0, the functional (1.5) reduces to (1.3). Referring to the variational calculus on path spaces of a Lie group, in the finite ¨ unel [U], considers right as well as left derivatives. Such derivatives dimensional case, Ust¨ are defined by the multiplication on the right or on the left by a deterministic path of bounded variation. In that case the left product corresponds, at the level of paths, to shifts by an element in the Cameron-Martin subspace of the Wiener space. On the other hand, the right product corresponds to a rotation of the path in the Wiener space. Starting with a metric on the Lie algebra which is invariant for the ad transformation (the differential of Ad(g) at the identity), both left and right variations are well defined from the measure theoretic point of view and Malliavin’s calculus of variations can be used. Unfortunately, in our case, as referred to in [A-K], an invariant metric for the ad(g) transformation does not exist in G 0V . We shall deal with G 0V -valued stochastic flows and for each g ∈ G 0V a suitable “tangent space” of variations will be defined. Let g ∈ G 0V and f be a C 2 function defined on M. The action of g on f is:  ∗  g f (ξ ) = f (g(ξ )). In this work we shall consider M = T2 , the two dimensional torus that we identify with [0, 2π ]×[0, 2π ], and denote by G 0V the space of volume preserving maps from T2 to T2 . Our construction is based on the fundamental fact that the generator of our (infinite dimensional) process actually coincides with the finite-dimensional Laplacian when computed on functions of the torus (Theorem 2.2). It is not clear that this property can be generalized in three dimensions; nevertheless there is no conceptual reason preventing a priori the development of our construction in any dimension. Let Ak , Bk , k ∈ Z2 be divergence free vector fields defined in local coordinates by Ak = A1k ∂1 + A2k ∂2 with A1k = k2 cos(k · θ ), A2k = −k1 cos(k · θ ) and Bk = Bk1 ∂1 + Bk2 ∂2 with Bk1 = k2 sin(k · θ ), Bk2 = −k1 sin(k · θ ),

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  Ak Bk , : k = 0 is a |k| |k| 2 2 complete system of the space of divergence free vector fields u in L (T ) such that u(θ )dθ = 0. T2

where θ = (θ1 , θ2 ) ∈ T 2 and k · θ = k1 θ1 + k2 θ2 . Then

2. Brownian Motion on the Group of Homeomorphisms Let xk (t) = (xk1 (t), xk2 (t)), t ≥ 0, be a sequence of R2 -valued independent standard Brownian motions defined on the probability space (, Ft , P). We define a Brownian motion on the space of divergence free vector fields on the two dimensional torus by  1  1 2 d x(t) = A d x (t) + B d x (t) (2.1) k k k k |k|β k =0

with β ≥ 3. Using standard probabilistic techniques, this series can be shown to converge uniformly in [0, T ] × T2 a.e. and the stochastic process x(t) belongs to H α , for 0 < α < β − 2. Let us consider the Stratonovich stochastic differential equation with respect to the filtration Ft ,

  dg(t) = ◦d x(t) (g(t)) (2.2) g(0) = e, where e denotes the identity of the group, or, more explicitly dg 1 (t) =

 1  A1 (g(t)) ◦ d xk1 (t) + Bk1 (g(t)) ◦ d xk2 (t) , |k|β k k =0

 1  dg 2 (t) = A2k (g(t)) ◦ d xk1 (t) + Bk2 (g(t)) ◦ d xk2 (t) . β |k|

(2.3)

k =0

Lemma 2.1. Equation (2.2) can be written in the Itô form as follows: dg 1 (t) =

 1  A1k (g(t)) · d xk1 (t) + Bk1 (g(t)) · d xk2 (t) , β |k| k =0

 1  dg 2 (t) = A2k (g(t)) · d xk1 (t) + Bk2 (g(t)) · d xk2 (t) , β |k|

(2.4)

k =0

i.e., the Itô contraction term vanishes. Proof. We have:       d A1k (g(t) = ∂1 A1k (g(t)) ◦ dg 1 (t) + ∂2 A1k (g(t)) ◦ dg 2 (t)  1    = ∂1 A1k (g(t)) A1m (g(t)) ◦ d xm1 (t) + Bm1 (g(t)) ◦ d xm2 (t) |m|β m =0

+



m =0

  1  ∂2 A1k (g(t)) A2m (g(t)) ◦ d xm1 (t) + Bm2 (g(t)) ◦ d xm2 (t) . β |m|

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Since d xm1 · d xk1 = δmk dt and d xm2 · d xk1 = 0 we obtain      1  d A1k (g(t) · d xk1 (t) = β ∂1 A1k (g(t))A1k (g(t))dt + ∂2 A1k (g(t))A2k (g(t))dt |k| 1  = β −(k2 )2 k1 sin(k · θ ) cos(k · θ )dt |k| + (k2 )2 k1 sin(k · θ ) cos(k · θ )dt = 0. All other Itô contractions can be shown to be zero in an analogous way.



From the classical theory of stochastic flows (cf. [K]), for β > 3, the solution g(t) of Eq. (2.2) is well defined as a stochastic flow of diffeomorphisms. For β = 3, we can follow [F1] to prove that the quadratic variation of the stochastic 1 process x(t)(θ ) − x(t)(θ ) can be estimated by C|θ − θ |2 log |θ−θ | for |θ − θ | small enough. This estimate enables to prove the existence of the process g(t). More precisely, we have: Theorem 2.1. For β = 3, the solution g(t) of the stochastic differential equation (2.2) exists and is a continuous process with values in the space of homeomorphisms on T2 preserving the volume measure. Proof. We follow the methodology of [F1]. We fix θ and denote by gin (t)(θ ) the solution of the following finite-dimensional s.d.e.:  1  dγ1n (t) = A1 (γ n (t)) · d xk1 (t) + Bk1 (γ n (t)) · d xk2 (t) , |k|3 k k =0 |k|≤2n

dγ2n (t)

=

 k =0 |k|≤2n

1  2 n Ak (γ (t)) · d xk1 (t) + Bk2 (γ n (t)) · d xk2 (t) , 3 |k| γ n (t)−γ n+1 (t)

(2.5)

with initial condition (γ1n (0), γ2n (0)) = (θ1 , θ2 ). Denote ηi (t) = i 25 i ; we have

 2 k j  2 1 cos(k · γ n+1 (t)) − cos(k · γ n (t)) dηi (t) · dηi (t) = 10 6 2 |k| k =0 |k|≤2n

 2 + sin(k · γ n+1 (t)) − sin(k · γ n (t)) 

+

k =0 2n +1≤|k|≤2n+1

≤

1 210

+

k 2j  2 n+1 2 n+1 cos (k · γ (t)) + sin (k · γ (t)) |k|6

 2 kj k =0 |k|≤2n

 γ n+1 (t) − γ n (t) 2 k · 4 sin |k|6 2

 k∈Z2 2n +1≤|k|≤2n+1

. 6

k 2j |k|

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Now  k 2j k =0 |k|≤2n

|k|6

 sin

2

γ n+1 (t) − γ n (t) k· 2

≤ C|η(t)|2 log

1 |η(t)|

and 

k 2j

k =0 2n +1≤|k|≤2n+1

|k|6



≤

k =0 2n +1≤|k|≤2n+1

1 ≤ C2−n , |k|4

 where C = |k|≥1 |k|1 3 . By Itô’s formula, for p ≥ 1, 2p

2 p−1

dηi (t) = 2 pηi

2 p−2

(t) · dηi (t) + p(2 p − 1)ηi

we have  2p  2p EFt ηi (t + ) − ηi (t) ≤ K 0 p




2p

t+

(t)dηi (t) · dηi (t),

 EFt |η(s)|2 p log

t

1 ds + K p 2−n . |η(s)|2 p

2p

Defining the function ϕ(t) = E(η1 (t) + η2 (t)), we obtain  1 ϕ (t) ≤ K 0 p · E |η(s)|2 p log + K p 2−n , |η(s)|2 p therefore ϕ(t) ≤ C p 2−nδ p (t) , where δ p (t) is a constant verifying lim p→+∞ δ p (t) = 0 and limt→0+ δ p (t) = 1. By the martingale maximal inequality,  E sup |g n (t)(θ ) − g n+1 (t)(θ )|2 p ≤ C p 2−nδ p (T ) . 0≤t≤T

Using Borel-Cantelli we deduce that g(t)(θ ) = lim g n (t)(θ ) exists uniformly in t ∈ [0, T ]. n

Following [F1], one can show that g(t) satisfies Eq. (2.4) and that it is its unique solution. The following estimate holds  2p E sup |g(t)(θ ) − g(t)(θ )| ≤ C p |θ − θ |2 pδ p (T ) . 0≤t≤T

Using this inequality, for fixed small T , we can apply Kolmogorov theorem to show that g(t)(·) is H¨older continuous. Following [F1] and [M], the flow property can be used to prove that the stochastic process g(t) lives in the space of volume preserving homeomorphisms.



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261

Definition 2.1. Let f be a function in C 2 defined on T2 . On a functional F(g)(θ ) = f (g(θ )), θ ∈ T2 , the infinitesimal generator of the process g(t) is defined by  1 L(F)(θ ) = lim E (g(t))∗ f (θ ) − f (θ ) . (2.6) t→0 t This infinitesimal generator corresponds to the usual Laplacian operator  on the torus, more precisely, we have: Theorem 2.2. Let L be the infinitesimal generator of the stochastic process g(t). Then there exists a positive constant c such that, for F(g)(θ ) = f (g(θ )), L(F) = c f,

f ∈ C 2 (T2 ).

Proof. Itô’s formula reads  1     d f (g(t)) = Ak f (g(t)) · d xk1 (t) + Bk f (g(t)) · d xk2 (t) |k|β k =0

+

    1  1   Ak Ak f (g(t))dt + Bk Bk f (g(t))dt . 2 |k|2β k =0

We have 1  |k|2β

  Ak Ak f (θ ) =

  1  k2 cos(k · θ )∂1 k2 cos(k · θ )∂1 f − k1 cos(k · θ )∂2 f |k|2β   − k1 cos(k · θ )∂2 k2 cos(k · θ )∂1 f − k1 cos(k · θ )∂2 f 1  = 2β (k2 )2 cos2 (k · θ )∂12 f − 2k1 k2 cos2 (k · θ )∂1 ∂2 f |k| + (k1 )2 cos2 (k · θ )∂22 f

and    1   1  Bk Bk f (θ ) = 2β k2 sin(k · θ )∂1 k2 sin(k · θ )∂1 f − k1 sin(k · θ )∂2 f 2β |k| |k|   − k1 sin(k · θ )∂2 k2 sin(k · θ )∂1 f − k1 sin(k · θ )∂2 f 1  = 2β (k2 )2 sin2 (k · θ )∂12 f − 2k1 k2 sin2 (k · θ )∂1 ∂2 f |k| + (k1 )2 sin2 (k · θ )∂22 f . Therefore the infinitesimal generator is given by 1 1  (k2 )2 ∂12 f + (k1 )2 ∂22 f − 2k1 k2 ∂1 ∂2 f . 2 |k|2β k =0

Since  k =0

 1  1 1 2 k k = 0 and (k ) = (k2 )2 , 1 2 1 |k|2β |k|2β |k|2β

the result follows from taking c =

k =0

1 2

1 2 k =0 |k|2β (k1 ) .

k =0



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F. Cipriano, A. B. Cruzeiro

3. Stochastic Differential Equations on the Group of Homeomorphisms Let us consider u : [0, T ] → GV0 . For each t, u(t) is a divergence free vector field on T2 . We can associate to u(t) the following Ft stochastic differential equation:   ν dgu (t) = u(t)dt + c ◦ d x(t) (gu (t)), (3.1) gu (0) = e, where c denote the constant defined in Theorem 2.2. This equation can be written in local coordinates as follows:   ν 1 1 1 d x (t) (gu (t)), dgu (t) = u (t)dt + c   ν 2 d x (t) (gu (t)), dgu2 (t) = u 2 (t)dt + c gu (0) = e.

(3.2)

The method to solve this stochastic equation depends on the regularity of the underlying Brownian motion and of the drift. When β = 3 (the most irregular case) we can use Girsanov transformation as in [F2] with u(t) ∈ H 2 . In this case, if u belong to C([0, T ]; GV2 ), one can show the existence of a stochastic process gu (t), solution of the s.d.e. (3.1), with values in the space of homeomorphisms of the torus preserving the Lebesgue measure. In the case where only L 2 regularity is available, we prove the following Theorem 3.1. Let u belong to the space L 2 ([0, T ]; GV0 ); then there exists a stochastic process gu (t), weak solution of the s.d.e. (3.1), with values in G 0V . Proof. Since u ∈ HV0 , we can write   Ak (θ ) Bk (θ ) 1 2 u k (t) u(t, θ ) = + u k (t) |k| |k|

(3.3)

k =0

 with k =0 |u 1k |2 + |u 2k |2 < +∞. Let us consider the following smooth aproximation u n of the vector field u,   Ak (θ ) Bk (θ ) n 1 2 u k (t) + u k (t) u (t, θ ) = |k| |k|

(3.4)

k =0 |k|≤n

and a smooth finite dimensional approximation x n (t) of the Brownian motion x(t) defined in (2.1),  1  n 1 2 Ak (θ )d xk (t) + Bk (θ )d xk (t) . d x (t, θ ) = (3.5) |k|β k =0 |k|≤n

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263

Then there exists a smooth global stochastic flow, strong solution of the stochastic differential Eq. (3.1) with u replaced by u n and x(t) replaced by x n (t) ([K]). More precisely, there exists g n (t)(θ ) such that 
t
t ν n n n (3.6) u (τ, g (τ )(θ ))dτ + d x n (τ, g n (τ )(θ )). g (t)(θ ) = θ + c 0 0 We consider the sequence of measures ν n , defined on the space C([0, T ]; G 0V ) as the laws of g n . We prove that this sequence is tight. It is sufficient to show that: lim sup ν n (y(0) L 2 > R) = 0,  n max y(t) − y(s) L 2 ≥ ρ = 0, ∀ρ > 0. lim sup ν

R→∞ n δ→0 n

|t−s|≤δ s,t∈[0,T ]

(3.7) (3.8)

Since g n (0)(θ ) = θ , condition (3.7) is clearly satisfied. Concerning condition (3.8), we have   1 n n n ν max y(t) − y(s) L 2 ≥ ρ ≤ E max g (t) − g (s) L 2 |t−s|≤δ |t−s|≤δ ρ s,t∈[0,T ] s,t∈[0,T ] 
t 
t         1 n n n n    ≤ E max  u (τ, g (τ )(θ ))dτ  + E max  d x (τ, g (τ )(θ ))  2 . |t−s|≤δ |t−s|≤δ ρ s s L2 L s,t∈[0,T ]

s,t∈[0,T ]

Using the invariance with respect to Lebesgue measure of the flow g n (t)(·), we obtain 
t   √   n n  ≤ δu L 2 ([0,T ];T2 ) . E max  u (τ, g (τ )(θ ))dτ   |t−s|≤δ s,t∈[0,T ]

L2

s

Also by the invariance of Lebesgue measure and the H¨older continuity of the Brownian motion, we have 
t 
t         n n  d x n (τ, θ )  = E max ≤ Cδ 2α E max  d x (τ, g (τ )(θ ))     |t−s|≤δ s,t∈[0,T ]

L2

s

|t−s|≤δ s,t∈[0,T ]

s

L2

with 0 < α ≤ 1. Condition (3.8) follows. The space L 2 endowed with the weak topology is a relatively compact space and the σ -algebras generated by the Borelian sets defined by the weak and the strong topologies are the same. Therefore, there exists a subsequence νn k of νn and a measure ν such that νn k → ν, with respect to the weak topology on the space of measures on C([0, T ]; L 2 ). To simplify the notation, we still denote such subsequence by νn . By Skorohod’s theorem, there exists a probability space (, F, P) and a family of stochastic processes g˜ ωn (t), gω (t), ω ∈  with laws, respectively, νn and ν such that for a.e. ω, g˜ ωn (·) → gω (·) in the space C([0, T ]; L 2 ). For any continuous function f defined on T2 and a.e. ω ∈ , we have

f (gωn (t)(θ ))dθ = f (θ )dθ. T2

T2

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F. Cipriano, A. B. Cruzeiro

The left-hand side integral can be identified in law with
f (g˜ ωn (t)(θ ))dθ. T2

n

For a.e. ω, the convergence of g˜ n to g implies the existence of a subsequence g˜ ω j (t)(θ ) n of g˜ ωn (t)(θ ) such that g˜ ω j (t)(θ ) → gω (t)(θ ) for a.e. θ . Using Lebesgue’s theorem

nj f (g˜ ω (t)(θ ))dθ → f (gω (t)(θ ))dθ T2

T2

which proves that g lives in G 0V . We now prove that the stochastic process gω (t)(θ ) is a weak solution of the s.d.e. (3.1). The process g n (t) can be identified in law with the solution of Eq. (3.6); we have: 2

t
t   n n  E u(τ, g(τ )(θ ))dτ   u (τ, g˜ (τ )(θ ))dτ − T2

0

0

2

t
t   n n n  ≤E u (τ, g˜ (τ )(θ ))dτ − u(τ, g˜ (τ )(θ ))dτ   0 T2 0 2

t
t   n  +E u(τ, g(τ )(θ ))dτ   u(τ, g˜ (τ )(θ ))dτ − =

I1n

T2 0 + I2n .

0

By the invariance of the Lebesgue measure and the fact that u n → u in L 2 ([0, T ]; T2 ) the integral I1n converges to zero as n → ∞. Applying Lusin’s theorem to the vector field u on [0, T ] × T2 , and considering, for a.e. ω, a subsequence of g˜ ωn (t)(θ ) that converges to gω (t)(θ ) uniformly in t and θ , there exists a subsequence of the integral I2n that converges to zero. Concerning the stochastic integral, for a.e. ω, we have: 2

t
t    d x n (τ, g˜ n (τ )(θ )) −  d x(τ, g(τ )(θ ))   T2 0 0 2

t
t   n n n  ≤ d x(τ, g˜ (τ )(θ ))  d x (τ, g˜ (τ )(θ )) − T2

0

0

2

t
t    d x(τ, g˜ n (τ )(θ )) −  + d x(τ, g(τ )(θ ))   2 T

0

0

= I1n (ω) + I2n (ω). Using again the invariance with respect to Lebesgue measure 2 2

t
   n    d x n (τ, θ ) − d x(τ, θ ) dθ = x (t, θ ) − x(t, θ ) dθ     T2

0

≤C

sup t∈[0,T ],θ∈T2

|x (t, θ ) − x(t, θ )| n

T2

which converges to zero as n → ∞. Since for a.e. ω, xω (t, θ ) are continuous random variables on [0, T ] × T2 , there exists a subsequence of the integral
  x(τ, g˜ n (τ )(θ )) − x(τ, g(τ )(θ ))2 dθ T2

that converges to zero.

Navier-Stokes Equation and Diffusions of Homeomorphisms

265

Therefore, taking the limit in L 2 ( × T2 ) of the integral equation (3.6), we conclude that g(τ )(θ ) satisfies the s.d.e. (3.1).

 Corollary 3.1. Suppose that u satisfy the hypothesis of Theorem 3.1 and let gu (t) be the solution of Eq. (3.1). Then the infinitesimal generator of this process, when computed at functionals of the form F(g)(θ ) = f (g(θ )), is given by Lu F = u · ∇ f + ν f,

∀ f ∈ C 2 (T2 ).

Proof. The proof is analogous to the proof of Theorem 2.2.

(3.9)



4. The Variational Principle Given a G 0V -valued stochastic process g(t) we define the action functional
T 1 1 S[g] = E Dg(t)2L 2 dt − EDg(T )2L 2 , 2 2 0

(4.1)

where D is the generalized derivative, defined for smooth functionals F by      1 Ft (4.2) F g(t + ), t +  − F g(t), t . D F(g(t), t) = lim E →0    Given v ∈ C 1 [0, T ]; GV∞ with v(0, ·) = 0, consider the following ordinary differential equation det (v) = v(t, ˙ et (v)), dt e0 (v) = e,

(4.3)

dv . Since v is divergence free, e· (v) is a G ∞ V - valued deterministic path. dt 0 Let us denote by P the set of continuous G V -valued Ft -semimartingales g(t) such that g(0) = e. The left product et (v) ◦ gu (t) of the diffusion gu (t) by an arbitrary deterministic path e· (v) is well defined on P. So, we define the derivative of a functional on P at a process g ∈ P using the left product by an arbitrary element of the following set:     H = et (v) : v ∈ C 1 [0, T ]; GV∞ and v(0, ·) = 0 . where v˙ =

This set can be considered as the “tangent” space to P, appropriate to a calculus of variation for (4.1). A small perturbation of g ∈ P in the direction h(t) = et (v) ∈ H will correspond to the product et (v) ◦ gu (t), where et (v) is the solution of (4.3) associated with the perturbation v of v. Definition 4.1. Let J be a functional defined on P taking values in R. We define its left and right derivatives in the direction of h(·) = e· (v) ∈ H at a process g ∈ P, respectively, by:  d J [e· (v) ◦ g(·)]=0 , d  d (D R )h J [g] = J [g(·) ◦ e· (v)]=0 . d (D L )h J [g] =

266

F. Cipriano, A. B. Cruzeiro

A process g ∈ P will be called a critical point of the functional J if (D L )h J [g] = (D R )h J [g] = 0, ∀h ∈ H. Theorem 4.1. Let u ∈ L 2 ([0, T ]; GV0 ) and gu (t) ∈ C([0, T ]; G 0V ) be a weak solution of Eq. (3.1). The stochastic process gu (t) is a critical point of the energy functional S defined in (4.1) if and only if the vector field u(t) verifies the Navier-Stokes equation: ⎧ ∂u ⎪ ⎨ + (u · ∇)u = νu + ∇ p ∂t . ∇ · u =0 ⎪ ⎩ u(T, θ ) = u T (θ )

(4.4)

Proof. Since the functional S is right invariant, the right derivative is not relevant. Let  > 0 and h(t) = et (v) be an arbitrary element in the “tangent” space H. We have
t (4.5) v(s, ˙ es (v))ds. et (v) = e +  0

Since et (0)(θ ) = θ for all t,  d et (v)=0 = d




t

v˙s (θ )ds = vt (θ ),

(4.6)

0

therefore  d  et (v) ◦ gu (t) =0 = vt ◦ gu (t) d

(4.7)

and 
T    d  S[e· (v) ◦ gu (·)] Dgu (t), D(v(t, gu (t))) L 2 dt =E d =0
0   − u T (θ )v(T, θ ) dθ.

(4.8)

T2

Using Itô’s formula       d Dgu (t), v(t, gu (t)) L 2 = d Dgu (t), v(t, gu (t)) L 2 + Dgu (t), dv(t, gu (t)) L 2   + d Dgu (t), dv(t, gu (t)) L 2 ,

and the expression of the contraction term,   d Dgu (t), dv(t, gu (t)) L 2 = 2ν




2  ∂v i ∂u i (t, gu (t)) (t, gu (t))dθ, ∂θ j T2 i, j=1 ∂θ j

(4.9)

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267

we deduce that
T     Dgu (t), D(v(t, gu (t))) L 2 dt = E Dgu (T ), v(T, gu (T )) L 2 E 0




T

−E 0


=

T2




T

u(T, θ )v(T, θ )dθ −



T   d Dgu (t), dv(t, gu (t)) L 2 dt dt − E 0
 ∂u + (u · ∇u) + νu (t, θ )v(t, θ )dθ dt T2 ∂t

D Dgu (t), v(t, gu (t))

0



L2

2 

∂v i ∂u i (t, θ ) (t, θ )dθ dt ∂θ j T2 i, j=1 ∂θ j
T

∂u + (u · ∇u) − νu (t, θ )v(t, θ )dθ dt. u(T, θ )v(T, θ )dθ − = T2 T2 ∂t 0 −2ν

Therefore

  d S[e· (v) ◦ gu (·)] = 0, ∀v ∈ H d =0

is equivalent to

T
 ∂u u(T, θ ) − u T (θ ) v(T, θ )dθ, + (u · ∇u) − νu (t, θ )v(t, θ )dθ dt = T2 ∂t T2 0 which corresponds to the weakformulation of the Navier-Stokes equation (4.4), since v is arbitrary in C 1 [0, T ]; GV∞ .

 5. Existence of the Critical Diffusion In this paragraph, we consider the action functional S defined in (4.1) on the set P of semimartingales. The next theorem proves the existence of a process in P that is a minimum of the functional. Theorem 5.1. The action functional S defined in (4.1) has a minimum g(t) on the subset of P with fixed final energy Dg(T ) L 2 . Proof. The functional is bounded below, let α be its infimum. We consider g n (t) a minimizing sequence of the functional S. This means that S[g n (·)] → α as n → ∞. Let us denote by u n the drift of g n (t). The sequence 2S[g n (·)] = u n  L 2 ([0,T ];T2 ) − u T  L 2 is bounded, therefore there exists a subsequence u n j of u n that converges with respect to the weak topology, more precisely there exists u ∈ L 2 ([0, T ]; T2 ) such that u n j → u, weakly in L 2 ([0, T ]; T2 ).

T  We obtain 0 T2 u(t, θ )∇ f (θ )dtdθ = 0, for all regular function f , which implies that u(t, ·) ∈ GV0 a.e. in t. The limit function u satisfies the assumptions of Theorem 3.1. Then we can construct a stochastic process gu (t) in P as solution of the stochastic differential equation (3.1). Since   S gu (·) ≤ S g n j (·) , ∀ j   we deduce that S gu (·) = α and gu (t) is a minimum.

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We may define a solution of the Navier-Stokes equation as the drift (or mean velocity) of the critical process obtained in the last theorem. We have, Corollary 5.1. The mean generalized velocity of the minimum of the action functional in Theorem 5.1 is a solution of the Navier-Stokes equation (4.4). Moreover, the generalized kinetic energy Dgu (t)2L 2 of such minimum gu (t) is a Ft -supermartingale. Proof. By construction gu (t), the minimum of the action functional obtained in Theorem 5.1, satisfies the stochastic differential equation (3.1). Let us consider an arbitrary deterministic path h(t) = et (v) ∈ H with v(T ) = 0. The final energy of all variations et (v) ◦ gu (t) coincide with the final energy of the minimum Dgu (T )2L 2 . Therefore (D L )h J [g] = 0 and, according to Theorem 4.1, u(t, θ ) satisfies Navier-Stokes equation. We prove that   E Fs Dgu (t)2L 2 ≤ Dgu (s)2L 2 , 0 ≤ s ≤ t ≤ T. Using Itô’s formula for the functional gu (t)(θ ), we have: dDgu (t)2L 2 = 2(d Dgu (t), Dgu (t)) L 2 + (d Dgu (t), d Dgu (t)) L 2 . Considering the expression of the stochastic contraction term (4.9), we derive
t   E Fs Dgu (t)2L 2 − Dgu (s)2L 2 = 2E (D Dgu (τ ), Dgu (τ )) L 2 s

+ 2ν∇u(τ, gu (τ ))2L 2
t
 ∂u  =2 + u · ∇u (τ, θ )u(τ, θ )dθ dτ 2 ∂t T s
t
|∇u(τ, θ )|2 dθ dτ. = −2ν s

T2



Since we have associated stochastic processes to solutions of the Navier-Stokes equation, a detailed study of these processes will determine various properties of NavierStokes flows. On the other hand, the existence of a variational principle should lead naturally to a study of the symmetries of our action functional, their probabilistic interpretation and geometrical implications for the same flow. Acknowledgements. The authors are grateful to Prof. J. C. Zambrini for his suggestions and very helpful discussions. We acknowledge the support of FCT, projects POCTI/0208/2003 and POCI / MAT /55977 / 2004.

References [A] [A-K] [C1]

Arnold, V.I.: Sur la géométrie différentielle des groupes de lie de dimension infinie et ses applications a l’hidrodynamique des fluides parfaits. Ann. Inst. Fourier 16, 316–361 (1966) Arnold, V.I., Khesin B.A.: Topological Methods in Hydrodynamics. New York: Springer-Verlag, 1998 Constantin, P.: An Eulerian-Lagrangian approach to the Navier-Stokes equations. Commun. Math. Phys. 216(3), 663–686 (2001)

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[C2]

269

Constantin, P., Iyer, G.: A Stochastic Lagrangian Representation of 3-Dimensional Incompressible Navier-Stokes Equations. Comm. Pure Appl. Math., in press, DOI 10.1002/cpa.20192, 2007 [F1] Fang, S.: Canonical Brownian motion on the diffeomorphism group of the circle. J. Funct. Anal. 196, 162–179 (2002) [F2] Fang, S.: Solving s.d.e.’s on Homeo (S 1 ). J. Funct. Anal. 216, 22–46 (2004) [G] Gawedzki, K.: Simple models of turbulent transport. XIV Intern. Congress on Mathem. Physics, RiverEdge, NJ: World Scientific, 2005 [Go] Gomes, D.A.: A variational formulation for the Navier-Stokes equation. Commun. Math. Phys. 257(1), 227–234 (2005) [I-F] Inoue, A., Funaki, T.: A new derivation of the Navier-Stokes equation. Commun. Math. Phys. 65, 83– 90 (1979) [K] Kunita, H.: Stochastic Flows and Stochastic Differential Equations. Cambridge: Cambridge University Press, 1990 [M] Malliavin, P.: The canonic diffusion above the diffeomorphism group of the circle. C. R. Acad. Sci. Paris, t. 329, Série I, 325–329 (1999) [N-Y-Z] Nakagomi, T., Yasue, K., Zambrini, J.C.: Stochastic variational derivations of the Navier-Stokes equation. Lett. Math. Phys. 160, 337–365 (1981) [S] Shkoller, S.: Geometry and curvature of diffeomorphism group with H1 metric and mean hydrodynamics. J. Funct. Anal. 160(1), 337–365 (1998) ¨ unel, A.S.: Stochastic analysis on Lie groups. Stoch. Anal. and Rel. Topics VI, Progr. Probab. [U] Ust¨ 42, Boston: Birkh¨auser 1998 [Y] Yasue, K.: A variational principle for the Navier-Stokes equation. J. Funct. Anal. 51(2), 133–141 (1983) Communicated by P. Constantin

Commun. Math. Phys. 275, 271–296 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0267-6

Communications in

Mathematical Physics

A Characterization of Right Coideals of Quotient Type and its Application to Classification of Poisson Boundaries Reiji Tomatsu Department of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan. E-mail: [email protected] Received: 6 November 2006 / Accepted: 16 November 2006 Published online: 26 June 2007 – © Springer-Verlag 2007

Abstract: Let G be a co-amenable compact quantum group. We show that a right coideal of G is of quotient type if and only if it is the range of a conditional expectation preserving the Haar state and is globally invariant under the left action of the dual discrete quantum group. We apply this result to the theory of Poisson boundaries introduced by Izumi for discrete quantum groups and generalize a work of Izumi-Neshveyev-Tuset on SUq (N ) for co-amenable compact quantum groups with the commutative fusion rules. More precisely, we prove that the Poisson integral is an isomorphism between the Poisson boundary and the right coideal of quotient type by a maximal quantum subgroup of Kac type. In particular, the Poisson boundary and the quantum flag manifold are isomorphic for any q-deformed classical compact Lie group. 1. Introduction Since Woronowicz introduced the axiomatic compact quantum groups [30], they have attracted a growing interest of many researchers as a framework to describe new types of symmetries. In this paper, we study two typical examples of ergodic actions of compact quantum groups, namely right coideals and Poisson boundaries. Let G be a compact quantum group. A right coideal is a von Neumann subalgebra of the function algebra on G which is globally invariant under the right translation action of G. Taking the fixed point algebra of a left action of a quantum subgroup gives an example of a right coideal. We say that such a right coideal is of quotient type. When G is an ordinary group, it is well-known that all the right coideals are of quotient type. However, when G is a quantum group, not all the right coideals are realized as quotients [18, 19, 23]. This fact presents a contrast between the quantum groups and the ordinary ones. Another difference occurs in behaviors of infinite tensor product actions. In the ordinary case, such an action is minimal and, in particular, the relative commutant of the fixed point algebra is trivial. However, this is not the case for quantum groups. In [10],

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Izumi has described this contrast by introducing the notion of a Poisson boundary of a dual discrete quantum group. More precisely, he has shown that the Poisson boundary is isomorphic to the relative commutant of the fixed point algebra. Moreover he has also studied the Poisson boundary of the dual of SUq (2) and shown a striking result that the Poisson boundary is isomorphic to the standard Podle´s sphere L ∞ (T \ SUq (2)) introduced in [18, 19]. On one hand, this result has led to a conjecture that the Poisson boundary is isomorphic to the quantum flag manifold for any q-deformed classical compact Lie group. For SUq (N ), the conjecture was confirmed affirmatively in [12]. Note that a q-deformed classical compact Lie group is co-amenable, that is, the dual discrete quantum group is amenable in the sense of [3]. On the other hand for non-amenable cases, the Poisson and Martin boundaries [17] of universal orthogonal discrete quantum groups are studied in [25] and [26]. In this paper, we first characterize when a right coideal is of quotient type and second apply the characterization to determine Poisson boundaries of amenable discrete quantum groups with the commutative fusion rules. A right coideal of quotient type has the following two properties. The first one is the expectation property, namely, existence of a normal conditional expectation preserving the Haar state from the function algebra. The second one is the coaction symmetry which means that the left action of the dual preserves the right coideal. Assuming amenability of the dual, we can prove the following theorem (Theorem 3.18). Theorem 1. Let G be a co-amenable compact quantum group and B ⊂ L ∞ (G) a right coideal. Then B is of quotient type if and only if B has the expectation property and the coaction symmetry. Next we study a Poisson boundary of an amenable discrete quantum group. In order to compute a Poisson boundary, we present an approach which differs from that of [12]. The key point of our proof is to construct an “inverse”of the Poisson integral. Although this strategy is the same as the one taken in [12], we do it by utilizing not the Berezin transforms but an invariant mean of a dual discrete quantum group. Then using Theorem 1, we show that the Poisson boundary is isomorphic to a right coideal of quotient type by a quantum subgroup. Moreover, we can specify the quantum subgroup which is the maximal quantum subgroup of Kac type with respect to inclusions. After this work was done, we learned from S. Vaes that the notion has been already introduced as the canonical Kac quotient in [20]. Our main result is the following theorem (Theorem 4.8). Theorem 2. Let G be a co-amenable compact quantum group. Assume that its fusion algebra is commutative. Then the following statements hold. (1) There exists a unique maximal quantum subgroup of Kac type H.
is an isomorphism. (2) The Poisson integral
: L ∞ (H \ G) → H ∞ (G)
= C. This yields the minimality In particular, G is of Kac type if and only if H ∞ (G) of an infinite tensor product action of G. For a q-deformed classical compact Lie group Gq , the maximal quantum subgroup of Kac type is the maximal torus T. Therefore, we obtain the following result (Corollary 4.11). Theorem 3. Let Gq be the q-deformation of a classical compact Lie group G. Then the q ) is an isomorphism. Poisson integral
: L ∞ (T \ Gq ) → H ∞ (G

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Notations. Let M be a von Neumann algebra with predual M∗ . We denote by M∗+ the set of positive functionals in M∗ . For a linear functional θ on M, we define a linear functional θ by θ (x) = θ (x ∗ ) for x ∈ M. For a weight φ on M, we set n φ = {x ∈ M | φ(x ∗ x) < ∞}, m φ = n ∗φ n φ . We denote by m +φ the set of positive elements in m φ . For a w

linear subspace X ⊂ M, we denote by X the weak closure of X . We denote by ⊗ the minimal tensor product for C ∗ -algebras and the spatial tensor product for von Neumann algebras. 2. Preliminaries We collect necessary results on compact (discrete) quantum groups. 2.1. Compact quantum groups. Our standard references are [1, 10, 30]. For the notion of a compact quantum group, we adopt the definition in [30, Def. 2.1] as follows:

Definition 2.1. A compact quantum group G is a pair (C(G), δG ) which satisfies the following conditions: (1) C(G) is a separable unital C ∗ -algebra. (2) The map δG : C(G) → C(G) ⊗ C(G) is a coproduct, i.e. it is a faithful unital ∗-homomorphism satisfying the coassociativity condition, (δG ⊗ id) ◦ δG = (id ⊗ δG ) ◦ δG . (3) The vector spaces δG (C(G))(C ⊗ C(G)) and δG (C(G))(C(G) ⊗ C) are dense in C(G) ⊗ C(G). Let h G be the Haar state on C(G) which satisfies the invariance condition, (id ⊗ h G )(δG (a)) = h G (a)1 = (h G ⊗ id)(δG (a)) for all a ∈ C(G). In this paper, we always assume that the Haar states are faithful. If the Haar state is tracial, we say that the compact quantum group is of Kac type [7]. Let (πh , L 2 (G), 1ˆ h ) be the GNS triple of h G , which consists of the representation, the Hilbert space and the GNS cyclic vector, respectively. We always omit πh and regard C(G) as a C ∗ -subalgebra w of B(L 2 (G)). We set a von Neumann algebra L ∞ (G) = C(G) . The multiplicative unitaries VG and WG are defined by WG∗ (x 1ˆ h ⊗ y 1ˆ h ) = δG (y)(x 1ˆ h ⊗ 1ˆ h ) for x, y ∈ C(G), VG (x 1ˆ h ⊗ y 1ˆ h ) = δG (x)(1ˆ h ⊗ y 1ˆ h ) for x, y ∈ C(G).

(2.1) (2.2)

Then we have the pentagon equalities, (WG )12 (WG )13 (WG )23 = (WG )23 (WG )12 , (VG )12 (VG )13 (VG )23 = (VG )23 (VG )12 . Using them, we can extend the coproduct δG to L ∞ (G) by δG (x) = VG (x ⊗ 1)VG∗ = WG∗ (1 ⊗ x)WG for x ∈ L ∞ (G).

(2.3) (2.4)

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The normal state h G (x) = (x 1ˆ h , 1ˆ h ), x ∈ L ∞ (G) is invariant under δG . Since h G on C(G) has the modular automorphism group [30, Theorem 2.4], h G is also faithful on L ∞ (G). Then the pair (L ∞ (G), δG ) is a von Neumann algebraic compact quantum group in the sense of [16]. Let H be a Hilbert space and v ∈ B(H )⊗ L ∞ (G) a unitary. If (id ⊗δG )(v) = v12 v13 , we say that v is a right unitary representation of G. A left unitary representation is similarly defined. For example, the unitaries WG and VG are left and right unitary representations, respectively. Let v ∈ B(H ) ⊗ L ∞ (G) be a unitary representation. Suppose that T ∈ B(H ) satisfies (T ⊗ 1)v = v(T ⊗ 1). If any such element T must be a scalar, v is said to be irreducible. Any unitary representation is completely decomposable, that is, it is a direct sum of irreducible ones. The set of the equivalence classes of all the irreducible representations is denoted by Irr(G). For π ∈ Irr(G), we choose a representation Hilbert space Hπ and an irreducible representation vπ = (vπi, j )i, j∈Iπ ∈ B(Hπ ) ⊗ C(G). We call an irreducible representation 1 ∈ C ⊗ L ∞ (G) the trivial representation and denote by 1 the equivalence class. We define a dense unital ∗-subalgebra A(G) ⊂ C(G) by A(G) = span{vπi, j | i, j ∈ Iπ , π ∈ Irr(G)}. It is clear that δG (A(G)) ⊂ A(G)⊗ A(G). We define the Hopf algebra structure, namely, the antipode κG and the counit εG on A(G) as follows. The invertible antimultiplicative map κG : A(G) → A(G) is defined by κG (vπi, j ) = vπ∗ j,i for i, j ∈ Iπ , π ∈ Irr(G).

(2.5)

The unital ∗-homomorphism εG : A(G) → C is defined by εG (vπi, j ) = δi, j for i, j ∈ Iπ , π ∈ Irr(G). In fact A(G) is a Hopf ∗-algebra, that is, κG (κG (x)∗ )∗ = x holds for any x ∈ A(G). For any finite dimensional unitary representation v ∈ B(H )⊗ A(G), we have (id⊗κG )(v) = v ∗ and (id ⊗ εG )(v) = 1, which follow from the complete decomposability of v. We introduce the Woronowicz characters { f zG }z∈C on A(G) [30, Theorem2.4]. The multiplicative functional f zG : A(G) → C satisfies the following properties: (i) f 0G = εG . (ii) For any a ∈ A(G), the function C  z → f zG (a) ∈ C is entirely holomorphic. (iii) ( f zG1 ⊗ f zG2 ) ◦ δG = f zG1 +z 2 for any z 1 , z 2 ∈ C. G (a), f G (a ∗ ) = f G (a). (iv) For any z ∈ C and a ∈ A(G), f zG (κ(a)) = f −z z −¯  z 2 G G (v) For any a ∈ A(G), κG (a) = ( f 1 ⊗ id ⊗ f −1 ) (δG ⊗ id)(δG (a)) .    (vi) For any a, b ∈ A(G), h G (ab) = h G b ( f 1G ⊗ id ⊗ f 1G ) (δG ⊗ id)(δG (a)) . h

The modular automorphism group {σt G }t∈R is given by   h σt G (x) = ( f itG ⊗ id ⊗ f itG ) (δG ⊗ id)(δG (x)) for all t ∈ R, x ∈ A(G). We define the following map τtG : A(G) → A(G) by  G  τtG (x) = ( f itG ⊗ id ⊗ f −it ) (δG ⊗ id)(δG (x)) for all t ∈ R, x ∈ A(G).

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Then {τtG }t∈R is a one-parameter automorphism group on A(G) and it is called the h scaling automorphism group. Note that any element of A(G) is analytic for {σt G }t∈R and {τtG }t∈R . Define a map RG : A(G) → A(G) by G (x)) for x ∈ A(G). RG (x) = κG (τi/2 2 = id. We call R the unitary Then RG is a ∗-antiautomorphism on A(G) with RG G G antipode. Actually RG commutes with τt for all t ∈ R, and it satisfies

G G = τ−i/2 ◦ RG . κG = RG ◦ τ−i/2

Since the Haar state h G is invariant under the ∗-preserving maps σt G , τtG and RG , we can extend them to the maps on C(G), moreover on L ∞ (G). Then the following relations among δG and them hold. h

h G ◦ τtG = h G = h G ◦ RG , δG ◦ σt

hG

δG ◦ τtG

=

(τtG

⊗ τtG ) ◦ δG ,

hG

= (σt

G ⊗ τ−t ) ◦ δG , op

δG ◦ RG = (RG ⊗ RG ) ◦ δG ,

(2.6) (2.7)

op

where δG (x) = δG (x)21 . Let v ∈ B(H ) ⊗ A(G) be a finite dimensional unitary representation. Set FvG = (id ⊗ f 1G )(v). Then FvG satisfies the following properties: (i) (ii) (iii) (iv) (v)

FvG is a non-singular positive operator on H . For any z ∈ C, (id ⊗ f zG )(v) = (FvG )z . For any t ∈ R, (id ⊗ τtG )(v) = ((FvG )it ⊗ 1)v((FvG )−it ⊗ 1). h For any t ∈ R, (id ⊗ σt G )(v) = ((FvG )it ⊗ 1)v((FvG )it ⊗ 1). Let w ∈ B(Hw ) ⊗ A(G) be a finite dimensional unitary representation of G. If a linear map T : Hv → Hw satisfies (T ⊗ 1)v = w(T ⊗ 1), then T Fv = Fw T .

For π ∈ Irr(G), we write FπG (or simply Fπ ) for FvGπ . Let Tr π be the non-normalized trace on B(Hπ ). Set Dπ = Tr π (Fπ ). Then we have the orthogonal relations, δ δ δ . h G (vπi, j vρ∗k, ) = Dπ−1 Fπ j, j δπ,ρ δi,k δ j, , h G (vπ∗i, j vρk, ) = Dπ−1 Fπ−1 i,i π,ρ i,k j, We decompose WG and VG into irreducible representations. For π ∈ Irr(G), we define two systems of matrix units {eπi, j }i, j∈Iπ and { f πi, j }i, j∈Iπ in B(L 2 (G)) by eπi, j (vρk, 1ˆ h ) = δπ,ρ δ j, vρk,i 1ˆ h for k,  ∈ Iρ , ρ ∈ Irr(G),

(2.8)

f πi, j (vρ∗k, 1ˆ h )

(2.9)

Then we have WG =





=

δπ,ρ δ j,k vρ∗i, 1ˆ h

for k,  ∈ Iρ , ρ ∈ Irr(G). 

vπi, j ⊗ f πi, j , VG =

π ∈Irr(G) i, j∈Iπ



π ∈Irr(G) i, j∈Iπ

Setting π = 1 at (2.8), we have e1 (x 1ˆ h ) = h G (x)1ˆ h for x ∈ L ∞ (G). The projection e1 = f 1 is minimal in B(L 2 (G)).

eπi, j ⊗ vπi, j .

(2.10)

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We will need some relations among WG , VG and the modular objects of L ∞ (G). Let ∆h G and JG be the modular operator and the modular conjugation of h G . We define a conjugate unitary JˆG by JˆG (x 1ˆ h ) = RG (x ∗ )1ˆ h for x ∈ C(G). We set a unitary UG = JG JˆG = JˆG JG . The implementing unitary PGit for τtG is defined by PGit (x 1ˆ h ) = τtG (x)1ˆ h for t ∈ R and x ∈ L ∞ (G). Then the following equalities are directly deduced from (2.1), (2.2), (2.6) and (2.7). WG =( JˆG ⊗ JG )WG∗ ( JˆG ⊗ JG ), −it

VG =(JG ⊗ JˆG )VG∗ (JG ⊗ JˆG ),

−it

WG =(PG ⊗ PG )WG (PG ⊗ PG ), WG =(PGit ⊗ ∆ithG )WG (PG−it ⊗ ∆−it h G ), it

it

−it

(2.11) −it

VG =(PG ⊗ PG )VG (PG ⊗ PG ), (2.12) it VG =(∆ithG ⊗ PG−it )VG (∆−it h G ⊗ PG ), (2.13) it

it

WG =( JˆG ⊗ JˆG )(VG )∗21 ( JˆG ⊗ JˆG ).

(2.14)

∞ We denote by L ∞ (G)fin ∗ the set of ω ∈ L (G)∗ which satisfies ω(vπi, j ) = 0, i, j ∈ Iπ all but finite elements π ∈ Irr(G). We simply write symbols by omitting G, if no confusion arises. For example, we write δ for δG .

2.2. Discrete quantum groups. The notions of a (dual) discrete quantum group have been studied in many papers, for example, [6, 16, 27, 29] and [31]. They have described essentially the same object. In this paper, we use a von Neumann algebraic quantum group presented in [16]. Definition 2.2. A discrete quantum group is a quintuplet (M, , ϕ, ψ, ε) which satisfies the following conditions: (1) M is a separable von Neumann algebra. (2)  : M → M ⊗ M is a coproduct, that is, it is a faithful normal unital ∗-homomorphism satisfying the coassociativity condition, ( ⊗ id) ◦  = (id ⊗ ) ◦ . (3) ϕ is a faithful normal semifinite weight on M satisfying the left invariance, ϕ((ω ⊗ id)((x))) = ω(1) ϕ(x) for all ω ∈ M∗+ , x ∈ m +ϕ . (4) ψ is a faithful normal semifinite weight on M satisfying the right invariance, ψ((id ⊗ ω)((x))) = ω(1) ψ(x) for all ω ∈ M∗+ , x ∈ m +ψ . (5) ε is a normal counit, that is, it is a normal character on M satisfying (ε ⊗ id) ◦  = id = (id ⊗ ε) ◦ . In fact, those weights ϕ, ψ are uniquely determined up to scalar multiplications and the counit ε is unique. From now, we simply write (M, ) for (M, , ϕ, ψ, ε) by omitting ϕ, ψ and ε once they are given. For a compact quantum group G, we construct the dual discrete quantum group as follows (see for example [1, 16]). We define the left group algebra and right group algebra by w

w

L(G) = {(ω ⊗ id)(WG ) | ω ∈ L ∞ (G)∗ } , R(G) = {(id ⊗ ω)(VG ) | ω ∈ L ∞ (G)∗ } .

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By (2.10), L(G) and R(G) are generated by { f πi, j } and {eπi, j }, respectively. Hence they are isomorphic to the von Neumann algebra direct sum of the matrix algebras {B(Hπ )}π ∈Irr(G) . Set fin ∞ fin L(G)fin = {(ω ⊗ id)(WG ) | ω ∈ L ∞ (G)fin ∗ }, R(G) = {(id ⊗ ω)(VG ) | ω ∈ L (G)∗ }.

They are an algebraic direct sum of the matrix algebras. We note the commutant property L(G) Jˆ. We define a minimal central projection corresponding to L(G) = R(G) = Jˆ π ∈ Irr(G) by 1π = i∈Iπ f πi,i . In particular, e1 = 11 . We denote by L(G)fin ∗ the subset of L(G)∗ which consists of ω ∈ L(G)∗ such that ω( f πi, j ) = 0, i, j ∈ Iπ all but finite elements π ∈ Irr(G). We prepare the coproducts  L and  R defined by  L (x) = WG (x ⊗ 1)WG∗ for x ∈ L(G),  R (x) = VG∗ (1 ⊗ x)VG for x ∈ R(G). We define the normal counit εˆ : L(G) → C by εˆ (x)e1 = xe1 for x ∈ L(G). In fact, there exist left, right invariant weights on L(G) and R(G) [16], and they are discrete quantum groups. Note that every discrete quantum group arises as the left (right) group algebra of a compact quantum group (see [1, 16] for duality theory). In this paper,
for the discrete quantum group (R(G),  R ). We simply write G we use the symbol G (or ) for  L and  R when it is not ambiguous. We define a positive operator F affiliated with L(G) by   Fπi,i f πi,i . F= π ∈Irr(G) i∈Iπ

Using (2.9), for all t ∈ R and x ∈ A(G) we have F it (x 1ˆ h ) = ( f −it ⊗ id)(δ(x))1ˆ h ,

Jˆ F it Jˆ(x 1ˆ h ) = (id ⊗ f it )(δ(x))1ˆ h .

Then we have ∆ith = F −it Jˆ F it Jˆ,

P it = F −it Jˆ F −it Jˆ.

(2.15)

The antipode Sˆ on L(G) is defined as follows. Since the map L ∞ (G)∗  ω → (ω ⊗ id)(WG ) ∈ L(G) is injective, we can define an invertible antimultiplicative map Sˆ : L(G)fin → L(G)fin by ˆ S((ω ⊗ id)(WG )) = (ω ⊗ id)(WG∗ ) for all ω ∈ L ∞ (G)fin ∗ . We define the unitary antipode Rˆ and the scaling automorphism group {τˆt }t∈R on L(G) by ˆ R(x) = J x ∗ J, τˆt (x) = F it x F −it for x ∈ L(G), t ∈ R. Using (2.11), (2.12) and (2.15), we have Sˆ = Rˆ ◦ τˆ−i/2 on , Rˆ and τˆ are as follows:

(2.16)

L(G)fin . The relations among

ˆ ◦ op ,  ◦ τˆt = (τˆt ⊗ τˆt ) ◦ .  ◦ Rˆ = ( Rˆ ⊗ R)

(2.17)

ˆ π ) = 1π . Let µ : L(G)fin ⊗ L(G)fin → L(G)fin be the Take π ∈ Irr(G) satisfying S(1 multiplication map. We prove the well-known results for the readers’ convenience.

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Lemma 2.3. For any x ∈ L(G) and π ∈ Irr(G), one has     ˆ π π (x)) = εˆ (x)1π = µ ( Sˆ ⊗ id)(π π (x)) , µ (id ⊗ S)( where π π (x) = (x)(1π ⊗ 1π ).   ˆ π π (x)) = εˆ (x)1π . Since π π (1ρ ) is non-zero for finite Proof. We prove µ (id ⊗ S)( elements ρ ∈ Irr(G), we may assume that x = (ω ⊗ id)(WG ), ω ∈ L ∞ (G)fin ∗ . Then we have ˆ π π (x)) = (id ⊗ S)( ˆ π π ((ω ⊗ id)(WG ))) (id ⊗ S)( ˆ = (id ⊗ S)((ω ⊗ id ⊗ id)((id ⊗ π π )(WG )))

ˆ = (id ⊗ S)((ω ⊗ id ⊗ id)((WG )12 (WG )13 (1 ⊗ 1π ⊗ 1π ))) = (ω ⊗ id ⊗ id)((WG )12 (WG )∗13 (1 ⊗ 1π ⊗ 1π )).

Hence

    ˆ π π (x)) = µ (ω ⊗ id ⊗ id)((WG )12 (WG )∗13 (1 ⊗ 1π ⊗ 1π )) µ (id ⊗ S)( = (ω ⊗ id)(WG WG∗ (1 ⊗ 1π )) = ω(1)1π = εˆ (x)1π .   Similarly we can prove µ ( Sˆ ⊗ id)(π π (x)) = εˆ (x)1π . 

Lemma 2.4. Let K be a Hilbert space and V ∈ B(K ) ⊗ L(G) a unitary representation of (L(G), ), that is, V is a unitary satisfying (id ⊗ )(V) = V12 V13 . Then for any π ∈ Irr(G), one has ˆ π ) = V∗π , (id ⊗ S)(V where Vπ = V(1 ⊗ 1π ). Proof. Set x = (ω ⊗ id)(V) ∈ L(G), ω ∈ B(K )∗ . For any π ∈ Irr(G), we have   ˆ (ω ⊗ id ⊗ id)((id ⊗ π π )(V)) ˆ π π (x)) = (id ⊗ S) (id ⊗ S)( ˆ = (ω ⊗ id ⊗ S)((V π )12 (Vπ )13 )   ˆ π )13 . = (ω ⊗ id ⊗ id) (Vπ )12 (id ⊗ S)(V   ˆ π π (x)) = εˆ (x)1. This implies By the previous lemma, we have µ (id ⊗ S)( ˆ π )) = ω(1) for any ω ∈ B(K )∗ . Hence (id ⊗ S)(V ˆ π ) = V∗ .  (ω ⊗ id)(Vπ (id ⊗ S)(V π 2.3. Amenability. We recall the notion of amenability of a discrete quantum group. For details of the theory, readers are referred to [2–4, 22] and references therein. A discrete quantum group (M, ) is said to be amenable if there exists a state m ∈ M ∗ such that m((ω ⊗ id)((x))) = ω(1)m(x) for all ω ∈ M∗ and x ∈ M. The state m is called a left invariant mean. It is known that (L(G),  L ) is amenable if and only if (R(G),  R ) is amenable. If it is the case, we say that G is co-amenable. Note that the counit εG is norm-bounded if and only if G is co-amenable [4, Theorem 4.7], [22, Theorem 3.8]. The amenability is also equivalent to the universality of C(G), that is, for any C ∗ -algebra B, any ∗-homomorphism σ : A(G) → B extends to a ∗-homomorphism σ : C(G) → B [3, Theorem 3.6].

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2.4. Right G-action α and left G-action β. We prepare two maps α : B(L 2 (G)) → 2 ∞ 2 B(L (G)) ⊗ L (G) and β : B(L (G)) → R(G) ⊗ B(L 2 (G)) which will be frequently used in our study. A map α is defined by α(x) = VG (x ⊗ 1)VG∗ for x ∈ B(L 2 (G)). Then α is a right action of G on B(L 2 (G)), that is, (α ⊗ id) ◦ α = (id ⊗ δG ) ◦ α holds. The conditional expectation E α = (id ⊗ h) ◦ α maps B(L 2 (G)) onto L(G) = {x ∈ B(L 2 (G)) | α(x) = x ⊗ 1}. The other one, β is defined by β(x) = VG∗ (1 ⊗ x)VG for x ∈ B(L 2 (G)).
on B(L 2 (G)), that is, (id ⊗ β) ◦ β = ( R ⊗ id) ◦ β holds. Then β is a left action of G We also call β a coaction of G. Note that both the actions α, β preserve not only L ∞ (G) but also R(G). 2.5. Right coideals. We introduce the notion of a right coideal. Our basic references for the theory of right coideals are [11] and [23]. Definition 2.5. Let G be a compact quantum group. Let B ⊂ L ∞ (G) be a von Neumann subalgebra. We say that B is a right coideal if δ(B) ⊂ B ⊗ L ∞ (G). A left coideal in L(G) is similarly defined. For a right coideal B ⊂ L ∞ (G) and a left coideal C ⊂ L(G), we define
= C ∩ L ∞ (G).
B = B ∩ L(G), C We prove a quantum group version of [11, Theorem 4.6] as follows. Lemma 2.6. Let G be a compact quantum group. Let B ⊂ L ∞ (G) be a right coideal and C ⊂ L(G) a left coideal.
⊂ L ∞ (G) is a right coideal. (1)
B ⊂ L(G) is a left coideal and C (2) The map B →
B is a lattice anti-isomorphism between the set of right coideal of
L ∞ (G) and left coideal of L(G). The inverse map is given by C → C. Proof. (1) It is similarly proved as in [11, Theorem 4.6].
is injective. Let C ⊂ L(G) be a left coideal. (2) First we show that the map C → C



∞

∞
Then C = (C ∩ L (G)) = C ∨ L (G) . We can adapt the proof of [11, Theorem 4.6
= C L ∞ (G) w . Recall the right G-action α on B(L 2 (G)) and the con(ii)] to deduce C
J = J C J L ∞ (G) w . ditional expectation E α : B(L 2 (G)) → L(G). We apply E α to J C Since J C J ⊂ J L(G)J ⊂ L(G) and the restriction of E α to L ∞ (G) is the Haar state
J ) = J C J . Hence the map C → C
is injective. Set B = C.
Then h, we have E α (J C

holds. The injectivity of the map C → C
yields
B ∩ L ∞ (G) = C B = C. Next we show that the map B →
B is injective. Let Bi ⊂ L ∞ (G), i = 1, 2 be right coideals. Then
Bi = (Bi ∩ L(G)) = Bi ∨ R(G). Let Bi
δ G = δ(Bi ) ∨ C ⊗ R(G) be Bi . the crossed product. Then we have WG (Bi
δ G)WG∗ = C ⊗ (Bi ∨ R(G)) = C ⊗
Hence
B1 =
B2 if and only if B1
δ G = B2
δ G. Then the biduality theorem (see [1, Théorème 7.5] or [24, Theorem 2.6]) implies that it is equivalent to B1 ⊗ B(L 2 (G)) = B2 ⊗ B(L 2 (G)), and B1 = B2 . Hence the map B →
B is injective. 

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2.6. Quantum subgroups. For the definition of quantum subgroups, we follow that of [19], in which matrix pseudogroups [28] are treated. Definition 2.7. Let G and H be compact quantum groups. (1) Suppose that there exists a surjective ∗-homomorphism rH : A(G) → A(H) such that δH ◦ rH = (rH ⊗ rH ) ◦ δG . Then we say that the pair {H, rH } is an algebraic quantum subgroup of G. (2) Suppose that there exists a surjective ∗-homomorphism rH : C(G) → C(H) such that δH ◦ rH = (rH ⊗ rH ) ◦ δG . Then we say that the pair {H, rH } is a quantum subgroup of G. In the above cases, the map rH is called a restriction map. We present basic properties on (algebraic) quantum subgroups in the following lemmas. Those are probably well-known for specialists, but we prove them for the sake of the readers’ convenience. Lemma 2.8. Let G and H be compact quantum groups. (1) If {H, rH } is a quantum subgroup of G, then it is an algebraic quantum subgroup of G. (2) Suppose G is co-amenable. Then any algebraic quantum subgroup of G is naturally regarded as a quantum subgroup. Proof. (1) Let π ∈ Irr(G). Then (id ⊗ rH )(vπ ) ∈ B(Hπ ) ⊗ C(H) is a finite dimensional unitary representation of H. Hence it is a finite direct sum of irreducible representations of H. This implies (id ⊗ rH )(vπ ) ∈ B(Hπ ) ⊗ A(H). Hence rH (A(G)) ⊂ A(H). Next we show the converse inclusion. Take any ρ ∈ Irr(H). Let wρ ∈ B(K ρ ) ⊗ A(H) be a corresponding irreducible unitary representation. Consider the bounded linear map θ : C(G) → B(K ρ ) defined by θ (x) = (id ⊗ h H )(wρ∗ (1 ⊗ rH (x))). Since rH is surjective, θ is a non-zero map. By density of A(G) ⊂ C(G), there exists π ∈ Irr(G) such that (id ⊗ θ )(vπ ) = 0. This shows that the unitary representation (id ⊗ rH )(vπ ) contains wρ . Hence all the entries of wρ are contained in rH (A(G)), and A(H) ⊂ rH (A(G)). (2) Let rH : A(G) → A(H) be a restriction map. Since C(G) is universal, the map extends to rH : C(G) → C(H). The image contains a total subspace A(H) in C(H), and rH is surjective. By continuity of rH , the relation δH ◦ rH = (rH ⊗ rH ) ◦ δG holds on C(G). Hence {H, rH } is a quantum subgroup of G.  Lemma 2.9. Let {H, rH } be an algebraic quantum subgroup of G. On A(G), (1) εH ◦ rH = εG , (2) τtH ◦ rH = rH ◦ τtG for all t ∈ R, (3) RH ◦ rH = rH ◦ RG . Proof. (1) Take any π ∈ Irr(G). Since (id ⊗ rH )(vπ ) is a unitary representation of H, (id ⊗ εH ◦ rH )(vπ ) = 1 = (id ⊗ εG )(vπ ). This implies εH ◦ rH = εG . (2) Take any π ∈ Irr(G). Since (id ⊗ rH )(vπ ) is a unitary representation of H, (id ⊗ κH ◦ rH )(vπ ) = (id ⊗ rH )(vπ )∗ = (id ⊗ rH ◦ κG )(vπ ). This implies rH ◦ κG = 2 = τ G and κ 2 = τ H , we have r ◦ τ G = τ H ◦ r . Let π ∈ Irr(G) κH ◦ rH . Since κG H H −i −i −i −i H and set w = (id ⊗ rH )(vπ ). Then we have   (FπG ⊗ 1)w((FπG )−1 ⊗ 1) = (id ⊗ rH ) (FπG ⊗ 1)vπ ((FπG )−1 ⊗ 1) G )(vπ ) = (id ⊗ rH ◦ τ−i

H H ◦ rH )(vπ ) = (id ⊗ τ−i )(w) = (id ⊗ τ−i

= (FwH ⊗ 1)w((FwH )−1 ⊗ 1).

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This shows kπ := (FwH )−1 FπG ∈ B(Hπ ) is an intertwiner of w, and in particular, FwH kπ = kπ FwH holds. Hence FπG = FwH kπ = kπ FwH . Since the positive operators FπG and (FwH )−1 commute, kπ is positive. In particular, we have Fπit = (FwH )it kπit for all t ∈ R. Hence for any π ∈ Irr(G), we have   (id ⊗ rH ◦ τtG )(vπ ) = (id ⊗ rH ) ((FπG )it ⊗ 1)vπ ((FπG )−it ⊗ 1) = ((FπG )it ⊗ 1)w((FπG )−it ⊗ 1)

= ((FwH )it kπit ⊗ 1)w(kπ−it (FwH )−it ⊗ 1) = ((FwH )it ⊗ 1)w((FwH )−it ⊗ 1)

= (id ⊗ τtH )(w)

= (id ⊗ τtH ◦ rH )(vπ ). Therefore the desired relation holds. (3) It follows from rH ◦ κG = κH ◦ rH and (2).  Let {H, rH } be an algebraic quantum subgroup of G. Since WG (1 ⊗ 1π ) ∈ A(G) ⊗ L(G)1π for any π ∈ Irr(G), we can define a left unitary representation of H, (rH ⊗ id)(WG ) ∈ L ∞ (H) ⊗ L(G) by  (rH ⊗ id)(WG ) = (rH ⊗ id)(WG (1 ⊗ 1π )). π ∈Irr(G)

Lemma 2.10. Let G and H be compact quantum groups. (1) Assume that {H, rH } is an algebraic quantum subgroup of G. Then there exists a t : L(H) → L(G) such that faithful normal unital ∗-homomorphism rH t )(WH ) = (rH ⊗ id)(WG ). (id ⊗ rH

(2) Assume that there exists a faithful normal unital ∗-homomorphism ι : L(H) → L(G) such that G ◦ ι = (ι ⊗ ι) ◦ H . t . Then there exists a restriction map rH : A(G) → A(H) such that ι = rH

Proof. (1) Since the map L ∞ (H)∗  ω → (ω ⊗ id)(WH ) ∈ L(H) is injective, we can t : L(H)fin → L(G) by define a linear map rH t ((ω ⊗ id)(WH )) = (ω ⊗ id)((rH ⊗ id)(WG )) for ω ∈ L ∞ (H)fin rH ∗ . t by the pentagon equalities (2.3) on W and It is easy to verify the multiplicativity of rH H t and π ∈ Irr(G). WG . We show that rH preserves the involutions. Take any ω ∈ L ∞ (H)fin ∗ Using the equality rH ◦ κG = κH ◦ rH , we have  ∗ t ((ω ⊗ id)(WH ))∗ (1 ⊗ 1π ) = (ω ⊗ id)((rH ⊗ id)(WG )) (1 ⊗ 1π ) rH = (ω ◦ rH ⊗ id)(WG∗ (1 ⊗ 1π )) = (ω ◦ rH ⊗ id)((κG ⊗ id)(WG )(1 ⊗ 1π ))   = (ω ◦ κH ⊗ id) (rH ⊗ id)(WG )(1 ⊗ 1π ) t = rH ((ω ◦ κH ⊗ id)(WH ))(1 ⊗ 1π )  ∗  t = rH (ω ⊗ id)(WH ) (1 ⊗ 1π ).

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t is a ∗-homomorphism. Since L(H) is a von Neumann algebra direct sum Hence rH t extends to a normal ∗-homomorphism from L(H) to of {B(K ρ )}ρ∈Irr(H) , the map rH t is unital. Take any π ∈ Irr(G) L(G). Then the desired equality holds. We show that rH ρ and ρ ∈ Irr(H). Let Nπ |H be the multiplicity of ρ in the unitary representation (rH ⊗ t : L(H)1 → L(G)1 is the N ρ times amplification, id)(WG (1⊗1π )). Then the map rH ρ π π |H t is unital. Since r is surjective, r t is injective. and rH H H (2) Since the linear map L(G)∗  ω → (id ⊗ ω)(WG ) ∈ L ∞ (G) is injective, we can define a linear map rH : A(G) → A(H) by

rH ((id ⊗ ω)(WG )) = (id ⊗ ω ◦ ι)(WH ) for ω ∈ L(G)fin ∗ . First we show that rH is multiplicative. Take any ω, θ ∈ L(G)fin ∗ . Then (ω ⊗ θ ) ◦ G ∈ L(G)fin , and ∗ rH ((id ⊗ ω)(WG )) rH ((id ⊗ θ )(WG )) = (id ⊗ ω ◦ ι)(WH )(id ⊗ θ ◦ ι)(WH ) = (id ⊗ ω ◦ ι ⊗ θ ◦ ι)((WH )12 (WH )13 ) = (id ⊗ ω ◦ ι ⊗ θ ◦ ι)((id ⊗ H )(WH ))   = (id ⊗ ω ⊗ θ ) (id ⊗ G )((id ⊗ ι)(WH )) = rH ((id ⊗ (ω ⊗ θ ) ◦ G )(WG )) = rH ((id ⊗ ω)(WG )(id ⊗ θ )(WG )). Next we show rH preserves the involutions. Let π ∈ Irr(G). Take ω ∈ L(G)fin ∗ such that ω is equal to zero on B(Hρ ) if ρ = π . Since G ◦ ι = (ι ⊗ ι) ◦ H , (id ⊗ ι)(WH ) is a unitary representation of the discrete quantum group (L(G), ). Set V = (id ⊗ ι)(WH ). ˆ π ) = V∗ . Then By Lemma 2.4, we have (id ⊗ S)(V π  ∗ ∗ rH ((id ⊗ ω)(WG )) = (id ⊗ ω ◦ ι)(WH ) = (id ⊗ ω ◦ ι)(WH∗ ) ˆ π) = (id ⊗ ω)(V∗π ) = (id ⊗ ω ◦ S)(V   ∗ ˆ = rH ((id ⊗ ω ◦ S)((W G )π )) = rH (id ⊗ ω)((WG )π )   = rH ((id ⊗ ω)(WG ))∗ . By taking summations on ω, we see that the above equality holds for any ω ∈ L(G)fin ∗ . Hence rH is a unital ∗-homomorphism. Let µ ∈ L(H)fin ˜ ∈ L(G)fin ∗ . Then there exists µ ∗ such that µ = µ˜ ◦ ι. Then we have rH ((id ⊗ µ)(W ˜ G )) = (id ⊗ µ)(WH ). Hence rH is surjective. Finally we show that rH is a restriction map. Let ω ∈ L(G)fin ∗ . Then   δH rH ((id ⊗ ω)(WG )) = δH ((id ⊗ ω ◦ ι)(WH )) = (id ⊗ id ⊗ ω ◦ ι)((WH )13 (WH )23 )   = (rH ⊗ rH ) (id ⊗ id ⊗ ω ◦ ι)((WG )13 (WG )23 ) = (rH ⊗ rH ) ◦ δG ((id ⊗ ω)(WG )). t by definition Hence {H, rH } is an algebraic quantum subgroup of G. It is clear that ι = rH of rH . 

On heredity of co-amenability to quantum subgroups, we have the following lemma. Lemma 2.11. Let G be a compact quantum group. Then the following statements are equivalent:

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(1) G is co-amenable. (2) Any quantum subgroup of G is co-amenable. (3) There exists a co-amenable quantum subgroup of G. Proof. The implication (2) to (3) is trivial. Suppose (3) holds. Let {H, rH } be a co-amenable quantum subgroup. Then εG = εH ◦ rH is bounded, and (1) holds. We have to show that (1) implies (2). Let {H, rH } be a quantum subgroup of G. By the previous t : L(H) → L(G) preserving the coproducts. lemma, there exists an embedding map rH t ∈ L(H)∗ is also a left ∗ Let m ∈ L(G) be a left invariant mean. Then the state m ◦ rH invariant mean on L(H). Hence the discrete quantum group (L(H), ) is amenable, and H is co-amenable.  Let {H, rH } be a quantum subgroup of G. We define a map γH : C(G)→C(H)⊗ C(G) by γH (x) = (rH ⊗ id)(δG (x)) for x ∈ C(G). Then γH is a left action of H on C(G), that is, γH satisfies (id ⊗ γH ) ◦ γH = (δH ⊗ id) ◦ γH . We set the fixed point algebra C(H \ G) = {x ∈ C(G) | γH (x) = 1 ⊗ x}. Since the action γH preserves the Haar state of G, it extends to an action on L ∞ (G). We denote the fixed point algebra by L ∞ (H \ G) as well. Then L ∞ (H\G) is the range of the conditional expectation E H = (h H ⊗id)◦γH . t : L(H)→ We often identify L(H) with the subalgebra of L(G) via the inclusion map rH L(G) defined in Lemma 2.10. By the identification, we have WH = (rH ⊗ id)(WG ). Similarly, we can regard R(H) as a subalgebra of R(G) and we then have (id ⊗ rH )(VG ) = VH . Then the left action of H on L ∞ (G) is given by γH (x) = WH∗ (1 ⊗ x)WH for x ∈ L ∞ (G). Lemma 2.12. In the above setting, one has L(H) ∩ L ∞ (G) = L ∞ (H \ G), L ∞ (H \ G) ∩ L(G) = L(H). Proof. By definition of rH , x ∈ L(H) ∩ L ∞ (G) if and only if WH∗ (1 ⊗ x)WH = 1 ⊗ x. It is equivalent to x ∈ L ∞ (H \ G). Hence L(H) ∩ L ∞ (G) = L ∞ (H \ G). By Lemma 2.6, we obtain the second equality.  Lemma 2.13. Let {H, rH } and {K, rK } be quantum subgroups of G. Assume that L ∞ (H\ G) = L ∞ (K \ G). Then there exists a ∗-isomorphism θ : C(H) → C(K) such that rK = θ ◦ rH on C(G). t (L(H)) = r t (L(K)). Applying Lemma 2.10 Proof. By the previous lemma, we have rH K t to the left group algebra rH (L(H)), we have a ∗-isomorphism θ : C(H) → C(K) such t )(W )) = (id ⊗ r t )(W ). This implies (θ ◦ r ⊗ id)(W ) = that (θ ⊗ id)((id ⊗ rH H K H G K (rK ⊗ id)(WK ), and rK = θ ◦ rH on C(G). 

3. Right Coideals of Quotient Type Definition 3.1. Let B ⊂ L ∞ (G) be a right coideal. We say that (1) B is of quotient type if there exists a quantum subgroup {H, rH } of G such that B = L ∞ (H \ G), (2) B has the expectation property if there exists a faithful normal conditional expectation E B : L ∞ (G) → B such that h ◦ E B = h, (3) B has the coaction symmetry if β(B) ⊂ R(G) ⊗ B. Typical examples of right coideals are given by taking quotients. In fact, they have the expectation property and the coaction symmetry as follows.

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Lemma 3.2. A right coideal of quotient type has the expectation property and the coaction symmetry. Proof. Let {H, rH } be a quantum subgroup of G. Set B = L ∞ (H \ G). It is easy to see that the conditional expectation E H preserves the Haar state h. Next we verify the coaction symmetry. Let x ∈ C(H \ G). Then we have (id ⊗ γH )(β(x)) = (id ⊗ (rH ⊗ id) ◦ δ)(β(x)) = (id ⊗ (rH ⊗ id) ◦ δ)(VG∗ (1 ⊗ x)VG ) = (id ⊗ rH ⊗ id)((VG )∗13 (VG )∗12 (1 ⊗ δ(x))(VG )12 (VG )13 ) = (VG )∗13 (VH )∗12 (1 ⊗ 1 ⊗ x)(VH )12 (VG )13 = (VG )∗13 (1 ⊗ 1 ⊗ x)(VG )13 = β(x)13 . Hence β(x) ∈ R(G) ⊗ L ∞ (H \ G). By weak continuity of β, β(L ∞ (H \ G)) ⊂ R(G) ⊗ L ∞ (H \ G).  In fact assuming the co-amenability G, we prove the converse statement of the previous lemma in Theorem 3.18. Let B ⊂ L ∞ (G) be a right coideal. We denote by L 2 (B) the norm closure of the space B 1ˆ h . Assume that B has the expectation property, that is, there exists a conditional expectation E B : L ∞ (G) → B such that E B preserves the Haar state h. Define the Jones projection e B : L 2 (G) → L 2 (B) by e B (x 1ˆ h ) = E B (x)1ˆ h for x ∈ L ∞ (G). Let B ⊂ L ∞ (G) ⊂ L ∞ (G) ∨ {e B }

=: B1 be the basic extension. The main properties of e B are as follows (see [13, Lemma 3.2] and [21, p. 312]). Lemma 3.3. With the above settings, one has (1) (2) (3) (4)

e B xe B = E B (x)e B for x ∈ L ∞ (G), B = L ∞ (G) ∩ {e B } , J e B = e B J , ∆ith e B = e B ∆ith for all t ∈ R, B1 = J B J .

Set
B = B ∩ L(G). Then
B is a left coideal of L(G) as is shown in Lemma 2.6. The following lemma is proved in [11, Theorem 4.6], which treats the Kac algebra case. The proof can be adapted to the quantum group case. Lemma 3.4. With the above setting, one has (1) e B ∈
B, (2) δ(E B (x)) = (E B ⊗ id)(δ(x)) for all x ∈ L ∞ (G), (3)
B ∩ L ∞ (G) = B. Proof. (1) Since B is a right coideal, we see that VG (L 2 (B)⊗L 2 (G)) ⊂ L 2 (B)⊗L 2 (G). Hence VG (e B ⊗ 1) = (e B ⊗ 1)VG (e B ⊗ 1). Since (J ⊗ Jˆ)VG (J ⊗ Jˆ) = VG∗ and B. J e B J = e B , VG (e B ⊗ 1) = (e B ⊗ 1)VG . Hence e B ∈ B ∩ L(G) =


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(2) Let x ∈ L ∞ (G). Since e B ∈ L(G) and VG ∈ R(G) ⊗ L ∞ (G), we have

(e B ⊗ 1)δ(x)(e B ⊗ 1) = (e B ⊗ 1)VG (x ⊗ 1)VG∗ (e B ⊗ 1) VG (e B ⊗ 1)(x ⊗ 1)(e B ⊗ 1)VG∗ = VG (E B (x)e B ⊗ 1)VG∗ VG (E B (x) ⊗ 1)VG∗ (e B ⊗ 1) δ(E B (x))(e B ⊗ 1). In particular, we have δ(E B (x))(1ˆ h ⊗ 1ˆ h ) = (E B ⊗ id)(δ(x))(1ˆ h ⊗ 1ˆ h ). Since 1ˆ h ⊗ 1ˆ h is a separating vector for L ∞ (G) ⊗ L ∞ (G), we have δ(E B (x)) = (E B ⊗ id)(δ(x)). (3) It follows from Lemma 2.6 or the direct argument as follows. It is clear that B⊂
B ∩ L ∞ (G). Since e B ∈
B and B = {e B } ∩ L ∞ (G), B =
B ∩ L ∞ (G). 
Next we consider a relation between B and B1 . Let α be the right G-action α on B(L 2 (G)) defined in §2.4. Since e B ⊗ 1 commutes with VG , B1 is globally invariant under the right action α. Let X be a globally invariant subspace in B1 . The set of the fixed point elements of X under α is denoted by X G . Note that X G = X ∩ L(G) holds. (E B ⊗ id)(δ(x))(e B ⊗ 1) = = = =

Lemma 3.5. The following equalities hold: ˆ B ) = eB . (1) R(e (2) (e B ⊗ 1)WG∗ (1 ⊗ e B ) = WG∗ (e B ⊗ e B ). w (3)
B = J B G J = C + J (L ∞ (G)e B L ∞ (G))G J . 1

ˆ B ) = J e∗ J = e B . Proof. (1) Since e∗B = e B and J e B = e B J , we have R(e B ∞ (2) Take any x, y ∈ L (G). Then we have (e B ⊗ 1)WG∗ (1 ⊗ e B )(x 1ˆ h ⊗ y 1ˆ h ) = (e B ⊗ 1)WG∗ (x 1ˆ h ⊗ E B (y)1ˆ h ) = (e B ⊗ 1)(δ(E B (y))x 1ˆ h ⊗ 1ˆ h ) = δ(E B (y))(E B (x)1ˆ h ⊗ 1ˆ h ) = W ∗ (e B ⊗ e B )(x 1ˆ h ⊗ y 1ˆ h ). G

Hence the desired equality holds. (3) Since B1 = J B J and J L(G)J = L(G), we have J
B J = J (B ∩ L(G))J = J B J ∩ J L(G)J = B1 ∩ L(G) = B1G . By Lemma 3.4, the ∗-subalgebra L ∞ (G) + L ∞ (G)e B L ∞ (G) is weakly dense in B1 . w Hence we have B1G = C + (L ∞ (G)e B L ∞ (G))G .  Lemma 3.6. One has (A(G)e B A(G))G = {(id ⊗ ω)((e B )) | ω ∈ L(G)fin ∗ }. Proof. Let π, σ ∈ Irr(G), i, j ∈ Iπ and k,  ∈ Iρ . Then E α (vπi, j e B vρ∗k, ) = (id ⊗ h)(α(vπi, j e B vρ∗k, ))   = vπi,m e B vρ∗k,n h(vπm, j vρ∗n, ) m∈Iπ n∈Iρ

=

 

δπ,ρ δm,n δ j, Dπ−1 Fπ j, j vπi,m e B vρ∗k,n

m∈Iπ n∈Iρ

= Dπ−1 Fπ j, j δπ,ρ δ j,

 m∈Iπ

vπi,m e B vπ∗k,m .

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R. Tomatsu

Hence (A(G)e B A(G))G = span



  vπi,m e B vπ∗k,m  i, k ∈ Iπ , π ∈ Irr(G) .

m∈Iπ

Using the description of WG in (2.10), we have  (e B )(1 ⊗ 1π ) = vπi, j e B vπ∗k, j ⊗ f πi,k . i, j,k∈Iπ

Hence (A(G)e B A(G))G = {(id ⊗ ω)((e B )) | ω ∈ L(G)fin ∗ }.  w

Lemma 3.7. One has B1G = (L ∞ (G)e B L ∞ (G))G . Proof. By Lemma 3.5, it suffices to show that the unit of B1 is equal to that of w (L ∞ (G)e B L ∞ (G))G . Let p ∈ B1G be the unit of the von Neumann subalgebra generated by (L ∞ (G)e B L ∞ (G))G . Set q = 1 − p. Then by the previous lemma, we have ˆ B ) = e B by Lemma 3.5, we have (e B )(q ⊗ 1) = 0. Since R(e op ˆ ˆ B )) = (1 ⊗ R(q))( ˆ ˆ ˆ R(e Rˆ ⊗ R)( (e B )) (1 ⊗ R(q))(e B ) = (1 ⊗ R(q))( op ˆ ˆ = ( R ⊗ R)( (e B )(1 ⊗ q)) = 0.

ˆ The equality (e B ) = WG (e B ⊗ 1)WG∗ yields (1 ⊗ R(q))W G (e B ⊗ 1) = 0, and ˆ (E B ⊗ id)(WG∗ (1 ⊗ R(q))W ) = 0. Since E is faithful, we have q = 0.  B G Lemma 3.8. One has w w B1G = {(id ⊗ ω)((e B )) | ω ∈ L(G)∗ } ,
B = {(ω ⊗ id)((e B )) | ω ∈ L(G)∗ } .

Proof. The left equality follows from Lemma 3.6 and the previous lemma. Take ω ∈ L(G)∗ . By (2.17) and Lemma 3.5, we have ˆ Rˆ ⊗ R)((e ˆ J (id ⊗ ω)((e B ))J = (id ⊗ ω ◦ R)(( B ))) = (ω ◦ Rˆ ⊗ id)((e B )) Hence the right equality holds since
B = J B1G J .



To construct a left invariant weight on
B, we make use of theory of spatial derivatives and operator valued weights which have been introduced in [5, 8] and [9]. We use [14] as

∞

our reference and freely use the notations there. Let E −1 B : B → L (G) be the operator ∞ valued weights associated with E B : L (G) → B. It is characterized by the following equality on spatial derivatives: dω ◦ E −1 dω

B = , dω dω ◦ E B

(3.1)

where ω and ω are faithful normal semifinite weights on L ∞ (G) and B, respectively.

The equality E −1 B (e B ) = 1 holds [13, Lemma 3.1]. We define the ∗-subalgebra B0 of B by
B0 = J (A(G)e B A(G))G J = {(ω ⊗ id)((e B )) | ω ∈ L(G)fin ∗ }.

(3.2)

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287

−1
Lemma 3.9. The operator valued weight E −1 B is finite on B0 . In particular, E B is semifinite on
B. ∗ ∞

Proof. Set a unitary WG
= (J ⊗ J )(WG )21 (J ⊗ J ) ∈ L(G) ⊗ L (G) . Then for any x ∈ L(G), we have ∗ ∗ WG
= (J ⊗ J )(WG )21 (1 ⊗ J x J )(WG )21 (J ⊗ J )
(1 ⊗ x)WG

ˆ ∗ ))(WG )∗21 (J ⊗ J ) = (J ⊗ J )(WG )21 (1 ⊗ R(x ˆ ∗ ))21 (J ⊗ J ) = (J ⊗ J )( R(x ˆ ˆ = ( Rˆ ⊗ R)(( R(x)) 21 ) = (x).

−1 ∞

Now take a positive functional ω ∈ L(G)fin ∗ . Then by L (G) -linearity of E B , we have −1 ∗ E −1
))
(1 ⊗ e B )WG B ((ω ⊗ id)((e B ))) = E B ((ω ⊗ id)(WG −1 ∗ = (ω ⊗ id)(WG
)
(1 ⊗ E B (e B ))WG

= ω(1) < ∞. Hence

E −1 B

is finite on
B0 . h



(L ∞ (G) )

∗ ∞

Define a state ∈ ∗ by h (x) = h(J x J ) for x ∈ L (G) . Set a faithful −1



normal semifinite weight ϕ = h ◦ E B on B . We denote by ϕ
B the restriction of ϕ on
B.

∞

Then following push-down lemma for E −1 B : B → L (G) is proved in a similar way to [11, Prop. 2.2].

Lemma 3.10. For any x ∈ n ϕ , e B x = e B E −1 B (e B x) holds. Lemma 3.11. One has (A(G)e B L ∞ (G))G = (A(G)e B A(G))G . Proof. It suffices to show (L ∞ (G)π e B L ∞ (G))G ⊂ L ∞ (G)π e B L ∞ (G)∗π , where L ∞ (G)π is the linear span of {vπi, j }i, j∈Iπ . Take any x ∈ L ∞ (G). Then  E α (vπi, j e B x) = vπi,k e B (id ⊗ h)((1 ⊗ vπk, j )δ(x)) ∈ L ∞ (G)π e B L ∞ (G)∗π , k∈Iπ

because (id ⊗ h)((1 ⊗ vπk, j )δ(x)) ∈ L ∞ (G)∗π for any x ∈ L ∞ (G).



Lemma 3.12. The weight ϕ
and ω ∈ L(G)∗ , B is left invariant, that is, for any x ∈ m ϕ
B one has (ω ⊗ ϕ
B )((x)) = ω(1)ϕ
B (x). fin Proof. Take θ ∈ L(G)∗ and set x = (θ ⊗ id)((e B )) ∈
B0 . By using (x) = θ (1). Then for any ω ∈ L(G) Lemma 3.9, we have E −1 ∗, B −1

∗ (ω ⊗ ϕ

)
(1 ⊗ x)WG B )((x)) = (ω ⊗ h ◦ E B )(WG −1

∗ = (ω ⊗ h )(WG
)
(1 ⊗ E B (x))WG = ω(1)θ (1) = ω(1)ϕ
B (x).

Hence the left invariance holds on
B0 .

the proof of

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R. Tomatsu

Take a bounded sequence {u n }n∈N in
B0 which strongly converges to 1. Let y ∈ n ϕ
B . Then u n y ∈ J A(G)J e B n ϕ . By Lemma 3.10, we see that e B n ϕ ⊂ e B L ∞ (G) , and u n y ∈ J (A(G)e B L ∞ (G))J for each n ∈ N. Since J u n y J ∈ J
B J = B1G , u n y ∈ ∞ G B0 . J (A(G)e B L (G)) J . By Lemma 3.11, we see that u n y ∈ J (A(G)e B A(G))G J =
Hence for any positive ω ∈ L(G)∗ , we have ∗ ∗ ∗ ∗ (ω ⊗ ϕ
B )((y u n u n y)) = ω(1)ϕ
B (y u n u n y).

(3.3)

Taking the limit as n → ∞ in (3.3), we have ∗ ∗ ∗ (ω ⊗ ϕ
B )((y y)) ≤ lim (ω ⊗ ϕ
B )((y u n u n y)) n→∞

∗ ∗ = lim ω(1)ϕ
B (y u n u n y) n→∞

∗ = ω(1)ϕ
B (y y) < ∞. ∗ In particular, the map x ∈
B → (ω ⊗ ϕ
B )((y x y)) is a normal functional. Hence again taking the limit as n → ∞ in (3.3), we have ∗ ∗ (ω ⊗ ϕ
B )((y y)) = ω(1)ϕ
B (y y).

. Therefore ϕ
B is left invariant on m ϕ
B



We summarize our arguments as follows. Theorem 3.13. Let B ⊂ L ∞ (G) be a right coideal. If B has the expectation property, then the left coideal
B has a left invariant faithful normal semifinite weight for the left action of L(G). Next we study a right coideal of G endowed with the coaction symmetry. Lemma 3.14. Let B ⊂ L ∞ (G) be a right coideal. Then B has the coaction symmetry if and only if (
B) ⊂
B⊗
B. Proof. We know (
B) ⊂ L(G) ⊗
B. Let x ∈ B and y ∈
B. Then we have (y)(x ⊗ 1) = WG (y ⊗ 1)WG∗ (x ⊗ 1) = (1 ⊗ UG )(VG )21 (1 ⊗ UG )(y ⊗ 1)(1 ⊗ UG )(VG )∗21 (1 ⊗ UG )(x ⊗ 1) = (1 ⊗ UG )(VG )21 (y ⊗ 1)β(x)21 (VG )∗21 (1 ⊗ UG ) and similarly (x ⊗ 1)(y) = (1 ⊗ UG )(VG )21 β(x)21 (y ⊗ 1)(VG )∗21 (1 ⊗ UG ). Hence (
B) ⊂
B⊗
B if and only if β(B) ⊂ R(G) ⊗ (
B ∩ L ∞ (G)) = R(G) ⊗ B.



Lemma 3.15. If a right coideal B ⊂ L ∞ (G) has the expectation property and the ˆ
coaction symmetry, then R( B) =
B. Proof. By Lemma 3.5 and Lemma 3.8, we have w ˆ
R( B) = J
B J = B1G = {(id ⊗ ω)((e B )) | ω ∈ L(G)∗ } .

Since B has the coaction symmetry, (e B ) ∈
B⊗
B by the previous lemma. Hence ˆ
ˆ
R( B) ⊂
B, and R( B) =
B. 

Right Coideals of Quotient and Classification of Poisson Boundaries

289

ˆ
Set a weight ψ
B = ϕ
B ◦ R on B. Using (2.17), we see that ψ
B is right invariant. Therefore (
B, ) is a quantum group in the sense of [16]. Clearly the restriction εˆ |
B is a normal counit on
B. Hence (
B, ) is a discrete quantum group. The counit εˆ |
is given B by cutting elements at the Jones projection e B as follows. B. In particular, the Lemma 3.16. The equality xe B = εˆ (x)e B = e B x holds for all x ∈
Jones projection e B is a minimal central projection of
B. Proof. By Lemma 3.5, (e B ⊗ 1)WG∗ (1 ⊗ e B ) = WG∗ (e B ⊗ e B ). Then we have (e B )(1 ⊗ e B ) = WG (e B ⊗ 1)WG∗ (1 ⊗ e B ) = e B ⊗ e B . Taking the adjoint of the above equality, we have (e B )(1 ⊗ e B ) = e B ⊗ e B = (1 ⊗ e B )(e B ). Take any ω ∈ L(G)∗ and set x = (ω ⊗ id)((e B )). Then we have xe B = (ω ⊗ id)((e B ))e B = (ω ⊗ id)((e B )(1 ⊗ e B )) = (ω ⊗ id)(e B ⊗ e B ) = ω(e B )e B = εˆ (x)e B . B by Similarly we obtain e B x = εˆ (x)e B . Then the desired equality holds for all x ∈
Lemma 3.8.  We summarize our arguments as follows. Theorem 3.17. Let B ⊂ L ∞ (G) be a right coideal. If B has the expectation property and the coaction symmetry, then the pair (
B, ) is a discrete quantum group. Under assumption on co-amenability, we obtain the following characterization of right coideals of quotient type. Theorem 3.18. Let G be a co-amenable compact quantum group and B ⊂ L ∞ (G) a right coideal. Then B is of quotient type if and only if B has the expectation property and the coaction symmetry. Proof. We have already proved the “only if” part in Lemma 3.2. So, it suffices to show the “if” part. By the previous theorem, (
B, ) is a discrete quantum group. Let H = (C(H), δH ) be a compact quantum group such that the discrete quantum groups (L(H), H ) and (
B, ) are isomorphic. We identify
B with L(H). By Lemma 2.10, H is represented as an algebraic quantum subgroup of G, that is, there exists the restriction t :
map rH : A(G) → A(H) such that rH B → L(G) is a given inclusion. Moreover by Lemma 2.8, {H, rH } is in fact a quantum subgroup of G. By Lemma 2.12 and Lemma 3.4, we have B =
B ∩ L ∞ (G) = L(H) ∩ L ∞ (G) = L ∞ (H \ G).  4. Application to Classification of Poisson Boundaries In the rest of this paper, we determine the Poisson boundary for a co-amenable compact quantum group with the commutative fusion rules. We also compute the Poisson boundary for a q-deformed classical compact Lie group.

290

R. Tomatsu

4.1. Poisson boundaries. We briefly recall the notion of the Poisson boundary for a discrete quantum group. We refer to [10, 12] and [17] for definitions of terminology. Let φπ ∈ B(Hπ )∗ be the right G-invariant state. Define a transition operator Pπ on R(G) by Pπ (x) = (id ⊗ φπ )( R (x)) for x ∈ R(G). For a probability measure µ on Irr(G), we set a non-commutative Markov operator  µ(π )Pπ . Pµ = π ∈Irr(G)

Then for a generating measure µ, we define an operator system
Pµ ) = {x ∈ R(G) | Pµ (x) = x}. H ∞ (G, It has the von Neumann algebra structure defined by
Pµ ), x · y = lim Pµn (x y) for x, y ∈ H ∞ (G, n→∞

where the limit is taken in the strong topology [10, Theorem 3.6]. The von Neumann
Pµ ) is called the (non-commutative) Poisson boundary of {R(G), Pµ }. algebra H ∞ (G, Assuming that the fusion algebra of G is commutative, we know that the Poisson boundary does not depend on the generating measure [12, Prop. 1.1], that is, we have
Pµ ) = {x ∈ R(G) | Pπ (x) = x for all π ∈ Irr(G)}. H ∞ (G,
We are interested only in such a compact quantum group, so we write simply H ∞ (G) ∞ ∞
for H (G, Pµ ). The Poisson integral
: L (G) → R(G) is defined by
= (id ⊗ h) ◦ β,

(4.1)


[10, Lemma 3.8]. It is G-G-equivariant
in the following which maps L ∞ (G) into H ∞ (G) sense: α ◦
= (
⊗ id) ◦ δ,  R ◦
= (id ⊗
) ◦ β. Now we consider how the inverse map of
could be constructed. That would be written as a similar form to (4.1), that is, α and some state ω on R(G) would take the place of β and h, respectively. Then we would consider the map R(G)  x →
(ω ⊗ id)(α(x)) ∈ L ∞ (G). The inverse of
(if it exists) should be G-G-equivariant as
is. Here, we have to realize what property of h derives the bi-equivariance of

in the proof of [10, Lemma 3.8]. While the G-equivariance follows by definition of the left action, the G-equivariance follows by right invariance of h. Hence the state ω has to satisfy the left invariance for the coproduct  R . From now we assume the amenability of (R(G),  R ). Let m ∈ R(G)∗ be a left invariant mean. Although m is non-normal in general, we can consider a unital completely positive map m ⊗ id : R(G) ⊗ M → M for any von Neumann algebra M. Indeed for any x ∈ R(G) ⊗ M, we define an element (m ⊗ id)(x) ∈ M = (M∗ )∗ by ω((m ⊗ id)(x)) = m((id ⊗ ω)(x)) for ω ∈ M∗ . Let M and N be von Neumann algebras and T : M → N a normal completely bounded map. Then (m ⊗ id N ) ◦ (id ⊗ T ) = T ◦ (m ⊗ id M ) holds. Here we need to assume the normalcy of T . In particular, we have (m ⊗ id)((1 ⊗ a)x(1 ⊗ b)) = a(m ⊗ id)(x)b for all x ∈ R(G) ⊗ M and a, b ∈ M. Now define a unital completely positive map  : R(G) → L ∞ (G) by (x) = (m ⊗ id)(α(x)) for x ∈ R(G).

Right Coideals of Quotient and Classification of Poisson Boundaries

291


in the next lemma. It turns out that  is normal on the Poisson boundary H ∞ (G) Moreover by Theorem 4.8, we will see that  is actually equal to
∗ defined in [12]. This means that  does not depend on the choice of an left invariant mean. Lemma 4.1. The following statements hold:
In particular,  is a faithful normal map. (1) h ◦  = εˆ on H ∞ (G).
(2)  is G-G-equivariant, that is, δ ◦  = ( ⊗ id) ◦ α, β ◦  = (id ⊗ ) ◦  R .
is Proof. (1) Since the fusion algebra of G is commutative, the action α on H ∞ (G) ∞
ergodic [10, Coro. 3.7]. Hence E α (x) = εˆ (x)1 for all x ∈ H (G). Then h((x)) = m(E α (x)) = m(ˆε(x)1) = εˆ (x). Since h and εˆ are faithful normal states [10, Th. 3.6], we conclude that  is a faithful normal map. (2) First we show the G-equivariance of . Let x ∈ R(G). Then we have δ((x)) = δ((m ⊗ id)(α(x))) = (m ⊗ id ⊗ id)((id ⊗ δ)(α(x))) = (m ⊗ id ⊗ id)((α ⊗ id)(α(x))) = ((m ⊗ id) ◦ α ⊗ id)(α(x)) = ( ⊗ id)(α(x)).
Next we show the G-equivariance of . The left invariance of m yields (id ⊗ m)( R (y)) = m(y)1 for all y ∈ R(G). Take any x ∈ R(G) and then (id ⊗ )( R (x)) = (id ⊗ m ⊗ id)((id ⊗ α)( R (x))) = (id ⊗ m ⊗ id)((VG )23 (VG )∗12 (1 ⊗ x ⊗ 1)(VG )12 (VG )∗23 ) = (id ⊗ m ⊗ id)((VG )∗13 (VG )∗12 (VG )23 (1 ⊗ x ⊗ 1)(VG )∗23 (VG )12 (VG )13 ) = VG∗ (id ⊗ m ⊗ id)(( R ⊗ id)(α(x)))VG = VG∗ ((id ⊗ m) ◦  R ⊗ id)(α(x))VG = VG∗ (1 ⊗ (m ⊗ id)(α(x)))VG = β((x)). 

((x)) = x. In particular,  is a faithful normal Lemma 4.2. For any x ∈ H ∞ (G), ∗-homomorphism.
By Lemma 4.1, we have Proof. Let x ∈ H ∞ (G).
((x)) = (id ⊗ h)(β((x))) = (id ⊗ h)((id ⊗ )( R (x))) = (id ⊗ εˆ )( R (x)) = x.
we have The multiplicativity of  is shown as follows. For any x ∈ H ∞ (G), x ∗ · x =
((x))∗ ·
((x)) ≤
((x)∗ (x)) ≤
((x ∗ · x)) = x ∗ · x. Hence we obtain
((x)∗ (x)) =
((x ∗ · x)), namely, (x)∗ (x) = (x ∗ · x). This implies that  is a ∗-homomorphism.  We determine the multiplicative domain of
.

292

R. Tomatsu


Lemma 4.3. Set B = (H ∞ (G)). (1) B is a right coideal with the expectation property and the coaction symmetry.
(2) The conditional expectation onto B is G-equivariant. (3) B coincides with the multiplicative domain of
. Moreover, the Poisson integral
is an isomorphism.
: B → H ∞ (G) Proof. (1) By the previous lemma, B ⊂ L ∞ (G) is a von Neumann subalgebra. Since
 is G-G-equivariant by Lemma 4.1, B is globally invariant under the actions α and β. Hence B is a right coideal with the coaction symmetry. Set E B =  ◦
. Then by the previous lemma, we see that E B is a faithful normal conditional expectation onto B. For x ∈ L ∞ (G), we have h(E B (x)) = h((
(x))) = εˆ (
(x)) = h(x). Hence B has the expectation property.

(2) Since
and  are G-equivariant, E B =
◦  is also G-equivariant. (3) Let D ⊂ L ∞ (G) be the multiplicative domain of
. It is easy to see that B ⊂ D. We show the converse inclusion. Let x ∈ D. Then by definition, we have
(x ∗ x) =
(x)∗ ·
(x) and
(x x ∗ ) =
(x) ·
(x)∗ . Applying  to both sides of the equalities, we have E B (x ∗ x) = E B (x)∗ E B (x) and E B (x x ∗ ) = E B (x)E B (x)∗ . This immediately
is a faithful normal ∗-homoyields that x ∈ B, and B = D. Hence
: B → H ∞ (G)

morphism. The surjectivity of
follows from
(B) =
((H ∞ (G))) = H ∞ (G).  Lemma 4.4. Let H be a quantum subgroup of G. Then H is of Kac type if and only
if the expectation E H : L ∞ (G) → L ∞ (H \ G) is G-equivariant, that is, it satisfies (id ⊗ E H ) ◦ β = β ◦ E H . Proof. For any x ∈ C(G), we have (id ⊗ E H )(β(x)) = (id ⊗ h H ◦ rH ⊗ id)((id ⊗ δ)(VG∗ (1 ⊗ x)VG )) = (id ⊗ h H ◦ rH ⊗ id)((VG )∗13 (VG )∗12 (1 ⊗ δ(x))(VG )12 (VG )13 ) = VG∗ (id ⊗ h H ⊗ id)((VH )∗12 (1 ⊗ (rH ⊗ id)(δ(x)))(VH )12 )VG . This is equal to β(E H (x)) = VG∗ (1 ⊗ E H (x))VG if and only if (id⊗h H ⊗ id)((VH )∗12 (1⊗(rH ⊗id)(δ(x)))(VH )12 ) = 1⊗(h H ⊗id)((rH ⊗id)(δ(x))). Multiplying 1 ⊗ 1 ⊗ y, y ∈ C(G) from the right, we have (id ⊗ h H ⊗ id)((VH )∗12 (1 ⊗ (rH ⊗ id)(δ(x)(1 ⊗ y)))(VH )12 ) = 1 ⊗ (h H ⊗ id)((rH ⊗ id)(δ(x)(1 ⊗ y))). Since the subspace δ(C(G))(C ⊗ C(G)) is dense in C(G) ⊗ C(G), we have for all x ∈ C(G) ⊗ C(G), (id ⊗ h H ⊗ id)((VH )∗12 (1 ⊗ (rH ⊗ id)(x))(VH )12 ) = 1 ⊗ (h H ⊗ id)((rH ⊗ id)(x))). More precisely, we have (id ⊗ h H )(VH∗ (1 ⊗ z)VH ) = h H (z)1 for all z ∈ C(H).
This shows that E H is G-equivariant if and only if the image of the Poisson integral of H consists of scalars. By [10, Cor. 3.9], it is equivalent to that H is of Kac type. 

Right Coideals of Quotient and Classification of Poisson Boundaries

293

Lemma 4.5. Let K be a quantum subgroup of Kac type in G. Then
◦ E K =
.
For any x ∈ L ∞ (G), we have Proof. By Lemma 4.4, E K is G-equivariant.
(E K (x)) = (id ⊗ h)(β(E K (x))) = (id ⊗ h)((id ⊗ E K )(β(x))) = (id ⊗ h)(β(x)) =
(x).  We prepare the notion of maximality for a quantum subgroup of Kac type as follows. Definition 4.6. Let G be a compact quantum group. We say that a quantum subgroup H of Kac type is maximal if for any quantum subgroup K of Kac type, we have L ∞ (H\G) ⊂ L ∞ (K \ G). Lemma 4.7. Let G be a compact quantum group. If there exists a maximal quantum subgroup of Kac type, it is unique in the following sense. Let {H, rH } and {K, rK } be maximal quantum subgroups of Kac type. Then there exists a ∗-isomorphism θ : C(H) → C(K) such that rK = θ ◦ rH . Proof. By definition, we have L ∞ (H \ G) = L ∞ (K \ G). Then by Lemma 2.13, we have the desired ∗-isomorphism.  Theorem 4.8. Let G be a co-amenable compact quantum group. Assume that its fusion algebra is commutative. Then the following statements hold: (1) There exists a unique maximal quantum subgroup of Kac type H.
is an isomorphism. (2) The Poisson integral
: L ∞ (H \ G) → H ∞ (G) Proof. (1) By using Theorem 3.18, Lemma 4.3 and Lemma 4.4, there exists a quantum subgroup H of Kac type such that B = L ∞ (H \ G). We show the maximality of H. Let K be another quantum subgroup of Kac type. By Lemma 4.5,
◦ E K =
. Since E B =  ◦
, we have E B ◦ E K = E B . Let eK be the Jones projection associated with E K . Then e B eK = e B , and hence eK e B = e B . It yields B ⊂ L ∞ (K \ G). Hence H is maximal. (2) It follows from Lemma 4.3 (3).  Remark 4.9. We have realized the existence of the maximal quantum subgroup of Kac type by studying the Poisson integral. However, the notion has been already introduced in [20, App. A], where it is called the canonical Kac quotient. The canonical Kac quotient is the function algebra on the maximal quantum subgroup of Kac type. The existence is proved for an arbitrary compact quantum group [20, Prop. A.1]. 4.2. q-deformed classical compact Lie groups. We determine the maximal quantum subgroup of Kac type in a q-deformed classical compact Lie group (0 < q < 1). In order to do so, we freely make use of terminology and results in [15] such as construction of quantum universal enveloping algebras, quantized function algebras and so on. n . We Let g be a complex classical simple Lie algebra with the simple roots {αi }i=1 denote by G the corresponding classical compact Lie group [15, Def. 1.2.4 of Chap. 3]. Let Uq (g) be the quantum universal enveloping algebra associated with g [15, Def. 7.1.1 of Chap. 2]. We equip Uq (g) with the Hopf ∗-algebra structure as in [15, Prop. 1.2.3 of Chap. 3]. We denote by W the Weyl group and by si the simple reflection with respect to αi .

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For a dominant integral weight λ, we denote by L q (λ) the irreducible Uq (g)-module with the highest weight λ. Fix an orthonormal basis {ξµr }r of each weight space L q (λ)µ with the weight µ. We define Cξλr ,ξ s ∈ Uq (g)∗ by µ

ν

Cξλµr ,ξνs (x) = (xξνs , ξµr ) for x ∈ Uq (g). Then we set A(Gq ) = span{Cξλr ,ξ s }λ,µ,ν,r,s . On the involution, we have µ

ν

0λ (Cξλµr ,ξνs )∗ = q (µ−ν,) Cξ−w r ,ξ s , −µ

(4.2)

−ν

n αi is the Weyl vector and w0 ∈ W is the longest element. Let where  = (1/2) i=1 κ and {τt }t∈R be the antipode and the scaling automorphism group of Gq introduced in §2.1. From (2.5), κ 2 = τ−i and (4.2), we obtain τt (Cξλµr ,ξνs ) = q i(µ−ν,2)t Cξλµr ,ξνs . We denote by C(Gq ) the C ∗ -completion of A(Gq ) with respect to the universal norm. Set qi = q (αi ,αi )/2 , i = 1, . . . , n. Let Uqi (su(2)) be the quantized universal enveloping algebra of the Lie algebra su(2) with the deformation parameter qi . The canonical embedding Uqi (su(2)) → Uq (g) induces the restriction map ri : A(Gq ) → A(SUqi (2)) [15, Subsect. 6.1 of Chap. 3]. Since C(Gq ) is a universal C ∗ -algebra, ri extends to the ∗-homomorphism C(Gq ) → C(SUqi (2)). Take a canonical infinite dimensional irreducible representation πi : C(SUqi (2)) → B(2 ) defined in [15, Prop. 4.1.1 of Chap. 3]. We note that the counit εi of C(SUqi (2)) factors through Im(πi ), that is, there exists a ∗-homomorphism ηi : Im(πi ) → C such that ηi ◦ πi = εi . Indeed, let p : B(2 ) → Q be the canonical surjection onto the Calkin algebra Q. Let S ∈ B(2 ) be the unilateral shift. By definition of πi , we see that p(Im(πi )) is a commutative C ∗ -algebra generated by a unitary p(S). Take the character ω : p(Im(πi )) → C defined by ω( p(S)) = 1. Then the character ηi := ω ◦ p : Im(πi ) → C has the desired property. Let {T, rT } be the maximal torus subgroup of Gq , where rT : C(Gq ) → C(T) is the restriction map. Any one dimensional ∗-homomorphism of C(Gq ) is given by πt := χt ◦rT , where χt : C(T) → C is the evaluation at t ∈ T. Then for an element w ∈ W \{e} with a reduced decomposition w = si1 · · · sik and t ∈ T, we define (k)

πw,t = (πi1 ◦ ri1 ⊗ . . . ⊗ πik ◦ rik ⊗ πt ) ◦ δGq , (k) (k) where δG : C(Gq ) → C(Gq )⊗(k+1) is recursively defined by δG = (δGq ⊗ id⊗(k−1) ) ◦ q q (k−1)

(1)

δGq and δGq = δGq . Then πw,t is an irreducible ∗-homomorphism of C(Gq ) and does not depend on the choice of the reduced decomposition of w up to equivalence [15, Th. 6.2.1 of Chap. 3]. Every irreducible ∗-homomorphism of C(Gq ) is equivalent to some πw,t [15, Th. 6.2.7 of Chap. 3]. Lemma 4.10. Let Gq be the q-deformation of a classical compact Lie group G. Then its maximal quantum subgroup of Kac type is the maximal torus T. Proof. Let {H, rH } be a quantum subgroup of Kac type. We will show that H is a subgroup of T. Assume that C(H) is not commutative. Then there exists an irreducible ∗-homomorphism ρ : C(H) → B(Hρ ) with dim Hρ ≥ 2. Set π = ρ ◦ rH . We

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may assume π = πw,t for some element w ∈ W \ {e} with a reduced expression w = si1 · · · sik and t ∈ T. Then we consider the irreducible ∗-homomorphism (id ⊗ ηi2 ⊗ · · · ⊗ ηik ⊗ id) ◦ π = (πi1 ◦ ri1 ⊗ πt ) ◦ δGq = πsi1 ,t , which factors through C(H). Hence we may assume π = ρ ◦ rH = πs j ,t for some 1 ≤ j ≤ n and t ∈ T. Since the scaling automorphism of C(H) is trivial, we have rH ◦ τt = rH for all t ∈ R by Lemma 2.9. Let ω j be the fundamental weight for α j . Setting λ = µ = ω j and ν = ω j − α j , ω ωj ωj we have τt (Cξωj ,ξω −α ) = q 2it j C ξω ,ξω −α . Hence rH (C ξω ,ξω −α ) = 0. In particular, ω

j

j

j

j

j

j

j

j

j

ω

πs j ,t (Cξωj ,ξω −α ) = 0. This is, however, in contradiction with πs j ,t (Cξωj ,ξω −α ) = 0 j j j j j j which is shown by direct computation. Therefore C(H) is a commutative C ∗ -algebra, that is, H is an ordinary compact group. For x ∈ H, we write χxH for the character ∗-homomorphism. For t ∈ T, we also use χtT as well. By the above arguments, for any x ∈ H there exists an element t (x) ∈ T such that χxH ◦ rH = χtT(x) ◦ rT . This shows that the map t : H → T is continuous. For any x, y ∈ H, we have χtT(x y) ◦ rT = χxHy ◦ rH = (χxH ⊗ χ yH ) ◦ δH ◦ rH = (χxH ⊗ χ yH ) ◦ (rH ⊗ rH ) ◦ δGq = (χtT(x) ⊗ χtT(y) ) ◦ (rT ⊗ rT ) ◦ δGq = (χtT(x) ⊗ χtT(y) ) ◦ δT ◦ rT = χtT(x)t (y) ◦ rT . Hence the map t : H → T is a group homomorphism. Next we show that t is injective. Suppose that x ∈ H satisfies χxH ◦ rH = χeT ◦ rT . Since it is equal to the counit ε of C(Gq ) by Lemma 2.9, we have χxH ◦ rH = χeT ◦ rT = ε = χeH ◦ rH . Hence x = e, and the map t is injective. Since the left action of x ∈ H is given by (χxH ◦ rH ⊗ id) ◦ δGq = (χtT(x) ◦ rT ⊗ id) ◦ δGq , we have L ∞ (T \ Gq ) ⊂ L ∞ (H \ Gq ). Therefore T is maximal.  By Theorem 4.8 and the previous lemma, we obtain the following corollary. Corollary 4.11. Let Gq be the q-deformation of a classical compact Lie group G. Then q ) is an isomorphism. the Poisson integral
: L ∞ (T \ Gq ) → H ∞ (G Acknowledgements. The author is grateful to Yasuyuki Kawahigashi and Masaki Izumi for encouragement and various useful comments. He would also like to thank Stefaan Vaes for informing him of canonical Kac quotients introduced in [20].

References 1. Baaj, S., Skandalis, G.: Unitaires multiplicatifs et dualité pour les produits croisé de C ∗ -algèbres. Ann. Sci. École Norm. Sup. (4) 26(4), 425–488 (1993) 2. Bédos, E., Conti, R., Tuset, L.: On amenability and co-amenability of algebraic quantum groups and their corepresentations. Canad. J. Math. 57(1), 17–60 (2005) 3. Bédos, E., Murphy, G., Tuset, L.: Co-amenability of compact quantum groups. J. Geom. Phys. 40(2), 130– 153 (2001) 4. Bédos, E., Murphy, G., Tuset, L.: Amenability and coamenability of algebraic quantum groups. Int. J. Math. Math. Sci. 31(10), 577–601 (2002) 5. Connes, A.: On the spatial theory of von Neumann algebras. J. Funct. Anal. 35(2), 153–164 (1980)

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6. Effros, E., Ruan, Z.-J.: Discrete quantum groups. I. The Haar measure. Internat. J. Math. 5(5), 681– 723 (1994) 7. Enock, M., Schwartz, J. M.: Kac algebras and duality of locally compact groups. Berlin: Springer-Verlag, 1992, pp. x+257 8. Haagerup, U.: Operator-valued weights in von Neumann algebras. I. J. Funct. Anal. 32(2), 175–206 (1979) 9. Haagerup, U.: Operator-valued weights in von Neumann algebras. II. J. Funct. Anal. 33(3), 339– 361 (1979) 10. Izumi, M.: Non-commutative Poisson boundaries and compact quantum group actions. Adv. Math. 169(1), 1–57 (2002) 11. Izumi, M., Longo, R., Popa, S.: A Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras. J. Funct. Anal. 155(1), 25–63 (1998) 12. Izumi, M., Neshveyev, S., Tuset, L.: Poisson boundary of the dual of SUq (n). Commun. Math. Phys. 262(2), 505–531 (2006) 13. Kosaki, H.: Extension of Jones’ theory on index to arbitrary factors. J. Funct. Anal. 66(1), 123–140 (1986) 14. Kosaki, H.: Type III factors and index theory. Lecture Notes Series 43, Seoul: Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, 1998, pp. ii+96 15. Korogodski, L.I., Soibelman, Y.S.: Algebras of functions on quantum groups. Part I. Mathematical Surveys and Monographs 56, Providence, RI: Amer. Math. Soc. 1998, pp. x+150 16. Kustermans, J., Vaes, S.: Locally compact quantum groups in the von Neumann algebraic setting. Math. Scand. 92(1), 68–92 (2003) 17. Neshveyev, S., Tuset, L.: The Martin boundary of a discrete quantum group. J. Reine Angew. Math. 568, 23–70 (2004) 18. Podle´s, P.: Quantum spheres. Lett. Math. Phys. 14(3), 193–202 (1987) 19. Podle´s, P.: Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2) and SO(3) groups. Commun. Math. Phys. 170(1), 1–20 (1995) 20. Sołtan, P.M.: Quantum Bohr compactification. Illinois J. Math. 49(4), 1245–1270 (2005) 21. Takesaki, M.: Conditional expectations in von Neumann algebras. J. Funct. Anal. 9, 306–321 (1972) 22. Tomatsu, R.: Amenable discrete quantum groups. J. Math. Soc. Japan 58(4), 949–964 (2006) 23. Tomatsu, R.: Compact quantum ergodic systems. http://arxiv.org/list/math.OA/0412012, 2004 24. Vaes, S.: The unitary implementation of a locally compact quantum group action. J. Funct. Anal. 180(2), 426–480 (2001) 25. Vaes, S., Vander Vennet, N.: Identification of the Poisson and Martin boundaries of orthogonal discrete quantum groups. http://arxiv.org/list/math.OA/0605489, 2006 26. Vaes, S., Vergnioux, R.: The boundary of universal discrete quantum groups, exactness and factoriality. http://arxiv.org/list/math.OA/0509706, 2005 27. Van Daele, A.: Discrete quantum groups. J. Algebra 180(2), 431–444 (1996) 28. Woronowicz, S.L.: Compact matrix pseudogroups. Commun. Math. Phys. 111(4), 613–665 (1987) 29. Woronowicz, S.L.: Tannaka-Kre˘ın duality for compact matrix pseudogroups. Twisted SU(N ) Groups. Invent. Math. 93(1), 35–76 (1988) 30. Woronowicz, S.L.: Compact quantum groups. In: Symétries quantiques (Les Houches, 1995), Amsterdam: North-Holland, 1998, pp. 845–884 31. Yamagami, S.: On unitary representation theories of compact quantum groups. Commun. Math. Phys. 167(3), 509–529 (1995) Communicated by Y. Kawahigashi

Commun. Math. Phys. 275, 297–329 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0294-3

Communications in

Mathematical Physics

Generalized Farey Trees, Transfer Operators and Phase Transitions Mirko Degli Esposti1 , Stefano Isola2 , Andreas Knauf3 1 Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato, 5, I-40127 Bologna, Italy.

E-mail: [email protected]

2 Dipartimento di Matematica e Informatica, Università di Camerino, via Madonna delle Carceri,

62032 Camerino, Italy. E-mail: [email protected]

3 Mathematisches Institut der Universität Erlangen-Nürnberg, Bismarckstr. 1 1/2, D-91054 Erlangen,

Germany. E-mail: [email protected] Received: 6 June 2006 / Accepted: 21 February 2007 Published online: 2 August 2007 – © Springer-Verlag 2007

Abstract: We consider a family of Markov maps on the unit interval, interpolating between the tent map and the Farey map. The latter is not uniformly expanding. Each map being composed of two fractional linear transformations, the family generalizes many particular properties which for the case of the Farey map have been successfully exploited in number theory. We analyze the dynamics through the spectral properties of the generalized transfer operator. Application of the thermodynamic formalism to the family reveals first and second order phase transitions and unusual properties like positivity of the interaction function.

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . 2. A One-Parameter Family of 1D Maps . . . . . . . . . 3. Dynamical Binary Trees and Coding . . . . . . . . . 3.1 A generalization of the Farey tree . . . . . . . . . 3.2 Traces and dynamical partition functions . . . . . 4. Polymer Model Analysis of the Markov Family . . . . 4.1 A 1D spin chain model . . . . . . . . . . . . . . 4.2 Positivity of the interaction and exponential sums 5. Thermodynamic Formalism . . . . . . . . . . . . . . 5.1 Partition function and transfer operator . . . . . . 5.2 Phase transitions . . . . . . . . . . . . . . . . . . 5.3 Fourier analysis of the transfer operator . . . . . 5.4 Twisted zeta functions . . . . . . . . . . . . . . .

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298 299 301 301 308 310 312 313 317 317 319 323 326

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1. Introduction The piecewise real-analytic map
F1 : [0, 1] → [0, 1], x →

x 1−x , 1−x x ,

0 ≤ x ≤ 1/2 1/2 < x ≤ 1

is known as the Farey map. From the ergodic point of view it is of interest, since it is expanding everywhere but at the fixed point x = 0 where it has slope one. This makes this map a simple model of the physical phenomenon of intermittency [PM]. From the point of view of number theory, F1 encodes the continued fraction algorithm as well as the Riemann zeta function. In particular it has an induced version given by the celebrated Gauss map [Ma]. In addition, several models of statistical mechanics have been considered in recent years in connection to Farey fractions and continued fractions [Kn2, Kn3, KO, FKO, LR]. Altogether this motivates a precise analysis of the dynamics induced by F1 . An effective tool in this analysis is provided by the transfer operator associated to the map (see [Ba] for an overview). For the map F1 the spectrum of the transfer operator when acting on a suitable space of analytic functions has been studied in [Rug, Is and Pr]1 and turns out to have a continuous component, in particular no spectral gap. As a consequence, the Farey map is ergodic w.r.t. the a.c. infinite measure dxx . Another interesting ergodic invariant measure for F1 is the Minkowski probability measure d? (for the question mark function [Mi]) which is singular w.r.t. Lebesgue measure and turns out to be the measure of maximal entropy for F1 . On the other hand, the Minkowski question mark function conjugates F1 with the much simpler tent map
2x, 0 ≤ x ≤ 1/2 F0 : [0, 1] → [0, 1], x → 2(1 − x), 1/2 < x ≤ 1 which is ergodic w.r.t. Lebesgue measure and is – from the point of view of number theory – connected with the base 2 expansion. In [GI] it was first noticed that F0 and F1 can be viewed as instances of a one-parameter family of interval expanding maps Fr which are continuous, real-analytic and accessible, being composed of two Möbius transformations. The present article now follows this line of research and connects it with aspects coming from thermodynamic formalism. In particular, one aim is to extend the theory developed in [Kn2] and [Kn3] to a more general class of trees encoding the dynamics of the maps Fr . It turns out that all these models share several interesting algebraic relations (see below). Besides that, the value of this approach consists in the fact that some delicate properties of the arithmetic case r = 1 can be approached by first studying the corresponding properties for r < 1 by means of spectral techniques, taking advantage of the existence of a spectral gap for the transfer operator, and then taking the limit r → 1. The paper is organized as follows. In Sect. 2 we introduce the one-parameter family of interval maps Fr and recall some of its basic properties obtained in [GI]. We then describe some relevant features of the Ruelle transfer operators associated to this family including its action on a suitable invariant Hilbert space of analytic functions. In Sect. 3 we describe how using the maps Fr we can construct a one-parameter family of binary 1 See also the recent preprint: “Spectral analysis of transfer operators associated to Farey fractions” by C. Bonanno, S. Graffi and S. Isola, where a conjecture put forward in [Is] has been proved.

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trees T (r ) interpolating between the dyadic tree and the Farey tree. As the parameter r ranges in the unit interval, the natural coding associated to real numbers in [0, 1] by each tree induces a Hölder continuous conjugation between the maps Fr and F0 , thus generalizing the Minkowski question mark function (Lemma 3.1). Moreover we show (Proposition 3.6) that T (r ) can be also constructed by a local generating rule which generalizes the mediant operation used to generate Farey fractions. A closed expression for the trace of the iterates of the transfer operator in terms of weighted sums over the leaves of the tree is obtained in Theorem 3.9, along with some consequences both on dynamical zeta functions and Fredholm determinants for the family Fr . The trees T (r ) turn out to be fundamental objects in establishing a direct connection between the transfer operators mentioned above and the partition functions of a class of spin chains introduced in Sect. 4. Using a polymer expansion technique we prove in Theorem 4.5 that when the parameter r is positive the interaction associated to the corresponding spin chain model is of ferromagnetic type. In the last section we establish explicit formulae for the iterates of the transfer operator (Propositions 5.1, 5.9 and 5.11) which are used to evaluate the canonical (and grand canonical) partition functions as well as some twisted sum with possible number theoretic significance. From this analysis it turns out that in the canonical setting our models undergo a phase transition whose features are described in Theorem 5.5. 2. A One-Parameter Family of 1 D Maps For r ∈ (−∞, 2), let Fr denote the piecewise real-analytic map Fr of the interval [0, 1] defined as
Fr,0 (x), if 0 ≤ x ≤ 1/2 , (2.1) Fr (x) := Fr,1 (x), if 1/2 < x ≤ 1 where Fr,0 (x) =

(2 − r )x (2 − r )(1 − x) and Fr,1 (x) = Fr,0 (1 − x) = · 1 − rx 1 − r + rx

Although some of the results obtained below hold true for a wider range of r values, in this paper we shall mainly restrict to r ∈ [0, 1] where this is a Markov family interpolating between the Farey map (r = 1) and the linear expanding tent map (r = 0). For r ∈ [0, 2) we have2   inf |Fr (x)| = Fr,0 (0) = −Fr,1 (1) = 2 − r =: ρ.

x∈[0,1]

(2.2)

This means that for 0 ≤ r < 1 the map Fr is uniformly expanding, i.e. |Fr | ≥ ρ > 1, thus providing an example of analytic Markov map [Ma]. On the contrary, for r = 1 one has |F  (x)| > 1 for x > 0 but F  (0) = 1. For r > 1 the origin becomes attractive and there is new repelling fixed point x ∗ with Fr (x ∗ ) = (2 − r )−1 . The left and right inverse branches of Fr are given by     1 1 ρ−ρx x = − , (2.3) r,0 : [0, 1] → 0, 21 , r,0 (x) = ρ + rx 2 2 ρ + rx 2 Notational warning: The parameters r and ρ, although simply related, are both useful to express the various quantities introduced in the sequel, and we therefore keep using both of them. Nevertheless, as long as the quantities dealt with below are well defined for all r < 2 we shall suppress this parameter.

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r=0

r=0.5

r=1

r=1.5

Fig. 2.1. The one-dimensional family Fr

and r,1 : [0, 1] →

1

 1 1 2 , 1 , r,1 (x) = 1 − r,0 (x) = 2 + 2



 ρ−ρx , ρ + rx

(2.4)

respectively. Given a complex weight s ∈ C we let Ps,r denote the Ruelle transfer operator associated to the map Fr . It acts on a function f : [0, 1] → C as   Ps,r f (x) : = |r,0 (x)|s f (r,0 (x)) + |r,1 (x)|s f (r,1 (x))      x ρs x f = + f 1− . (ρ + r x)2s ρ + rx ρ + rx

(2.5)

For s = 1 and r ∈ [0, 1] the fixed point of the Perron-Frobenius operator Pr ≡ P1,r corresponds to the density of an Fr -invariant absolutely continuous measure νr (d x). A direct calculation shows that Cr dx (2.6) νr (d x) = (1 − r + r x)  1 with Cr a positive constant. For r < 1 the choice Cr = r/log 1−r yields νr ([0, 1]) = 1. Note that Cr diverges as r 1, showing that for r = 1 the measure νr is not normalisable. We refer to [GI] for several results on the measure νr and a spectral analysis of the family of operators Pr when acting on a suitable Hilbert space of analytic functions (see [Rug, Is and Pr] for related results on the intermittent case r = 1 and s ∈ C). Let H (D1 ) be the Banach space of functions analytic in the disk 

D1 := x ∈ C | Re( x1 ) > 21 = {x ∈ C | |x − 1| < 1} and continuous on ∂ D1 . It is easy to check that H (D1 ) is invariant under the action of the transfer operator Ps,r for all r ∈ (−∞, 2) and s ∈ C. It is moreover a standard result (see, e.g., [Ma]) that as long as the map Fr is uniformly expanding Ps,r : H (D1 ) → H (D1 ) is of the trace-class for all s ∈ C. In particular, for r ∈ [0, 1) we have [GI]  2s−1  (x )|s |r,i ρ 1−s ρs 2 i trace (Ps,r ) = , (2.7) √  (x ) = ρ − 1 + √ 1 − r,i 1 + 4ρ 1 + 1 + 4ρ i i=0,1

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where xi (i = 0, 1) is the unique fixed point of r,i in the unit interval. Later on, using the family of binary trees associated to the maps Fr , we shall generalize this formula to all iterates of Ps,r (see Theorem 3.9). We end this section with some remarks on the general structure of the eigenfunctions of the operator Ps,r . The matrix     1−ρ ρ r −1 2−r ∈ P S L(2, R) (2.8) = Sr := 2−ρ ρ−1 r 1−r with Sr2 = Id and det Sr = −1 acts on C as the Möbius transformation (r − 1)x + 2 − r . Sˆr (x) := rx + 1 − r

(2.9)

Since r,i ◦ Sˆr = r,1−i , i = 0, 1, we have the implication Ps,r f = λ f, λ = 0

=⇒

Is,r f = f

for the involution (Is,r f )(x) :=

 1 ˆr (x) . f S (r x + 1 − r )2s

Therefore the eigenvalue equation is equivalent to the generalized three-term functional equation

1    ( ρ + ρ − 1)x + ρ 1 x −s +1 + ρ λ f (x) = f f ρ ((2 − ρ)x + ρ − 1)2s (2 − ρ)x + ρ − 1 which for r = 1 reduces to the Lewis-Zagier functional equation arising in the theory of modular forms [LeZa]. 3. Dynamical Binary Trees and Coding 3.1. A generalization of the Farey tree. For each r ∈ (−∞, 2) we construct a ‘dynamical’ binary tree T (r ) from the sequences −k Tn (r ) := ∪n+1 k=0 Fr (0).

(3.1)

The ordered elements of Tn can be written as ratios of irreducible polynomials over Z, with positive coefficients when written as functions of the variable ρ = 2 − r . For example,     0 1 0 1 1 T0 = , , T1 = , , , 1 1 1 2 1   0 1 1 1+ρ 1 , T2 = , , , , 1 2+ρ 2 2+ρ 1   1 1 + ρ 1 + 2ρ 1 + ρ + ρ 2 T3 \ T2 = , , , 2 + ρ + ρ 2 2 + 3ρ 2 + 3ρ 2 + ρ + ρ 2 and so on. The rooted tree T is now constructed as follows:

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1 2

1 2

3 4

1 4

3 8

1 8

1 16

3 16

5 16

9 16

11 16

13 16

2 5

1 4

7 8

5 8

7 16

2 3

1 3

15 16

1 5

2 7

3 8

3 4

3 5

3 7

4 7

5 8

5 7

4 5

Fig. 3.2. The dyadic tree (left) and Farey tree (right) 1

0

1

2

Fig. 3.3. Plot of the set Tn \ Tn−1 for n = 10 and 0 ≤ r ≤ 2

• for n ≥ 1 the n th row has 2n−1 vertices and coincides with Tn \ Tn−1 . The vertex 1/2 ∈ T1 is considered as the root; • edges connect each element in Tn \ Tn−1 to a pair of elements in Tn+1 \ Tn in such a way that no edges overlap. The edge pointing to the left is labelled by 0, the other edge by 1. We say that an element

p q

∈ T has rank n, written rank( qp ) = n, if it belongs to 

Tn \ Tn−1 . We say moreover that two elements qp , qp in Tn are neighbours whenever they are neighbours when Tn is considered as an ordered subset of [0, 1]. It is an easy task to  realize that for each pair of neighbours qp , qp in Tn one of them has rank n and the other has rank n − k for some 1 ≤ k < n. Since Fr is expansive for all r ∈ [0, 1] the vertex set of T (r ) is dense in [0, 1]. In particular (see Figs. 3.2 and 3.3), • T (1) is the Farey tree whose vertex-set is Q ∩ (0, 1). • T (0) is the dyadic tree whose vertex-set is the set of all dyadic rationals of the form k/2m . • For r > 1 the vertex set of T (r ) is not dense anymore and for r 2 accumulates to the single point 1/2.

Generalized Farey Trees, Transfer Operators and Phase Transitions

303

The following result is a straightforward consequence of the construction given above and developed in what follows (see Lemma 3.5). Lemma 3.1. For all r ∈ [0, 1] we have that • to every x ∈ [0, 1] there corresponds a unique sequence φr (x) ∈ {0, 1}N which represents the sequence of edges of an infinite path {xk }k≥1 on T (r ) with x1 = 1/2 and xk → x (if x is a vertex of T (r ) we extend the sequence leading to that vertex either with 01∞ or 10∞ ); • the map φr ◦ Fr ◦ φr−1 acts as the left-shift on  := {0, 1}N /ι, where ι(σ ) := σ with σ j := 1 − σ j ; • the homeomorphism h r := φ0−1 ◦ φr : [0, 1] → [0, 1] conjugates Fr to F0 , so that the measure dh r (x) is Fr -invariant and its entropy is equal to log 2. Remark 3.2. dh r (x) is the measure of maximal entropy for Fr , and for r = 0 is plainly singular w.r.t. Lebesgue (cfr. (2.6)).  −n then φ (x) = σ . Example 3.3. • If for r = 0 in binary notation x = 0. σ = ∞ 0 n=1 σn 2 • Instead, for r = 1, if in continued fraction notation x = [a1 , a2 , a3 , . . . ] then φ1 (x) = 0a1 1a2 0a3 · · · . In this case the conjugating function h 1 (x) = φ0−1 ◦ φ1 (x) is but the Minkowski question mark function [Mi], defined as ?(x) := (−1)k−1 2−(a1 +···+ak −1) k≥1

= 0.00 . . . 0 11 . . . 1 00 . . . 0 · · · .       a1 −1

a2

a3

For k ∈ N0 the point 1/2 has exactly 2k preimages w.r.t. the iterated map Frk . We enumerate them using the group Gk := (Z/2Z)k , the group elements σ = (σ1 , . . . , σk ) ∈ Gk being ordered lexicographically. Considered as a function of r the preimage (σ ) indexed by σ is a quotient qpkk (σ ) of polynomials. These are inductively defined by setting p0 := 1, q0 := 2, pk+1 (0, σ ) := pk (σ ), pk+1 (1, σ ) := (2 − r )qk (σ ) + (r − 1) pk (σ ),

(3.2)

qk+1 (0, σ ) := (2 − r )qk (σ ) + r pk (σ ), qk+1 (1, σ ) := (2 − r )qk (σ ) + r pk (σ ).

(3.3)

and

Here we use the shortcut σ for the group element (1 − σ1 , . . . , 1 − σk ). The identities pk (σ ) + pk (σ ) = qk (σ ) = qk (σ ),

(σ ∈ Gk ),

(3.4)

follow immediately from (3.2) and (3.3). Note that in terms of the left and right inverse branches of Fr (cfr (2.3) and (2.4)) we can write (using the shorthand i ≡ r,i , i = 0, 1)     pk (σ ) pk (σ ) pk+1 (0, σ ) pk+1 (1, σ ) = 0 , = 1 . (3.5) qk+1 (0, σ ) qk (σ ) qk+1 (1, σ ) qk (σ ) The induction step for the following arithmetic characterization for r = 0 and r = 1 is provided by (3.5).

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Fig. 3.4. The conjugating function h r−1 for some values of r ∈ [0, 1]. For r = 1 we see the graph of the (inverse of the) Minkowski function ?

Lemma 3.4. Let σ ∈ Gk be of the form σ = (0, 0, . . . 0, 1, 1, . . . 1, 0, 0, . . . 0, · · · u, . . . , u )             a1 −1

a2

a3

an −1

with u = 0 for n odd and u = 1 otherwise, and some positive integers ai , 1 ≤ i ≤ n, such that an > 1 and a1 = k + 1 for

n = 1 and

n

ai = k + 2 for n > 1.

i=1

We have r =0

=⇒

r =1

=⇒

pk (σ ) = 0. σ 1, qk (σ )  pk (σ ) 1/(a1 + 1), n = 1, = [a1 , . . . , an ], n > 1. qk (σ )

The connection between this coding of the leaves and the one naturally induced by the dynamics can be understood as follows: let the group isomorphisms k : Gk → Gk be given by

k (t1 , t2 , . . . , tk ) := (t1 , t1 + t2 , t1 + t2 + t3 , . . . , t1 + t2 + · · · + tk ) (mod 2).

Generalized Farey Trees, Transfer Operators and Phase Transitions

305

Clearly

k−1 (s1 , s2 , . . . , sk ) = (s1 , s1 + s2 , s2 + s3 , . . . , sk−1 + sk ) (mod 2). Lemma 3.5. For all r ∈ (−∞, 2) and σ ∈ Gk ,   pk (σ ) =  −1 (σ ) 21 , k qk (σ )

(3.6)

with the iterated inverse branch (t1 ,...,tk ) := t1 ◦ (t2 ,...,tk ) . Proof. The relation is obviously true for k = 1 and 1 = IdG1 . Assume now that (3.6) holds for a given k ∈ N, then from the relations: −1

k+1 (1, σ¯ ) = (1, σ1 , σ1 + σ2 , . . . , σk−1 + σk ) = (1, k−1 (σ ))

and −1

k+1 (0, σ ) = (0, σ1 , σ1 + σ2 , . . . , σk−1 + σk ) = (0, k−1 (σ )),

it follows from (3.5) that pk+1 (0, σ ) = 0 ◦  −1 (σ ) k qk+1 (0, σ )

    1 1 =  −1 (0,σ ) , k+1 2 2

pk+1 (1, σ¯ ) = 1 ◦  −1 (σ ) k qk+1 (1, σ¯ )

    1 1 =  −1 (1,σ¯ ) . k+1 2 2

and

  Lets us now consider again the sequence Tn defined in (3.1). The following result generalizes for T (r ) the mediant operation which generates the Farey tree T (1) [GKP]. Proposition 3.6. Let r ∈ (−∞, 2). For each pair of neighbours rank( qp )

= n − k and

 rank( qp )

= n, its child

p  q 

given by

p + ρ k p p  := q  q + ρk q satisfies p p  p <  <  and rank q q q



p  q 



Moreover, it holds p  q − p q  = ρ n−k , (the roles of p, q, p  , q  are plainly reversed if

p q

<

p q ).

= n + 1.

p q

<

p q

in Tn , with

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M. Degli Esposti, S. Isola, A. Knauf 



Proof. Let us assume that qp < qp with rank( qp ) = n −k and rank( qp ) = n. The opposite case is similar. We have p pn−1 (σ  ) p pn−k−1 (σ ) ≡ ≡ , (σ ∈ Gn−k−1 ), and , (σ  ∈ Gn−1 ). q qn−k−1 (σ ) q qn−1 (σ  ) 

Moreover, since qp and qp are neighbours we have σ  = σ σ ∗ , where σ ∗ ∈ Gk is given by σ ∗ = (1, 0 . . . , 0).  Now, the element having rank n + 1 which appears in Tn+1 between qp and qp has the 

(σ )   ∗ form qpnn (σ  ) , where σ ∈ Gn is given by σ = σ σ 0. Therefore, using Lemma 3.5 we have the expressions   1 pn−k−1 (σ ) , =  −1 (σ ) n−k−1 qn−k−1 (σ ) 2   pn−1 (σ  ) 1 =  −1 (σ ) ◦  −1 (σ ∗ ) ,  n−k−1 k qn−1 (σ ) 2   pn (σ  ) 1 =  −1 (σ ) ◦  −1 (σ ∗ 0) .  n−k−1 k qn (σ ) 2

A direct computation yields

k−1 i   1 + 2ρ + i=2 ρ 1 a  −1 (σ ∗ ) = =:   k−1 k 2 b 2 + 3ρ + 2 i=2 ρ i

and, setting

:= 21 , we get k   1 + 2ρ + i=2 ρi 1 a  a + ρk a = .  −1 (σ ∗ 0) =: =  k k 2 b b + ρ k b 2 + 3ρ + 2 i=2 ρi

a b





Therefore, to complete the proof it suffices to verify that the three ratios ab , ab , ab verify a  a + ρk a = b b + ρ k b if and only if, for any choice of (t1 , . . . , tl ) ∈ Gl , the ratios      a p p  a a p := (t1 ,...,tl ) , , := (t1 ,...,tl ) := (t1 ,...,tl ) q b q b q  b satisfy p + ρ k p p  = q  q + ρk q as well. On the other hand, this property can be easily verified by induction using the expressions a a a a + ρ(b − a) = = 0 and 1 . b 2a + ρ(b − a) b 2a + ρ(b − a) −1 The first statement now follows by taking l = n − k − 1 and (t1 , . . . , tl ) = n−k−1 (σ ). The second follows in a similar way.  

Generalized Farey Trees, Transfer Operators and Phase Transitions

For r ∈ [0, 2) the matrices I j :=



1 − jρ jρ 2−ρ ρ

307

 ,

j = 0, 1,

(3.7)

are in GL(2, R). As such they act on C as Möbius transformations, and the action of (0) (1) the two operators forming the transfer operator Ps,r = Ps,r + Ps,r of Fr on a function f : [0, 1] → C is  s     ( j) (Ps,r f )(x) = j (x) f  j (x) , j = 0, 1, (3.8) with (1 − jρ)x + jρ  j (x) := Iˆj (x) = · (2 − ρ)x + ρ

(3.9)

Using the involution S ≡ Sr introduced in (2.8) we shall see that the elements of T (r ) can be represented by means of a subgroup of GL(2, R) with generators     1ρ 1 0 · (3.10) and R := S L S = L := I0 = 0ρ 2−ρ ρ Note that I1 = S R = L S. For example we have LL =



1 0 2 + ρ − ρ2 ρ2



(3.11) 

and L R =

1 ρ 2 − ρ 2ρ

 ,

so that 1 ˆ L(1) = , 2

LL(1) =

1 1+ρ ,  L R(1) = . 2+ρ 2+ρ

More generally, we have the following characterization of the leaves of T (r ) as matrix products which generalize what is known for the arithmetic case r = 1 [Kn3]. Proposition 3.7. For all r ∈ (−∞, 2) the element pk (σ )/qk (σ ) of T (r ) can be unik quely presented as the product X = L i=1 Mi , where Mi = (1 − σi )L + σi R. More precisely, pk (σ ) = Xˆ (1). qk (σ ) Since det L = det R = ρ, we have det X = ρ k+1 . Proof. By Lemma 3.5, (3.9), (3.10) and (3.11) the element pk (σ )/qk (σ ) can be written as     pk (σ ) =  −1 (σ ) 21 = Iˆσ1 ◦ Iˆσ1 +σ2 ◦ · · · ◦ Iˆσk−1 +σk 21 k qk (σ )   = ( Lˆ ◦ Sˆ σ1 ) ◦ ( Lˆ ◦ Sˆ σ1 ◦ S σ2 ) ◦ · · · ◦ ( Lˆ ◦ Sˆ σk−1 ◦ Sˆ σk ) 21     = Lˆ ◦ Mˆ σ1 ◦ · · · ◦ Mˆ σk−1 Sˆ σk 21 = Xˆ 21 ,   since Sˆ σk 21 = Sˆ σk Iˆσ k (1) = Mˆ σk (1).  

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3.2. Traces and dynamical partition functions. Let us now define two maps T j : T (r ) → R, j = 0, 1, as     p p := trace(X ), T1 := trace(X S), (3.12) T0 q q  where qp = Xˆ (1) is the presentation from Prop. 3.7. For r = 1 the numbers T0 qp and    T1 qp are given by p  + q  and p  + q  , respectively, with qp = qp ++qp . Also note that if

p q

occurs in T (r ) at level n, namely

p q

∈ Tn \ Tn−1 , then

det(X ) = −det(X S) = ρ n .   Lemma 3.8. T0 qp + T1 qp = r p + ρq.  Proof. T0

p q

 p q

+ T1

= tr(X (1l + S)) = tr

(3.13)

 x11 x12   r ρ  x21 x21

rρ

(0)

(1)

= r p + ρq.  

We already know that the operator Ps,r = Ps,r + Ps,r when acting H (D1 ) is of the trace class for all r ∈ [0, 1) and s ∈ C. The above construction provides a closed n , n ≥ 1. expression for the trace of Ps,r Theorem 3.9. For all r ∈ [0, 1), s ∈ C and n ≥ 1 we have n trace (Ps,r )

=

p q ∈Tn \Tn−1

j=0,1

⎛ 

ρ ns T j2 ( qp ) − (−1) j 4ρ n

⎝

⎞2s−1 T j ( qp ) +



2 T j2 ( qp ) − (−1) j 4ρ n

⎠

.

Proof. We have 2n terms n (σ ) (σ1 ,...,σn ) (σ1 ) (σn ) )= trace (Ps,r ) with Ps,r := Ps,r · · · Ps,r , trace (Ps,r σ ∈Gn

and to each of them we can associate the matrix product Iσ1 · · · Iσn according to (3.8) and (3.9). On the other hand, the commutation rules (3.11) yield I I . . . I I = L S L . . . L S R = L R . . . R. 1 0 0 1 k

k−1

k

Using this fact it is not difficult to realize that each  term where I1 appears an even number of times can be expressed in the form X = L i Mi and, moreover, to each such term there corresponds exactly another term where the number of occurrences of I1 is odd and which can be written as X S. (σ ) Finally, for all r ∈ [0, 1) and s ∈ C the generic term Ps,r is a composition operator of  s the form f (x) → |ψ (x)| f (ψ(x)), where ψ(x) = (σ ) (x) is holomorphic and strictly contractive in a disk containing the unit interval, with a unique fixed point x¯ ∈ [0, 1]. Standard arguments then yield for its trace the expression |ψ  (x)| ¯ s /(1 − ψ  (x)) ¯ (see, e.g., [Ma], Sect. 7.2.2). The thesis now follows by direct computation putting together the above along with (3.12) and (3.13).  

Generalized Farey Trees, Transfer Operators and Phase Transitions

309

We now define the signed operator P˜ s,r : H (D1 ) → H (D1 ) as (0) (1) P˜ s,r = Ps,r − Ps,r .

(3.14)

An immediate consequence of the proof of Theorem 3.9 is the following Corollary 3.10. For all r ∈ [0, 1), s ∈ C and n ≥ 1 we have n ) trace (P˜ s,r

=

p q ∈Tn \Tn−1

j=0,1

⎛ 

(−1) j ρ ns T j2 ( qp ) − (−1) j 4ρ n

⎝

⎞2s−1 T j ( qp ) +



2 T j2 ( qp ) − (−1) j 4ρ n

⎠

so that n n trace (Ps,r ) + trace (P˜ s,r )

=2

p q ∈Tn \Tn−1



⎛

ρ ns T02 ( qp ) − 4ρ n

⎝

⎞2s−1 T0 ( qp ) +



2 T02 ( qp ) − 4ρ n

⎠

Furthermore, let us define a dynamical partition function n (s) as   (F n ) (x)−s . n (s) := r

·

(3.15)

x=Frn (x)

Another consequence of the above is the following Corollary 3.11. n n ) − trace (P˜ s+1,r ) n (s) = trace (Ps,r 4s ρ ns =  2s ·  p p p j=0,1 T j ( ) + T j2 ( ) − (−1) j 4ρ n ∈ T \ T n n−1 q q q

n (σ ) Proof. For σ ∈ Gn set |σ | = i=1 σi . The trace of the operator (−1)|σ | Ps,r has the |σ |  s  expression (−1) |ψ (x)| /(1 − ψ (x)), ¯ where ψ(x) = (σ ) (x) and x is the unique solution of (σ ) (x) = x in [0, 1]. The first identity now follows from the equality |ψ  (x)|s (−1)|σ | |ψ  (x)|s+1 − = |ψ  (x)|s , 1 − ψ  (x) ¯ 1 − ψ  (x) ¯ and the second by direct calculation.   Remark 3.12. If X = L n then T0 = 1 + ρ n and T1 = 1 + ρ + ρ 2 + · · · ρ n−1 .

 Therefore, as r 1 we have T0 → 2 and T1 → n 2 + 4. In particular T02 − 4ρ n = n diverges (we recall that for ρ n − 1 → 0 and the corresponding term in the trace of Ps,r r = 1 the spectrum of Ps,1 contains the interval [0, 1], see [Rug, Is]). On the other hand, one easily sees that for each n ≥ 1 this is the only term which diverges as r 1. Unlike traces, the function n (s) is well defined for r = 1.

310

M. Degli Esposti, S. Isola, A. Knauf n ) and (s) to form the Fredholm determinant One can store the numbers trace (Ps,r n

⎛

⎞ zn n ⎠ det(1 − z Ps,r ) := exp ⎝− ) trace (Ps,r n

(3.16)

n≥1

and the dynamical zeta function ⎛ ⎞ zn n (s)⎠ ζr (z, s) := exp ⎝ n

(3.17)

n≥1

respectively. As a consequence of the above we have the following Corollary 3.13. For r ∈ [0, 1) and s ∈ C, ζr (z, s) =

det(1 − z P˜ s+1,r ) · det(1 − z Ps,r )

(3.18)

For r ∈ [0, 1) the above determinants are entire functions of both s and z and therefore, for all s ∈ C, ζr (z, s) is meromorphic in the whole complex z-plane and analytic in {z ∈ C : z −1 ∈ / sp(Ps,r )}. 4. Polymer Model Analysis of the Markov Family The Fourier transform of a function f : Gk → C is fˆ : Gk → C, fˆ(t) := 2−k f (σ )(−1)σ ·t . σ ∈Gk

We now calculate the polynomials pˆ k (t), qˆk (t) for t ∈ Gk , using the language of polymer models. As in [GuK] we call the group elements t ∈ Gk a polymer if t = (t1 , . . . , tk ) contains exactly one or two ones. So the set of polymers in Gk is Pk := Pke ∪ Pko with even resp. odd polymers Pke := { p,r := δ + δr ∈ Gk : 1 ≤  < r ≤ k}, Pko := { p := δ ∈ Gk : 1 ≤  ≤ k}.

(4.1)

Slightly diverging from [GuK], we defined their supports by supp( p,r ) := {i :  ≤ i ≤ r } and supp( p ) := {1, . . . , }. Two polymers are called incompatible if their supports have nontrivial intersection. Thus we can uniquely decompose every group element t ∈ Gk as a sum t = γ1 + · · · + γn of mutually incompatible polymers γi ∈ Pk . The activities of the polymers are defined as the rational functions     2 − r supp( p,r ) r 2 − r supp( p ) z( p,r ) := − , z( p ) := − . (4.2) 2−r 4−r 4−r

Generalized Farey Trees, Transfer Operators and Phase Transitions

311

Proposition 4.1. For the polymer decomposition t = γ1 + · · · + γn(t) of t ∈ Gk the Fourier coefficients equal  pˆ k (t) =

 qˆk (t) =

4−r 2

4−r 2

k n(t) z(γi ),

(4.3)

i=1

k

n(t) |t|

(1 + (−1) )

z(γi ).

(4.4)

i=1

Proof. For k = 0 the above formulae reduce to pˆ 0 = p0 = 1, qˆ0 = q0 = 2. The induction step from k to k + 1 proceeds by decomposing the group elements in the form (τ, t) ∈ G1 × Gk ∼ = Gk+1 . Then by (3.2) and (3.3),  pˆ k+1 (τ, t) = 21 pˆ k (t) + (−1)τ +|t| ((2 − r )qˆk (t) + (r − 1) pˆ k (t))  =

1 2

 =

1 2

4−r 2 4−r 2

k n(t)

  z(γi ) 1 + (−1)τ +|t| (1 + (−1)|t| )(2 − r ) + (r − 1)

i=1



k n(t) z(γi ) · i=1

|t|

r + 1+(−1) (4 − 2r ) , τ + |t| = 0 (mod 2) 2 . 1+(−1)|t| 2−r − (4 − 2r ) , τ + |t| = 1 (mod 2) 2

For all (τ, t) this coincides with formula (4.3). Similarly we get  qˆk (τ, t) = 21 1 + (−1)τ +|t| [(2 − r )qˆk (t) + r pˆ k (t)] =

1 2

    4 − r k n(t) 1 + (−1)|t| (4 − 2r ) , 1 + (−1)τ +|t| z(γi ) r + 2 2 i=1

coinciding with (4.4).   This formula for the Fourier coefficients implies a symmetry of the denominator function qk which, unlike (3.4) is not immediate from its definition. Corollary 4.2. For all r ∈ [0, 2), k ∈ N and σ = (σ1 , . . . , σk ) ∈ Gk , qk (σk , σk−1 , . . . , σ2 , σ1 ) = qk (σ1 , σ2 , . . . , σk−1 , σk ). Proof. This statement is equivalent to the one qˆk (tk , tk−1 , . . . , t2 , t1 ) = qˆk (t1 , t2 , . . . , tk−1 , tk )

(t ∈ Gk )

for the Fourier coefficients. Since anyhow qˆk (t) = 0 for odd |t|, in formula (4.4) for qˆk only activities z( pl,r ) of even polymers pl,r appear. Unlike the odd polymers, they have the symmetry z( pk−r +1,k−l+1 ) = z( pl,r ), which leads to the above statement.  

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r 0

7 6 5 4 3 2 1 100

200

300

r 0.2

7 6 5 4 3 2 1 400

100

500

200

300

400

500

400

500

400

500

r 0.6

r 0.4 7 6 5 4 3 2 1

7 6 5 4 3 2 1 100

200

300

400

500

100

200

r 0.8

300

r 1. 7 6 5 4 3 2 1

7 6 5 4 3 2 1 100

200

300

400

500

100

200

300

Fig. 4.5. The energies Q k = log qk for k = 10 and different values of r ∈ [0, 1]

4.1. A 1D spin chain model. In the same spirit as [Kn2] and [Kn3] we now interpret the sequences σ ∈ Gk as different configurations of a chain of k classical binary spins with energy function Q k := log qk : Gk → R.

(4.5)

The corresponding (canonical) partition function will be

Z nC (s) := 1 +

0≤k
exp (−s Q k (σ )) ≡



q −s .

(4.6)

p q ∈Tn (r )\{0}

Plots of the function Q k for different values of r are reported in Fig. 4.5. The canonical partition function can be expressed in a more standard way using the following Definition 4.3. For all k ∈ N we inductively define polynomials p ck (σ ), q ck (σ ) ∈ Z[ρ] (σ ∈ Gk ) by setting p c1 (0) := 0,

p c1 (1) := 1, q c1 (0) := 1, q c1 (1) := 2,

Generalized Farey Trees, Transfer Operators and Phase Transitions

313

and for σ ∈ Gk with


max{i|σi = 1}, σ ∈ Gk \{0}, r : Gk → N, r (σ ) := 0, σ =0
c τ =0 q k (σ ), q ck+1 (σ, τ ) := , ρ k−r (σ ) q ck (σ ) + ρ k−r (σ ) q ck (σ ), τ = 1
c τ =0 p k (σ ), . p ck+1 (σ, τ ) := ρ k−r (σ ) q ck (σ ) + ρ k−r (σ ) (q ck (σ ) − p ck (σ )), τ = 1 Example 4.4. p c2 (0, 1) = 1, p c2 (1, 1) = 1 + ρ, q c2 (0, 1) = q c2 (1, 1) = 2 + ρ. The relations between these new polynomials and the old ones (see (3.2)–(3.3)) are given by pk (σ ) = p ck+1 (σ, 1) and qk (σ ) = q ck+1 (σ, 1), which can be seen to be another way to state Proposition 3.6. Finally, using the new denominators just introduced, we can write  −s qnc (σ ) . Z nC (s) =

(4.7)

(4.8)

σ ∈Gn

4.2. Positivity of the interaction and exponential sums. The negative Fourier coefficients − Qˆ k (t) can then be interpreted as interaction coefficients in the sense of statistical mechanics. Theorem 4.5. The interaction is ferromagnetic for r ∈ [0, 1], that is − Qˆ k (t) ≥ 0,

(t ∈ Gk \{0}).

Proof. We introduce the notation of polymer models (see, e.g., Gallavotti, Martin-Löf and Miracle-Solé [GMM], Simon [Si] and Glimm and Jaffe [GJ]). In an abstract setting one starts with a finite set P of polymers. Two given polymers γ1 , γ2 ∈ P may or may not be incompatible. Incompatibility is assumed to be a reflexive and symmetric relation on P. Thus one may associate to a n–polymer X := (γ1 , . . . , γn ) ∈ P n an undirected graph G(X ) = (V (X ), E(X )) without loops with vertex set V (X ) := {1, . . . , n}, vertices i = j being connected by the edge {γi , γ j } ∈ E(X ) if γi and γ j are incompatible. Accordingly the n–polymer X is called connected if G(X ) is path-connected and disconnected if it has no edges (E(X ) = ∅). The corresponding subsets of P n are n resp. D n , with D 0 := P 0 := {∅} consisting of a single element. Moreover called C! !∞ !∞ ∞ n ∞ := n ∞ := n P := ∞ n=0 P with the subsets D n=0 D and C n=1 C . We write n |X | := n if X ∈ P . Statistical weights or activities z : P → C of the polymers are |X | multiplied to give the activities z X := i=1 z(γi ) of multi-polymers. A system of statistical mechanics is called polymer model if its partition function Z has the form Z=

X ∈D ∞

zX . |X |!

(4.9)

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Then the free energy is given by log(Z ) =

n(X ) zX, |X |! ∞

(4.10)

X ∈C

with n(X ) := n + (X ) − n − (X ), n ± (X ) being the number of subgraphs of G(X ) connecting the vertices of G(X ) with an even resp. odd number of edges. In the present context we index by the number k of spins and use the polymer set Pk from (4.1). Then the map Dk∞ → Gk , (γ1 , . . . , γn ) →

n

γi

i=1

is a set-theoretic isomorphism between the disconnected multi-polymers in {1, . . . , k} and the abelian group Gk . Similarly (using a superscript e for objects derived from the subset Pke ⊂ Pk of even polymers) the image of Dk∞,e → Gk , (γ1 , . . . , γn ) →

n

γi

i=1

is the subgroup of Gk whose elements have an even number of ones. In terms of this notation and with (4.4), Qˆ k (t) = 2−k log(qk (σ ))(−1)σ ·t σ ∈Gk

= 2−k

σ ∈Gk

⎛ log ⎝



⎞ qˆk (s)(−1)s·σ ⎠ · (−1)σ ·t

s∈Gk

⎤ ⎡    X z˜ σ ⎥ 4−r ⎢ σ ·t + 2−k = δt,0 · log(2) + k log log ⎣ ⎦ · (−1) 2 |X |! ∞,e σ ∈Gk

X ∈Dk

with the redefined single-polymer activities z˜ σ (γ ) := z σ (γ ) · (−1)σ ·γ ,

(γ ∈ Pk , σ ∈ Gk ).

(4.11)

By (4.10) and (4.11) we get    4−r ˆ Q k (t) − δt,0 · log(2) + k log 2 n(X ) z˜ σX · (−1)σ ·t = 2−k |X |! ∞,e σ ∈Gk X ∈C k

=



X =(γ1 ,...,γ|X | )∈Ck∞,e

=



∞,e X =(γ1 ,...,γ|X | )∈Ck |X | i=1 γi =t

|X | n(X ) X −k z ·2 (−1)σ ·(t+ i=1 γi ) |X |!

n(X ) X z , |X |!

σ ∈Gk

(4.12)

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 using the identity σ ∈Gk (−1)σ ·s = 2k δs,0 . As shown in [GuK] as a consequence of Thm. 4, for the graph G = (V, E),
0, G not connected sign(n(G)) = . −(−1)|V | , G connected So, noticing the negative signs of the activities in (4.2), all terms on the r.h.s. of (4.12) are nonpositive, proving the ferromagnetic property.   We now rewrite the Fourier coefficients qˆk (t) (t ∈ Gk ) of Proposition 4.1, using the previously defined map ψk : Gk → Gk , ψk (t1 , . . . , tk ) := (t1 , t1 + t2 , . . . , t1 + · · · + tk ) (mod 2). This is a group automorphism with inverse ψk−1 (s1 , . . . , sk ) = (s1 , s1 + s2 , s2 + s3 , . . . , sk−1 + sk ) (mod 2). Proposition 4.6. For parameter values r ∈ [0, 2), k ∈ N0 and t ∈ Gk ,  qˆk (t) = (1 + (−1)|t| )(−1)t,ψk (t) exp c0 k + c1 |ψk (t)| + c2 t, ψk (t)     2−r r with constants c0 (r ) := ln 4−r , c (r ) := ln (r ) := ln , c 1 2 2 4−r 4−r .

(4.13)

Proof. Our starting point is formula (4.4):  4 − r k n(t)  |t| qˆk (t) = 1 + (−1) z(γi ) 2

(t ∈ G k ),

(4.14)

i=1

with the activities z(γi ) defined in (4.2). • For r < 2 and even |t| we get sign(qˆk (t)) = (−1)n(t) , since then   2 − r supp(γi ) r z(γi ) = − <0 2−r 4−r (for odd |t| the Fourier coefficient qˆk (t) vanishes anyhow).  As (ψk (t))i = 1 iff i−1 =1 t is odd, ti (ψk (t))i = 1 for every second i with ti = 1. So n(t) = t, ψk (t)

(t ∈ Gk ).

• Returning to the assumption that |t| is even, we note that |ψk (t)| =

n(t) n(t) (supp(γi ) − 1) = −n(t) + supp(γi ). i=1

i=1

We now can write (4.14) in the form  qˆk (t) = 1 + (−1)|t| (−1)n(t) ⎛ ⎛ ⎞ ⎞       n(t) n(t) 4 − r r 2 − r ⎠. + +⎝ exp ⎝k ln ln supp(γi )⎠ ln 2 2−r 4−r i=1

Substituting (4.15), we obtain formula (4.13).  

i=1

(4.15)

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Corollary 4.7. For k ∈ N and t ∈ Gk we have qˆk (t) = 0 for |t| odd, and for |t| even, setting σi := (−1)(ψk (t))i , i = 1, . . . , k,

k−1  k |qˆk (t)| = ck exp c˜2 σi σi+1 + c˜1 (i)σi , i=1

with c˜2 = c˜1 (1) = c˜1 (k) = − 41 ln  (4−r )3 1 2 ln r (2−r ) .

 4−r  r

i=1

< 0 and c˜1 (2) = · · · = c˜2 (k − 1) =

Proof. We change from additive to multiplicative notation of the group elements, that is, from ti ∈ {0, 1} to (−1)ti ∈ {1, −1}. Then the exponent in (4.13) can be written as linear combinations of the terms (−1)ti +tk , (−1)ti and t–independent constants. In terms of s := ψk (t) we get, using si = 21 (1 − (−1)si ) ( ) c0 k + c1 |ψk (t)| + c2 t, ψk (t) = c0 k + c1 |s| + c2 ψk−1 (s), s 

k 1 si (−1) c1 = c0 k + k− 2 i=1

c2 + 3k + 1 + (−1)s1 + (−1)sk − 4 4  k k−1 si si si+1 × (−1) + (−1) (−1) i=1

i=1



k (−1)si = c˜0 (k) + c˜1I (−1)s1 + (−1)sk + c˜1I I i=1

+c˜2

k−1

(−1)si (−1)si+1

i=1

with c˜0 (k) := kc0 + 21 kc1 + (3k − 1) c42 , c˜1I := c42 ,     (4 − r )3 1 4−r 1 c2 and c˜2 := = − ln < 0. c˜1I I := −( 21 c1 + c2 ) = ln 2 r (2 − r ) 4 4 r   Remark 4.8. In this sense qˆk (t) equals, up to a sign, the Boltzmann factor of a 1D antiferromagnetic Ising system, whose two-body interaction is of nearest neighbor form and translation invariant. In the sense discussed in [GuK], the function qˆk equals the correlation function of the spin system at inverse temperature −1, up to a normalisation factor. So the antiferromagnetic character (the negative sign of c˜2 ) of the interaction is due to negativity of the inverse temperature. Several mathematicians, beginning with Kac (see his Comments in Pólya [Po], pp. 424–426), Newman [Ne] and Ruelle [Ru2], conjectured the existence of a Ising spin system related to the Riemann zeta function. One motivation for that conjecture is the

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317

Lee-Yang circle theorem of statistical mechanics. It states that all zeroes of the partition function Z (h) := exp(h|X |) ax y X ⊂

x∈X y∈−X

of a ferromagnetic (ax y = a yx ∈ [−1, 1]) Ising model occur at imaginary values of the external magnetic field h (see [Ru2] for a proof). The significance – if any – of the above corollary, remains to be clarified. 5. Thermodynamic Formalism 5.1. Partition function and transfer operator. We now establish a direct correspondence between the partition function Z n of the spin chain (along with some generalizations of it) and the transfer operator of the map Fr . To this end we first extend the tree T (r ) by considering the tree T˜ (r ) having 11 as root node and T (r ) as its left sub-tree starting at the second row. Each row is then completed by reflecting the corresponding row of T (r ) w.r.t the middle column and acting on each leaf with the transformation Sˆr defined in (2.9). Using the above terminology, the n th row Rn of T˜ (r ) is given for n > 1 by     ˆ Rn := Tn (r ) \ Tn−1 (r ) ∪ Sr Tn (r ) \ Tn−1 (r ) . (5.1) Note that T˜ (1) coincides with the classical Stern-Brocot tree. Proposition 5.1. For all r ∈ [0, 2), x ∈ R+ , s ∈ C and n ≥ 1 we have n 1)(x) = 2 ρ ns ( p r x + ρq)−2s . (Ps,r

(5.2)

p q ∈Rn

Proof. For n = 1 we have (Ps,r 1)(x) =

2ρ s . (r x + ρ)2s

Suppose that (5.2) holds true. Then n+1 (Ps,r 1)(x)

= 2ρ

(n+1)s

p q ∈Rn

= 2 ρ (n+1)s

1 (ρ + r x)2s



p q ∈Rn

We now note that p(r − 1) + ρq = 1 pr + ρq

* pr x ( ρ+r x

1 + + ρ q)2s ( pr −

+

1 pr x ρ+r x

+ ρ q)2s

1 (( p + ρq)r x + ρ 2 q)2s  1 . + (( p(r − 1) + ρq)r x + ρ( pr + ρq))2s   p and q

   p p + ρq = Sr 1 . ρq q

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M. Degli Esposti, S. Isola, A. Knauf 1 1

2 1

1 2

2 3

1 3

1 4

2 5

3 1

3 2

3 4

3 5

4 3

5 3

5 2

4 1

Fig. 5.6. The Stern-Brocot tree

Therefore, if qp ∈ Rn ∩ T (r ) then p(rpr−1)+ρq ∈ Rn+1 ∩ T (r ). On the other hand, if +ρq  p p p ˜ ˆ ∈ Rn ∩ (T (r ) \ T (r )), then  = Sr ( ) ∈ Rn ∩ T (r ) and therefore 1 ( p ) = 0 ( p ). q

q

q

q

q

This allows to conclude that the last line in the above expression is (5.2) with n replaced by n + 1 and the proof follows by induction.   Corollary 5.2. For all r ∈ [0, 2), k ∈ N0 and s ∈ C,

qk−2s (σ ) =

σ ∈Gk

1 −(k+1)s k+1 ρ (Ps,r 1)(1) 2

(5.3)

(with ρ = 2 − r ) and therefore 2

C (2s) Z n−1

=1+

n−1

k ρ −ks (Ps,r 1)(1).

(5.4)

k=0

The proof of this corollary follows at once from identity (5.2) along with the following lemma, whose proof amounts to an elementary calculation. Lemma 5.3. If 1> then 1<

p := Sˆr q

p ∈ T (r ), q

  p (r − 1) p + (2 − r )q = ∈ T˜ (r ) \ T (r ) q r p + (1 − r )q

Generalized Farey Trees, Transfer Operators and Phase Transitions

319

and r p  + ρ q  = r p + ρ q. Therefore r p  + ρ q  is the denominator of both descendants of

p q

in T (r ).

5.2. Phase transitions. If we define a ‘grand-canonical’ ensemble as in [Kn3] where the partition function is given for k ∈ N0 by Z kG (s) := exp (−s Q k (σ )) ≡ q −s , k ≥ 0, (5.5) σ ∈Gk

p q ∈Tk+1 (r )\Tk (r )

then Z nC (s) = 1 +

n−1

Z kG (s),

n ∈ N.

(5.6)

k=0

Due to Corollary 5.2, the existence of a spectral gap for all r ∈ [0, 1) and standard arguments of thermodynamic formalism [Ru1] all ‘grand canonical’ thermodynamic functions are analytic for all s ∈ C and there is no phase transition. On the other hand, in the canonical setting, using (5.4) and observing that the denominators qk (σ ) are monotonically decreasing functions of r, the limit limn→∞ Z nC (s) exists and is finite for Re s large enough. This suggests that in this framework there is indeed a phase transition. More specifically, let  1 C Fn (s, r ) := log 2Z n−1 (2s) , n ∈ N, s ∈ R, r ∈ [0, 1], n and F(s, r ) := lim Fn (s, r ).

(5.7)

n→∞

In terms of the canonical expectations  · n,s,r the mean magnetization is defined as , n 1 σk (−1) . (5.8) M(s, r ) := lim Mn (s, r ), where Mn (s, r ) := n→∞ n k=1

n,s,r

Remark 5.4. We remind the reader of the notions of order of a phase transition: for a free energy density F : R → R, s ∈ R is called a phase transition of order n ∈ N if F is n − 1 times, but not n times continuously differentiable at s. Theorem 5.5. 1. The limit in (5.7) exists and F ∈ C(R × [0, 1]). 2. Set   λs ≡ λs,r := spec rad Ps,r .

(5.9)

Then the function scr : [0, 1] → R of r , defined as the smallest positive real solution of the equation λs/2,r = (2 − r )s/2 is real analytic, increasing and convex, with scr (0) = 1 and scr (1) = 2.

(5.10)

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M. Degli Esposti, S. Isola, A. Knauf

3. For r ∈ [0, 1] there is a phase transition at s = scr (r ). More precisely, for each r ∈ [0, 1], s → F(s, r ) is real analytic on R \ {scr (r )} and F(s, r ) > 0 for s < scr (r ) whereas F(s, r ) = 0 for s ≥ scr (r ). If 0 ≤ r < 1, the phase transition is of first order, whereas if r = 1, it is of order two. 4. The limit (5.8) exists for all s ∈ R, r ∈ [0, 2) and  0, s < scr (r ), M(s, r ) = (5.11) 1, s > scr (r ). Proof. 1. To prove the first assertion, note that since s is real, Eq. (5.4) yields

1/n n−1 −ks k exp (Fn (s, r )) = 1 + ρ (Ps,r 1)(1) . k=0

The existence of the limit for r < 1 is a simple consequence of the spectral decomposition of the transfer operator Ps,r : Hs → Hs , r ∈ [0, 1), s ∈ R, k Ps,r f = λks h s νs ( f ) + Nsk f,

k ≥ 1,

(5.12)

where ∗ νs = λs νs Ps,r h s = λs h s , Ps,r k and λ−k s  Ns → 0, k → ∞. For r = 1 the existence of the limit has been shown in [Kn4]. The continuity follows from standard convexity arguments. 2. The family of operators Ps,r : H (D1 ) → H (D1 ) is an analytic family of type A in the sense of Kato ([Ka], Chapter 7). Therefore, for s ∈ R, s < scr (r ), the function λs defined in (5.9) is real analytic and monotonically decreasing. The other claimed properties of the function scr (r ) can be easily obtained by differentiating twice the logarithm of (5.10). 3. The first statement of point (3) of the theorem is now a direct consequence of what was just proved and the identity (5.4). The proof that the phase transition for r = 1 is of second order is contained in [CK] and [PS]. So let r < 1 and δ := scr −s ≥ 0. The real analytic function g(δ) := ρ −s λs is convex and increasing in δ, with g(0) = 1. Using once more (5.12) and setting as := h s (1)νs (1), we can write n−1 k=0

k ρ −ks (Ps,r 1)(1) =

g n (δ) − 1 (as + Rn (δ)) , g(δ) − 1

(5.13)

with Rn (δ) = O(1) in both limits δ → 0 and n → ∞. Therefore, using the expansion g(δ) = 1 + aδ + o(δ), a > 0, we get F(s, r ) = aδ + o(δ) as δ → 0, which is what we needed. 4. Concerning the last statement, note first that from the identity (3.4) it follows that the “grand canonical” energy function Q k = log qk : Gk → R has the symmetry Q k (σ ) = Q k (σ¯ ). Moreover, if we denote with 0k = (0, 0, . . . , 0) ∈ Gk , it is immediate to see that for an arbitrary function f on Gn , we have σ ∈Gn

f (σ ) = f (0n ) + f (10n−1 ) +

n−1 m=1 τ ∈Gm

f (τ, 1, 0n−m−1 ).

Generalized Farey Trees, Transfer Operators and Phase Transitions

321

Therefore putting these observations together with (4.8) and (5.5) we can write

n  1  c −s C σk Z n (s)Mn (s, r ) = (−1) qn (σ ) n k=1 σ ∈Gn m   n−1 τ 2 −s n − m − 2 + j=1 (−1) j 2 + = 1+ 1− (qm (τ ))−s n n m=1 τ ∈Gm

  n−1 2 −s n − m − 2 G 2 + Z m (s) = 1+ 1− n n m=1

n−1 n−m−2 G Z m (s). = 1+ n

(5.14)

m=0

If s > scr , then the limit limn→∞ Z nC (s) is finite and the factors (n − m − 2)/n in (5.14) go to one. Comparison of (5.6) with (5.14) immediately implies that limn→∞ Mn (s, r ) exists and equals 1. If instead s < scr , then the ferromagnetic property (Thm. 4.5) implies Mn (s, r ) ≥ 0.

(5.15)

We follow a similar strategy as in the proof of Lemma 9 of [CK], but nowbased on the transfer operator analysis. By (5.12) and (5.9) for 2s < scr and ε ∈ 0, 21 (1 − ρ s /λs ) (which is a nonempty interval since s → ρ s /λs monotonically increases to 1 for 2s = scr ) µ± := (1 ± ε)

λs > 1. ρs

(5.16)

Using again the spectral decomposition (5.12) of the transfer operator and Corollary 5.2,  G 2 Z nG (2s) − µn−l ± Z l (2s)   n−l −(l+1)s n+1 l+1 l+1 a = ρ −(n+1)s as λn+1 + (N 1)(1) − µ ρ λ + (N 1)(1) s ± s s s s n−l     n+1 n+1 s l+1 (Ns 1)(1) (Ns 1)(1) µ± ρ λs as + − = as + ρs λn+1 λs λl+1 s s    n+1  λs (Nsn+1 1)(1) (Nsl+1 1)(1) n−l a a . = + − ± ε) + (1 s s ρs λn+1 λl+1 s s There exist δ ∈ (0, 1) and C ≥ 1 with  k  . .  (Ns 1)(1)  .Nsk .  ≤ ≤ Cδ k , k ∈ N.   λks λks Thus there exists a n min such that for all n ≥ n min and for all l = 0, . . . , n,    λ (n+1)  s G n−l l+1 2 Z nG (2s) − µn−l a a >0 Z (2s) ≥ − − ε) + Cδ (1 s s − l ρs

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M. Degli Esposti, S. Isola, A. Knauf

and similarly  2

G Z nG (2s) − µn−l + Z l (2s)



 ≤

λs ρs

(n+1)  as + Cδ n+1 − as (1 + ε)n−l < 0.

In other words, rescaling 2s → s and the constants µ± accordingly, we have for all s < scr , l−n G G G µl−n + Z n (s) ≤ Z l (s) ≤ µ− Z n (s),

l ∈ {0, . . . , n}.

(5.17)

So we have for n ≥ n min , with inequality (5.16)

n−1  n−1 1 − µ−n + C G l−n+1 G Z l (s) ≥ µ+ (s) = Z G (s).(5.18) Z n (s) = 1 + Z n−1 −1 n−1 1 − µ + l=0 l=0 Now we use the upper bound in (5.17) for the grand canonical partition function Z lG : n−1 n−1 n−m−2 G l −1 G Z m (s) = 1 + Z n−1−l (s) n n m=0 l=0

n−1  n−1 G (s) Z n−1 l − 1 −l d −l+1 G µ− ≤ 1 + Z n−1 (s) · − µ− = 1 + · n n dµ−

Z nC (s)Mn (s, r ) = 1 +

l=0

= 1+ = 1+

l=0

d µ− − µ−n+1 − · n dµ− 1 − µ−1 − * + −n+1 −n −1 µ (1 − µ )(1 − 2µ ) 1 − − − G + Z n−1 (s) . 2 µ− − 1 n (1 − µ−1 − ) G (s) Z n−1

With the lower bound (5.18) for Z nC (s) we obtain

 /  −n−1 µ 1 − µ−n 1 1 + − G −1 Mn (s, r ) ≤ (Z n−1 (s)) + + ; 2 (µ− − 1) n (1 − µ−1 1 − µ−1 + − ) G (s) = ∞, and µ > 1, this implies since limn→∞ Z n−1 −

lim sup Mn (s, r ) ≤ 0. n→∞

 

Together with (5.15) we see that the limit in (5.8) exists and equals 0.

Remark 5.6. Note that (only) for r = 1 (due to arithmetical quibbles) we have n Ps,1 1(0) = 1 +

n−1

k Ps,1 1(1)

k=0

and therefore C n 2 Z n−1 (2s) = Ps,1 1(0).

This makes the ‘canonical’ and ‘grand canonical’ settings equivalent at all temperatures −1 . for r = 1. But for r = 1 this equivalence fails below scr

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323

Example 5.7. For r = 0 we find Z nC (s) =

2s − 1 − 2n(1−s) so that 2s − 2 s

lim Z nC (s) =

n→∞

2s − 1 2s − 2

s

for Re s > 1 (Eq. (5.10) becomes 21− 2 = 2 2 ). For real s the free energy is given by  1 (1 − s) log 2, C F(s, 0) = lim log Z n (s) = n→∞ n 0,

s<1 s ≥ 1.

Example 5.8. For r = 1 one finds [Kn1] lim Z nC (s) =

n→∞

ϕ(n) n≥1

ns

=

ζ (s − 1) , ζ (s)

Re s > 2,

where ζ (s) is the Riemann zeta function. Moreover one can show that Z nC (2) ∼

n , n → ∞. 2 log n

The free energy F(s, 1) is real analytic for s < 2 and [PS] F(s, 1) ∼

2−s as s 2. − log (2 − s)

5.3. Fourier analysis of the transfer operator. Up to now we mainly analysed the action of the transfer operator on positive functions, related to the Perron-Frobenius eigenfunction. Now we are interested in the spectral gap and its disappearance for r 1. n to e (x) := e2πimx . Therefore we extend Proposition 5.1 by applying Ps,r m Proposition 5.9. For all r ∈ [0, 2), x ∈ R+ , s ∈ C, m ∈ Z and n ≥ 1 we have   n Ps,r em (x) = ρ ns

em p q ∈Rn



n 0 (x, p/q) pr x+ρq



( pr x



n 1 (x, p/q) pr x+ρq 2s + ρq)

+ em

,

(5.19)

where the functions n 0 and n 1 satisfy: n 0 (x, p/q) + n 1 (x, p/q) = p r x + ρ q, ∀x ∈ R+ .

(5.20)

More specifically, n 0 (x, p/q) = µ x + ρ ν, n 1 (x, p/q) = ( pr − µ)x + ρ(q − ν) for some choice of numbers 0 ≤ µ ≤ pr and 0 ≤ ν ≤ q.

(5.21) (5.22)

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Proof. For n = 1 we find



em   Ps,r em (x) = ρ s whereas for n = 2, 

⎡



2 Ps,r em (x) = ρ 2s ⎣

 em  em

+

x r x+ρ





(r x

x+ρ r x+2ρ

(r −1)x+ρ r x+ρ 2s + ρ)

+ em



 + em

(r −1)x+ρ r x+2ρ 2s )

,



(r x + 2ρ  ⎤ ((3−r )r −1)x+ρ 2 x + e m 2 2 (3−r )r x+ρ (3−r )r x+ρ ⎦. ((3 − r )r x + ρ 2 )2s

Hence formula (5.19) holds for n = 1 with the choice µ = 1, ν = 0 and for n = 2 with the choice µ = 1, ν = 1. If we set   p n i (x, p/q) Vi x, , µ, ν := , i = 0, 1, q pr x + ρq we find, for i = 0, 1,     p + ρq p Vi 0 (x), , µ, ν = Vi x, , µ + rρν, ρν , q ρq and

    p(r − 1) + ρq p Vi 1 (x), , µ, ν = Vi x, , µ(r − 1) + ν, µ + ν . q pr + ρq

The proof now proceeds by induction along the same lines as for Proposition 5.1.   A more precise characterization of the integers µ and ν appearing in the definition of the functions n 0 and n 1 would be of some interest (see however below, Proposition 5.11). Nevertheless, property (5.20), along with Lemma 5.3, is sufficient to have the following extension of Corollary 5.2: Corollary 5.10. For all r ∈ [0, 2), k ∈ N0 , s ∈ C and m ∈ Z,    pk (σ ) k+1 = 21 ρ −(k+1)s Ps,r qk−2s (σ ) em em (1). qk (σ )

(5.23)

σ ∈Gk

The proof of Proposition 5.19 extends at once to arbitrary complex functions f : [0, 1] → C. On the other hand we shall formulate and prove this more general result in a direct way, without employing the extended tree T˜ (r ). Proposition 5.11. For all r ∈ [0, 2), k ∈ N0 , s ∈ C and f : [0, 1] → C,   −2s k+1 1 −(k+1)s Ps,r qk (σ ) − (1 − x)r pk (1 − σ ) f (x) = . 2ρ 1 2

i=0,1

 f

σ ∈Gk

pk (σ ) − (1 − x)sk+1 (σ, i) qk (σ ) − (1 − x)tk+1 (σ, i)

 (x ∈ [0, 1]),

(5.24)

Generalized Farey Trees, Transfer Operators and Phase Transitions

325

with functions sk ≡ sk,r and tk ≡ tk,r : Gk → R defined by s0 := 1, t0 := 0, sk+1 (0, σ ) sk+1 (1, σ ) tk+1 (0, σ ) tk+1 (1, σ )

:= sk (σ ), := (r − 1)sk (σ ) + (2 − r )tk (σ ), := r sk (σ ) + (2 − r )tk (σ ), := r sk (σ ) + (2 − r )tk (σ ).

Proof. By definition the transfer operator acts like (Ps,r f )(x) =

  ρs f (0 (x)) + f (1 (x)) 2s (ρ + r x)

(x ∈ [0, 1]),

(5.25)

with ρ = 2 − r . It is well–defined for the range 0 ≤ r < 2. • For k = 0 we have as arguments of f in (5.24), 0 (x) =

1 − (1 − x) 1 − (r − 1)(1 − x) and 1 (x) = 2 − r (1 − x) 2 − r (1 − x)

which in this case gives the formula, since s1 (0) = 1, s1 (1) = r − 1 and t1 (0) = t1 (1) = r. • For k ∈ N0   pk (σ ) − (1 − x)sk+1 (σ, i) 0 qk (σ ) − (1 − x)tk+1 (σ, i) [(2 − r )qk (σ ) + (r − 1) pk (σ )] − (1 − x)[sk+1 (σ, i)] = [(2 − r )qk (σ ) + r pk (σ )] − (1 − x)[r sk+1 (σ , i) − (2 − r )tk+1 (σ , i)] qk+1 (0, σ ) − (1 − x)sk+2 (0, σ, i) = pk+1 (0, σ ) − (1 − x)tk+2 (0, σ, i) and, with i  := 1 − i,   pk (σ ) − (1 − x)sk+1 (σ , i  ) 1 qk (σ ) − (1 − x)tk+1 (σ , i  ) [(2 − r )qk (σ ) + (r − 1) pk (σ )] − (1 − x)[(r − 1)sk+1 (σ , i  ) + (2 − r )tk+1 (σ , i  )] = [(2 − r )qk (σ ) + r pk (σ )] − (1 − x)[r sk+1 (σ , i  ) − (2 − r )tk+1 (σ , i  )] qk+1 (1, σ ) − (1 − x)sk+2 (1, σ, i) . = pk+1 (1, σ ) − (1 − x)tk+2 (1, σ, i) • Concerning the (k + 1)th iterates of the factor reads, substituting first pk (σ ) = qk (σ ) − pk (σ ),

ρs (ρ+r x)2s

in (5.25), the induction step

 −2s ρs q (σ ) − (1 −  (x))r p (σ ) k 0 k (ρ + r x)2s   −2s = ρ s (2 − r )qk (σ ) + r pk (σ ) − (1 − x)r [(2 − r )qk (σ ) + (r − 1) pk (σ )  −2s = ρ s qk+1 (0, σ ) − (1 − x)r pk+1 (1, σ ) ,

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since qk+1 (0, σ ) = (2−r )qk (σ )+r pk (σ ) and pk+1 (1, σ ) = (2−r )qk (σ )+(r −1) pk (σ ). Similarly  −2s ρs q (σ ) − (1 −  (x))r p (σ ) k 1 k (ρ + r x)2s  −2s  = ρ s (2 − r )qk (σ ) + r pk (σ ) − (1 − x)r pk (σ )  −2s = ρ s qk+1 (1, σ ) − (1 − x)r pk+1 (0, σ ) ,  since qk+1 (1, σ ) = (2 − r )qk (σ ) + r pk (σ ), too and pk+1 (0, σ ) = pk (σ ).  Corollary 5.12. For all r ∈ [0, 2), k ∈ N0 , s ∈ C and f : [0, 1] → C,    pk (σ ) k+1 −2s 1 −(k+1)s . Ps,r f (1) = (qk (σ )) f 2ρ qk (σ ) σ ∈Gk

5.4. Twisted zeta functions. For m ∈ Z define: 2πi m qp Z n(m) (s) := q −s e .

(5.26)

p q ∈Tn (r )\{0}

Then by the above 2 Z n(m) (2s) = 1 +

n

 k ρ −ks Ps,r em (1).

(5.27)

k=0

Example 5.13. For m = r = 1 we have for Re s > 2, lim Z n(1) (s) =

n→∞

q≥1

since the Möbius function µ

 p

µ(q)

np





qs



(−1) = 0,

satisfies



µ(q) =

e

np,

2πi

p q

=

1 , ζ (s)

n p ≤ 1, otherwise,

, q ∈ N.

0< p≤q gcd( p,q)=1

Remark 5.14. Defining ζ0 (s) := 2s /(2s − 1) from Example 5.7 we have lim Z nC (s) =

n→∞

ζ0 (s − 1) , ζ0 (s)

Re s > 1.

(5.28)

Re s > scr ,

(5.29)

One could guess that for all r ∈ [0, 1] it holds lim Z nC (s) =

n→∞

ζr (s − 1) , ζr (s)

Generalized Farey Trees, Transfer Operators and Phase Transitions

327

with ζ1 (s) = ζ (s) and more generally, for r ∈ [0, 1], 1 := lim Z n(1) (s), n→∞ ζr (s)

Re s > scr .

On the other hand, a simple direct verification shows that for r = 0, 1 this is not the case. Note that by the above we have Re s > scr , ∞  2 k =1+ ρ −ks Ps,r e1 (1). ζr (2s)

(5.30)

k=0

We conclude by discussing further ζ ’s for general integer values of m restricting to the case r = 1. For real x we set ex : C → C, ex (c) := e2πi xc . The multiplicative group of units in the ring Z/qZ is denoted by U (Z/qZ). It is of cardinality ϕ(q). We are interested in the functions µ(m) : N → C, µ(m) (q) := em/q ( p) (m ∈ Z, q ∈ N). p∈U (Z/q Z)

Lemma 5.15. Denoting by µ the Möbius function,   ϕ(q) q µ µ(m) (q) =  q gcd(m, q) ϕ gcd(m,q)

(m ∈ Z, q ∈ N).

Remark 5.16. In particular µ(m) is integer–valued, multiplicative, µ(−m) = µ(m) , µ(0) = ϕ and µ(1) = µ. Proof. We set q  := q/ gcd(m, q), m  := m/ gcd(m, q). Then µm (q) =

p∈U (Z/q Z)

=

ϕ(q) ϕ(q  )

em  /q  ( p) =

ϕ(q) ϕ(q  )

e1/q  ( p  ) =

p  ∈U (Z/q  Z)



em  /q  ( p  )

p  ∈U (Z/q  Z)

ϕ(q) µ(q  ). ϕ(q  )

The third equality is due to the fact that for m  relatively prime to q  multiplication by m  only permutes the elements of U (Z/q  Z).   Next we consider the Dirichlet series ζ (m) (s) :=

∞

µ(m) (n)n −s .

n=1

As |µ(m) (n)| ≤ n, we know that these series converge absolutely for Re(s) > 2. This is in fact also the abscissa of unconditional convergence if m=0. Proposition 5.17. For m ∈ Z\{0} the Dirichlet series ζ (m) (s) converges absolutely for Re(s) > 1.

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Proof. This follows from Lemma 5.15 and the estimate

ϕ



ϕ(q) q gcd(m,q)

≤ m.

(5.31)

Equation (5.31) can be proven by taking the prime powers pa for m, since ϕ is multiplicative. In that case we can also assume that q = p b , and the inequality follows from ϕ( p c ) = ϕ( p) p c−1 , valid for c ≥ 1.   References [Ba] [CK] [FKO] [GMM]

[GI] [GJ] [GKP] [GuK] [GR] [HW] [Is] [Ka] [KO] [Kn1] [Kn2] [Kn3] [Kn4] [LR] [LeZa] [Ma] [Mi] [Ne] [Ob] [Po]

Baladi, V.: Positive Transfer Operators and Decay of Correlations. In: Advanced Series in Nonlinear Dynamics, Vol. 16, Singapore: World Scientific, 2000 Contucci, P., Knauf, A.: The phase transition of the number-theoretical spin chain. Forum Math. 9, 547–767 (1997) Fiala, J., Kleban, P., Özlük, A.: The phase transition in statistical models defined on Farey fractions. J. Stat. Phys. 110, 73–86 (2003) Gallavotti, G., Martin-Löf, A., Miracle-Solé, S.: Some Problems Connected with the Description of Coexisting Phases at low Temperatures in the Ising Model. In: Statistical Mechanics and Mathematical Problems (Batelle, 1971), Lenard, A., ed., Lecture Notes in Physics 20, Berlin, Heidelberg, New York: Springer, 1973 Giampieri, M., Isola, S.: A one-parameter family of analytic Markov maps with an intermittency transition. Disc. Cont. Dyn. Syst. 12, 115–136 (2005) Glimm, J., Jaffe, A.: Quantum Physics. 2nd ed. New York: Springer, 1987 Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics. Reading, MA: AddisonWesley, 1990 Guerra, F., Knauf, A.: Free energy and correlations of the number-theoretical spin chain. J. Math. Phys. 39, 3188–3202 (1998) Gradshteyn, I., Ryzhik, I.: Table of integrals, series and products. New York: Academic Press, 1965 Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers. Oxford: Oxford Univ. Press, 1979 Isola, S.: On the spectrum of Farey and Gauss maps. Nonlinearity 15, 1521–1539 (2002) Kato, T.: Perturbation Theory for Linear Operators. Berlin: Springer, 1995 Kleban, P., Özlük, A.: A farey fraction spin chain. Commun. Math. Phys. 203, 635–647 (1999) Knauf, A.: Number theory, dynamical systems and statistical mechanics. Rev. Math. Phys. 11, 1027–1060 (1999) Knauf, A.: The number-theoretical spin chain and the riemann zeros. Commun. Math. Phys. 196, 703–731 (1998) Knauf, A.: On a ferromagnetic spin chain. Commun. Math. Phys. 153, 77–115 (1993) Knauf, A.: On a ferromagnetic spin chain. part ii: thermodynamic limit. J Math Phys 35, 228– 236 (1994) Lanford, O.E., Ruedin, L.: Statistical mechanical methods and continued fractions. Helv. Phys. Acta 69, 908–948 (1996) Lewis, J., Zagier, D.: Period functions and the Selberg zeta function for the modular group. In: The Mathematical Beauty of Physics, Adv. Series in Math. Phys. 24, River Edge, NJ: World Sci. Publ., 1997, pp. 83–97 Mayer, D.H.: Continued fractions and related transformations. In: Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Bedford, T., Keane, M., Series, C., eds., Oxford: Oxford University Press, 1991 Minkowski, H.: Zur Geometrie der Zahlen. Gesammelte Abhandlungen, Vol. 2. Hilbert, D., ed., Leipzig: Teubner, 1911 Newman, Ch.: Gaussian correlation inequalities for ferromagnets. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 33, 75–93 (1975) Oberhettinger, F.: Tables of Bessel Transforms. Berlin-Heidelberg New York: Springer-Verlag, 1972 Pólya, G.: Collected Papers, Vol. II: Locations of Zeros. Boas R.P. ed., Cambridge: M.I.T. Press, 1974

Generalized Farey Trees, Transfer Operators and Phase Transitions

[PM] [Pr] [PS] [Ru1] [Ru2] [Rug] [Si]

329

Pomeau, Y., Manneville, P.: Intermittency transition to turbulence in dissipative dynamical systems. Commun. Math. Phys. 74, 189–197 (1980) Prellberg, T.: Towards a complete determination of the spectrum of the transfer operator associated with intermittency. J. Phys. A. 36, 2455–2461 (2003) Prellberg, T., Slawny, J.: Maps of intervals with indifferent fixed points: thermodynamic formalism and phase transitions. J. Stat. Phys. 66, 503–514 (1992) Ruelle, D.: Thermodynamic Formalism. Reading, MA: Addision-Wesley Publ. Co, 1978 Ruelle, D.: Is our mathematics natural? The case of equilibrium statistical mechanics. Bull. AMS 19, 259–268 (1988) Rugh, H.H.: Intermittency and regularized Fredholm determinants. Invent. Math. 135, 1–24 (1999) Simon, B.: The Statistical Mechanics of Lattice Gases. Vol. I. Princeton NJ: Princeton University Press, 1993

Communicated by J.L. Lebowitz

Commun. Math. Phys. 275, 331–372 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0297-0

Communications in

Mathematical Physics

Renormalization Proof for Massive ϕ44 Theory on Riemannian Manifolds Christoph Kopper1 , Volkhard F. Müller2 1 Centre de Physique Théorique, CNRS, UMR 7644, Ecole Polytechnique, F-91128 Palaiseau, France.

E-mail: [email protected]

2 Fachbereich Physik, Technische Universität Kaiserslautern, D-67653 Kaiserslautern, Germany.

E-mail: [email protected] Received: 11 September 2006 / Accepted: 30 January 2007 Published online: 11 August 2007 – © Springer-Verlag 2007

Abstract: In this paper we present an inductive renormalizability proof for massive ϕ44 theory on Riemannian manifolds, based on the Wegner-Wilson flow equations of the Wilson renormalization group, adapted to perturbation theory. The proof goes in hand with bounds on the perturbative Schwinger functions which imply tree decay between their position arguments. An essential prerequisite is precise bounds on the short and long distance behaviour of the heat kernel on the manifold. With the aid of a regularity assumption (often taken for granted) we also show that for suitable renormalization conditions the bare action takes the minimal form, that is to say, there appear the same counterterms as in flat space, apart from a logarithmically divergent one which is proportional to the scalar curvature.

1. Introduction Among the different schemes devised to prove the perturbative renormalizability of a local quantum field theory, the one based on the Wegner-Wilson differential flow equations of the Wilson renormalization group shows a distinctive characteristic: it circumvents completely the combinatoric complexity of generating Feynman diagrams and the subsequent cumbersome analysis of Feynman integrals within general overlapping divergences. Initiated by Polchinski [Pol], this approach to renormalization has now been adapted to a wide variety of physically interesting instances. Partial reviews of the rigorous work which started from [KKS] may be found in [Kop1, Sal, Kop2, Mü]. It is tempting to extend the approach via Wilson’s flow equation further to prove the perturbative renormalizability of a quantum field theory defined on curved spacetime. There is a caveat, however. Using functional integration, one actually deals with a quantum field theory defined on a “Euclidean section” of curved spacetime, i.e. on a Riemannian manifold. In contrast to flat space there is no Wick rotation of Lorentzian curved spacetime, in general. Nevertheless, beyond static spacetimes, on particular nonstatic ones the

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analytic continuation of a quantum field theory to a corresponding Euclidean formulation has been rigorously shown recently: Bros, Epstein and Moschella [BEM] considered a quantum field theory on the anti-de Sitter (AdS) spacetime within a Wightman-type approach. As a consequence of certain spectral assumptions they show that the n-point correlation functions admit an analytic continuation to tuboidal domains (of n copies) of the complexified covering space of the AdS spacetime. This continuation includes the Euclidean AdS spacetime and satisfies there Osterwalder-Schrader positivity. Euclidean AdS spacetime is a Riemannian manifold with constant negative curvature.1 Moreover, Birke and Fröhlich [BiFr], establishing in an algebraic approach Wick rotation of quantum field theories at finite temperature, also presented a reconstruction of quantum field theories on specific curved spacetimes from corresponding imaginary-time formulations, using group-theoretical techniques. Our work starts straight off considering a Riemannian manifold as a given “spacetime”. This manifold is assumed to be geodesically complete and to have all its sectional curvatures confined by a negative lower and a positive upper bound. We then study perturbative renormalizability of massive ϕ44 -theory defined on such a manifold by analysing the generating functional L ,0 of connected (free propagator) amputated Schwinger functions (CAS). From the physical point of view it seems justified to restrict to this class of manifolds, since in situations where curvature becomes large or where singularities appear the treatment of gravity as a classical background effect becomes questionable anyway. As there is no translation symmetry, the CAS and thus the system of flow equations relating them have to be dealt with in position space. Establishing bounds involving these CAS we heavily rely on global lower and upper bounds for the heat kernel on the manifold, found in the mathematical literature. Around the beginning of the eighties a considerable amount of work was carried out to formulate quantum field theory perturbatively on curved spacetime. Based on the intuition that ultraviolet divergences involve arbitrarily short wavelengths, an approximating local momentum space representation of the Feynman propagator in curved spacetime was developed in [Bir, BPP, BuPn] for the φ 4 -theory and generalized in [BuPr, Bun1]. Combined with dimensional regularization, the euclideanized φ 4 -theory was then shown to be renormalizable with local counterterms in one- and two-loop order. Furthermore, choosing the same general approach, Bunch [Bun2] has demonstrated the BPHZ renormalization of the φ 4 -theory on euclideanized curved spacetime, by taking into account the power counting singular contributions in the asymptotic expansion of the propagator around its euclidean form. A different kind of generally applicable dimensional regularization scheme has been given by Lüscher [Lü], who applies it to the φ 4 -theory on an arbitrary compact four-dimensional manifold with positive metric and to the Yang-Mills gauge theory on S 4 . He also shows the renormalizability of the φ 4 -theory by local counterterms at the one- and two-loop levels. Further references on work before 1982 can be found in the monograph [BiDa]. More recently, the perturbative construction of the φ 4 -theory has been performed in an algebraic setting by Brunetti, Fredenhagen, Hollands and Wald [BrFr, HoWa1, HoWa2]. These authors adapted the renormalization method of Epstein and Glaser to construct the algebra of local, covariant quantum fields of the φ 4 -model on a globally hyperbolic curved spacetime to any order of the perturbative expansion, making use of techniques from microlocal analysis. The crucial notion of a local and covariant quantum field introduced in [HoWa1, HoWa2] has been further formalized by Brunetti, Fredenhagen and Verch [BFV]. 1 It is called ‘hyperbolic space’ in the mathematical literature.

Renormalization Proof for Massive ϕ44 Theory on Riemannian Manifolds

333

This paper is organized as follows: In Sect. 2 we collect and slightly adapt global bounds on the heat kernel found in the mathematical literature, which are pertinent to our treatment. The action considered and the system of perturbative flow equations satisfied by the CAS is set up in Sect. 3. To establish bounds on the CAS, being distributions, they have to be folded first with test functions. In Sect. 4 a suitable class of test functions is introduced, together with tree structures with the aid of which the bounds to be derived on the CAS are going to be expressed. In Sect. 5 we state the boundary and the renormalization conditions used to integrate the flow equations of the irrelevant and relevant terms, respectively. The flow equations permit one to be quite general in this respect, compassing basically all situations of interest. Section 6 is the central one of this paper. We state and prove inductive bounds on the Schwinger functions which, being uniform in the cutoff, directly lead to renormalizability. Beyond that they imply tree decay of the Schwinger functions between their external points. The last section is devoted to the proof that the bare action of the theory may be chosen minimally, i.e. with position independent counterterms apart from one (logarithmically divergent) term which is proportional to the scalar curvature of the manifold. Here we have to make the assumption that geometric quantities on the manifold have a smooth expansion (to lowest orders) w.r.t. contributions of curvature terms of increasing mass dimension. An Appendix shows the notations and conventions used. 2. The Heat Kernel We consider geodesically complete simply connected Riemannian manifolds M of dimension n without boundary, whose sectional curvatures are bounded between two constants −k 2 and κ 2 . The related heat kernel then has the following properties: K (t, x, y) ∈ C ∞ ((0, ∞) × M × M), 0 < K (t, x, y) < ∞, K (t, x, y) = K (t, y, x),
K (t, x, y) d V (y) = 1, M
K (t1 + t2 , x, y) = K (t1 , x, z) K (t2 , z, y) d V (z).

(1) (2) (3) (4) (5)

M

Stochastic completeness (4) holds due to the assumed bounded curvature, cf. [Tay, Ch.6, Prop.2.3]. Mathematicians have established quite sharp pointwise bounds on the respective heat kernels of various classes of manifolds. We are going to make explicit now some of these bounds, because we will rely on them in the subsequent construction. Some bounds are known to hold for 0 < t < T , others for 0 < t or T < t. We will write tδ = t (1 + δ), where the parameter δ satisfies 0 < δ < 1 and may be chosen arbitrarily small. Furthermore c, C, collectively denote constants which depend on δ, n and - if involved in the claim - on k 2 , T . For notations see the Appendix. On complete Riemannian manifolds of dimension n with nonnegative Ricci curvature the heat kernel satifies the lower and upper bounds [LiYa, Dav1] 

c |B(x, t 1/2 )| |B(y, t 1/2 )|

d 2 (x,y)

e− 4t (1−δ) ≤ K (t, x, y)≤ 

C |B(x, t 1/2 )| |B(y, t 1/2 )|

d 2 (x,y)

e− 4t (1+δ) , (6)

valid for all x, y ∈ M and t > 0.

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In the case of negative Ricci curvature bounded below let E ≥ 0 denote the bottom of the spectrum of the operator −. Then it holds, see [Dav2, Theorems 16 and 17]: If δ > 0, there exists a constant cδ such that  d 2 (x, y)  − 1  (7) K (t, x, y) ≤ cδ |B(x, t 1/2 )| |B(y, t 1/2 )| 2 exp − 4t (1 + δ) for 0 < t < 1 and for all x, y ∈ M, whereas  − 1  d 2 (x, y)  2 K (t, x, y) ≤ cδ |B(x, 1)| |B(y, 1)| exp (δ − E)t − 4t (1 + δ)

(8)

for 1 ≤ t < ∞ and for all x, y ∈ M. Moreover, a lower bound of the form appearing in (6) holds here, too, however restricted to 0 < t < T , [Var]. In addition, given bounded sectional curvature −k 2 ≤ SecM ≤ κ 2 , there is the lower bound

d 2 (x, y) ˜ (9) K (t, x, y) ≥ c exp − Et − C t for all x, y ∈ M and for t > T , with constants c, C > 0 and E˜ > E, possibly much larger, [Gri, Ch. 7.5]. On a Cartan-Hadamard manifold of dimension n, i.e. a geodesically complete simply connected noncompact Riemannian manifold with nonpositive sectional curvature, and assuming that the sectional curvature is bounded below by −K 2 , we also have for all x, y ∈ M, and for 0 < t, [Gri, Ch. 7.4],

(n − 1)2 2 d 2 (x, y) C . (10) exp − K t− K (t, x, y) ≤ min(1, t n/2 ) 4 4tδ For later use we extract from these bounds particular versions valid for four-dimensional complete Riemannian manifolds whose sectional curvatures may range between two constants −k 2 and κ 2 . From volume comparison, cf. e.g. [Cha, Sect.3.4], follows: i) If all sectional curvatures of M have values in [−k 2 , 0], k > 0, fixed, then π2 2 π2 2 t ≤ |B(x, t 1/2 )| ≤ t h 4 (k t 1/2 ) 2 2 with the positive increasing function h 4 (r ) =

(11)

cosh(3r ) − 9 cosh r + 8 , h 4 (0) = 1, 3r 4

ii) if all sectional curvatures of M have values in [0, κ 2 ], κ > 0, fixed, then2 π2 2 π2 2 t s4 (κ t 1/2 ) ≤ |B(x, t 1/2 )| ≤ t , for κ t 1/2 < π, 2 2 with the positive decreasing function, 0 < r < π , s4 (r ) =

cos(3r ) − 9 cos r + 8 , s4 (0) = 1. 3r 4

2 The restriction on t accounts for the injectivity radius of the manifold.

(12)

Renormalization Proof for Massive ϕ44 Theory on Riemannian Manifolds

335

Taking (6) together with (12), as well as taking (10) or (7) and the lower bound from (6) 3 together with (11), we obtain, restricting4 to 0 < t < T : C c d 2 (x, y) d 2 (x, y) ) ≤ K (t, x, y) ≤ ). exp(− exp(− t2 4t (1 − δ) t2 4t (1 + δ)

(13)

The constants c, C depend on k 2 , κ 2 , δ, T , but do not depend on t. As a consequence of this lower and upper bound we obtain under the same conditions d s (x, y) K (t, x, y) ≤ c  t s/2 K (t δ  , x, y)

for t ≤ T,

(14)

with δ  > δ. For 1 ≤ s ≤ 3 we also need the bound |∇ s K (t, x, y)| ≤ C t −s/2 K (tδ , x, y)

(15)

based on [CLY, Dav3] and valid for 0 < t < T . Here ∇ s denotes a covariant derivative of order s w.r.t. x and the norm is that of (A.32). The constant C here also depends on the norm of the covariant derivatives of the curvature tensor up to order s − 1. From (15) and the heat equation it follows directly that |∂t K (t, x, y)| ≤ C t −1 K (tδ , x, y).

(16)

Finally we note the following recently proven bound on the logarithmic derivative of the heat kernel [SoZh] which holds for RicM ≥ −k 2 and for 0 < t < T : |∇ K (t, x, y)| 1 d 2 (x, y) ≤ O(1) 1/2 (1 + ). K (t, x, y) t t

(17)

In closing this section we remark that the restriction to manifolds M of the kind considered is not dictated by the validity of our methods of proof. It rather seems to be a choice which is reasonable and interesting on physical grounds. 3. The Action and the Flow Equations The regularized (free) propagator is given in terms of the heat kernel by
t 2  ε,t C (x, y) = dt  e−m t K (t  , x, y). ε

(18)

Its derivative w.r.t. t is denoted as Ct (x, y) := ∂t C ε,t (x, y) = e−m

2

t

K (t, x, y).

(19)

We assume 0 < ε ≤ t < ∞ so that the flow parameter t takes the role of a long distance cutoff, whereas ε is a short distance regularization. The full propagator is recovered for ε = 0 and t → ∞. For finite ε and in finite volume the positivity and regularity properties of C ε,t permit to define the theory rigorously from the functional integral
1 1 ε,ε ε,t ε,t e−
(L (ϕ)+I ) = dµε,t (φ) e−
L (φ + ϕ) , L ε,t (0) := 0, (20) 3 Remember the statement after (8). 4 The restriction is necessary both for the upper and lower bounds.

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where the factors of
have been introduced to allow for a consistent loop expansion in the sequel. In (20) dµε,t (φ) denotes the Gaussian measure with covariance
C ε,t (x, y). The test function ϕ here is supposed to be in the support of the Gaussian measures dµt  ,t  (φ), ε ≤ t  ≤ t  < ∞, which implies in particular that it is in C ∞ (M). The normalization 1 ε,t factor e−
I is due to vacuum contributions. It diverges in infinite volume so that we can take the infinite volume limit only when it has been eliminated. We do not make the finite volume explicit here since it plays no role in the sequel. The functional L ε (ϕ) := L ε,ε (ϕ) is the bare (inter)action including counterterms, viewed as a formal power series in
. The superscript ε indicates the UV cutoff. For shortness we will pose in the following, with x, y ∈ M, x = (x1 , · · · , xn ) ∈ M×n ,



:= x


M

d V (x),

:=

x

n
 i=1 M

d V (xi ),

and ˜ δ(x, y) := |g|−1/2 (x) δ(x, y). As is known from lowest order calculations [Bir, BPP, Lü], in curved spacetime there will appear an additional counterterm of the type x R(x) ϕ 2 (x) which is proportional to the scalar curvature R(x) of the spacetime manifold M considered. So the bare interaction for the symmetric ϕ44 theory would be λ L (ϕ) = 4! ε




1 ϕ (x) + 2 x




{(a ε + ξ ε R(x))ϕ 2 (x)

4

x

+bε g µν (x) ∂µ ϕ(x) · ∂ν ϕ(x) +

2 ε 4 c ϕ (x)}, 4!

(21)

where λ > 0 is the renormalized coupling, and the cutoff dependent parameters a ε , ξ ε , bε , cε - which remain to be fixed and which are directly related to the mass, curvature, wave function, and coupling constant counterterms5 - will fulfill a ε , ξ ε , bε , cε = O(
).

(22)

It seems to us that there is no a priori reason to restrict to bare interactions of this form. In fact, since there is no translation invariance in curved space time, all counterterms and even the coupling λ itself may be position dependent. Quite generally the bare action is not a directly observable physical object, and the constraints on its form stem from the symmetry properties of the theory which are imposed, on its field content and on the form of the propagator. The symmetry properties depend in particular on the renormalization conditions which fix the physical (relevant) parameters of the theory. They might be position dependent: e.g. a local scattering experiment performed at different places at the same external momenta might give different cross sections and, as a consequence of 5 Since it is not necessary in the flow equation framework to introduce bare fields in distinction from renormalized ones (our field is the renormalized one in this language), there is a slight difference, which is to be kept in mind only when comparing to other schemes.

Renormalization Proof for Massive ϕ44 Theory on Riemannian Manifolds

337

this, the renormalized coupling, fixed in terms of the cross section, would be position dependent. It is therefore natural to admit more general bare interactions6

λ(x) 4 1 L ε (ϕ) = {a ε (x) ϕ 2 (x) + u µ,ε (x) ϕ(x) ∇µ ϕ(x) ϕ (x) + 4! 2 x x 2 ε µν +bˆ (x)g (x) ∂µ ϕ(x) · ∂ν ϕ(x) + cε (x)ϕ 4 (x)}. 4! Here λ(x), a ε (x), bˆ ε (x), cε (x) are general scalars and u µ,ε (x) is a general vector, all functions are supposed to be smooth, and |λ(x)| (of course) uniformly bounded on M. When calculating the two point function Lε (x1 , x2 ) = δ/δϕ(x1 ) δ/δϕ(x2 ) L ε (ϕ)|ϕ≡0 from this bare action one obtains 1 ˜ 2 , x1 ) − (∇µ u µ, )(x1 ) δ(x ˜ 2 , x1 ) L2 (x1 , x2 ) = a (x1 ) δ(x 2 1 1 ˜ 2 , x1 ). −|g(x2 )|− 2 ∂µ(2) bˆ ε (x2 ) g µν (x2 ) |g(x2 )| 2 ∂ν(2) δ(x

(23)

This means that u µ, only contributes via its divergence, and that the contribution ∼ ∇µ u µ, can be absorbed in a . On the other hand the particular tensor coupling bˆ ε (x)g µν (x) can be generalized - without changing symmetry properties - into a smooth symmetric (2, 0)-tensor field b µν,ε (x). In (23) the product bˆ ε (x)g µν (x) is then replaced by b µν,ε (x), and we recognise that in (23) the generalisation 1

1

(b) := |g(x)|− 2 ∂µ b µν (x) |g(x)| 2 ∂ν

(24)

of the Laplace-Beltrami operator , (A.4), appears. We thus adopt a bare interaction of the form

λ(x) 4 1 2 ϕ (x)+ {a˜ ε (x)ϕ 2 (x)+bµν,ε (x) ∂µ ϕ(x) · ∂ν ϕ(x)+ cε (x)ϕ 4 (x)} L ε (ϕ) = 4! 2 4! x x (25) with smooth scalar functions a˜ ε (x), cε (x) and a smooth symmetric tensor field bµν,ε (x) - which remain to be fixed and which are of (at least) order
. The flow equation (FE) is obtained from (20) on differentiating w.r.t. t. It is a differential equation for the functional L ε,t : δ δ ε,t
δ 1 δ  , Ct L ε,t −  L ε,t , Ct L . 2 δϕ δϕ 2 δϕ δϕ

∂t (L ε,t + I ε,t ) =

(26)

By , we denote the standard inner product in L 2 (M, d V (x)). The FE can also be stated in integrated form 1

e−
(L

ε,t (ϕ)+I ε,t )

1 ε,ε = e
F (ε,t) e−
L (ϕ) .

The functional Laplace operator F (ε, t) is given by F (ε, t) = 6 One could of course be even more general.

1 δϕ , C ε,t δϕ 2

(27)

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C. Kopper, V. F. Müller

using the notation δϕ(x) = δ/δϕ(x). We may expand L ε,t (ϕ) w.r.t. the number of fields ϕ setting L ε,t n (ϕ) :=

1 ∂ n ε,t L (κϕ)|κ=0 . n! ∂κ n

(28)

The functional L ε,t (ϕ) can also be expanded in a formal powers series w.r.t.
, and in a double series w.r.t.
and the number of fields L ε,t (ϕ) =

∞


l L lε,t (ϕ) =

l=0

∞

l=0


l

∞

L ε,t n,l (ϕ),

L ε,t 2,0 (ϕ) ≡ 0.

Corresponding expansions for a˜ ε (x), b µν,ε (x), cε (x) are a˜ ε (x) = We can then rewrite (26) in loop order l as ∂t L ε,t n,l (ϕ)

1 = 2 −


x,y

(29)

n=2



˜ lε (x)
l , l≥1 a

etc.

 Ct (x, y) δϕ(x) δϕ(y) L ε,t n+2,l−1 (ϕ)

n 1 +n 2 =n+2 l1 +l2 =l

 ε,t (δϕ(x) L ε,t n 1 ,l1 (ϕ)) δϕ(y) L n 2 ,l2 (ϕ) .

(30)

From L lε,t (ϕ) we obtain the connected amputated Schwinger functions of loop order l as ε,t Lε,t n,l (x 1 , . . . , x n ) := δϕ(x1 ) . . . δϕ(xn ) L l |ϕ≡0 .

(31)

It is straightforward to realize that the Lε,t n,l are distributions 1) which are completely symmetric w.r.t. permutations of the arguments (x1 , . . . , xn ), 2) and which fall off rapidly with the distances d(xi , x j ). These facts follow from (25), (27) and the properties of the regularized propagator (6), (13). The distributional character of the Lε,t n,l is related to the fact that we consider ˜ i , z) to the amputated Schwinger functions. Thus there is associated a factor of δ(x external line joining xi to an internal vertex at z which is integrated over. Therefore the distributional character is different according to whether one or several (up to three) external lines end in a given z-vertex. From the point of view of the FE the distributional character is a consequence of the boundary conditions, see (25) and (60), (61). Note that by (1) the propagators C ε,t which join different vertices are smooth functions of their position arguments for ε > 0. The two-point function is the most divergent object as regards its flow for small t. But the distributional structure of its regularized version is particularly simple. In fact it follows from (25) and (27) that Lε,t 2,l (x 1 , x 2 ) can be written as a sum over regularized Feynman amplitudes with vertices from L ε and with regularized propagators which satisfy (1), (6). Thus the only distributional singularities appearing ε,ε ˜ 2 , x1 ). Thus Lε,t (x1 , x2 ) can be written as a ˜ 2 , x1 ) and (bl ) x2 δ(x are of the form δ(x 2,l linear combination of these two contributions and a smooth function f (x1 , x2 ) of rapid decrease in d(x1 , x2 ).

Renormalization Proof for Massive ϕ44 Theory on Riemannian Manifolds

339

The FE for the Schwinger functions derived from (30) takes the following form: ∂t Lε,t n,l (x 1 , . . . , x n ) =

1 2




x,y

Ct (x, y) Lε,t n+2,l−1 (x 1 , . . . , x n , x, y)

(32)



 ε,t Ln 1 +1,l1 (x1 , . . . , xn 1 , x) Lε,t (y, x , . . . , x ) n +1 n 1 n 2 +1,l2

−

 .

sym

l1 +l2 =l, n 1 +n 2 =n

Here sym means symmetrization - i.e. summing over all permutations7 of (x1 , . . . , xn ) modulo those which only rearrange the arguments of a factor.8 For the renormalization proof we also need the FE for the Schwinger functions derived w.r.t. the UV cutoff ε. Integrating the FE over t  between ε and t and then deriving w.r.t. ε we obtain ∂ε Lε,t n,l (x 1 , . . . , x n ) = ∂ε Lε,ε n,l (x 1 , . . . , x n ) −

1 2


x,y

 Cε (x, y) Lε,ε n+2,l−1 (x 1 , . . . , x n , x, y)



 ε,ε Ln 1 +1,l1 (x1 , . . . , xn 1 , x) Lε,ε − (y, x , . . . , x ) n +1 n 1 n 2 +1,l2





x,y

(33)

sym

l1 +l2 =l, n 1 +n 2 =n

1 + 2



ε

t

  dt Ct  (x, y) ∂ε Lε,t n+2,l−1 (x 1 , . . . , x n , x, y) 



  ε,t   ∂ε Ln 1 +1,l1 (x1 , . . . , xn 1 , x) Lε,t − n 2 +1,l2 (y, x n 1 +1 , . . . , x n )

sym

l1 +l2 =l, n 1 +n 2 =n

 .

Integrating the FE instead from t < 1 to t = 1 and then deriving w.r.t. ε we get ε,1 ∂ε Lε,t n,l (x 1 , . . . , x n ) = ∂ε Ln,l (x 1 , . . . , x n ) 

1  1 dt  Ct  (x, y) ∂ε Lε,t − n+2,l−1 (x 1 , . . . , x n , x, y) 2 x,y t 

  ε,t   ∂ε Ln 1 +1,l1 (x1 , . . . , xn 1 , x) Lε,t − n 2 +1,l2 (y, x n 1 +1 , . . . , x n ) l1 +l2 =l, n 1 +n 2 =n

sym

(34)

 .

4. Test Functions and Tree Structures The distributional character of the Lε,t n,l necessitates the introduction of test functions against which they will be integrated. Later on we will only use a subclass of the test functions introduced in the subsequent definition, see (42). 7 By our choice of normalization there is no normalization factor to be divided out. 8 This may be implemented by counting only the configurations in which the permuted position variables ε,t appearing in Lε,t n 1 +1 and Ln 2 +1 appear in lexicographic order.

340

C. Kopper, V. F. Müller

Definition 1. For n ∈ IN we set Hn := {ϕ( xn ) = ϕ1 (x1 ) . . . ϕn (xn ) | ϕi ∈ C ∞ (M) ∩ L ∞ (M)}. We wrote xn = (x1 , . . . , xn ) and we shall write9 x2,n = (x2 , . . . , xn ) ∈ M×(n−1) . For ϕ ∈ Hn−1 we set
(x , ϕ) := Lε,t xn ) ϕ(x2,n ). (35) Lε,t n,l 1 n,l ( x2,n

The regularized Schwinger functions are obviously linear w.r.t. the test functions: ε,t ε,t Lε,t n,l (x 1 , a ϕ1 + b ϕ2 ) = a Ln,l (x 1 , ϕ1 ) + b Ln,l (x 1 , ϕ2 ), a, b ∈ C,

(36)

and it also follows from (25) and (27) and the properties of the regularized propagator (1), (10) that they satisfy Lε,t n,l (x 1 , ϕ) ∈ H1 . We will also consider Schwinger functions multiplied by products of factors σ (x j , x1 )µ , (A.21). Definition 2. We introduce a smooth (external) covector field ωµ (x) and form the bi-scalar insertions E (i) ≡ E(xi , x1 ; ω) := σ (xi , x1 )µ ωµ (x1 ),

i = 2, · · · , n,

(37)

and, more generally, for r ∈ IN, (r ) (r ) E (i) ≡ E(xi , x1 ; ω(r ) ) := σ (xi , x1 )µ1 . . . σ (xi , x1 )µr ωµ (x1 ) 1 ...µr

(38)

(r )

with a smooth (external) symmetric covariant tensor field ωµ1 ...µr (x) of rank r . We have, because of (A.22), |E(xi , x1 ; ω(r ) )| ≤ | ω(r ) (x1 )| d r (xi , x1 ) with the norm | ω(r ) (x1 )| according to (A.32). For r ∈ IN we then pose
(r ) ε,t E(xi , x1 ; ω(r ) ) Lε,t xn ) ϕ(x2,n ). Ln,l (x1 , E (i) ϕ) := n,l (

(39)

(40)

x2,n

Mostly we will suppress ω in the notation as we did in (40). Furthermore, for given x1 , x2 ∈ M we consider the products
(r ) 3−r (r ) F(12) Lε,t (x , x , ϕ) := d (x , x )E(x , x ; ω ) Lε,t xn ) ϕ(x3,n ) (41) 1 2 2 1 n,l 1 2 n,l ( x3,n

for r = 0, 1, 2, with E ≡ 1 if r = 0, and with ϕ(x3,n ) ≡ 1 (and no integration) if n < 3. 9 By the bosonic symmetry of the Lε,t all bounds are independent of the particular role assigned to the n,l coordinate x1 , which can be exchanged with any other coordinate.

Renormalization Proof for Massive ϕ44 Theory on Riemannian Manifolds

341

Definition 3. i) A graph G(V, P) is defined as a set of vertices V and a set P of unordered pairs p of vertices called lines/edges. Two lines are connected if they share a vertex in common. A graph is connected if for each pair of vertices (i, j) there exists a path of connected lines connecting i to j. A tree is a connected graph G(V, P) with |P| = |V | − 1. For a tree one can prove that the path of connected lines connecting i to j is unique. A rooted tree is a tree where one vertex in V has been chosen to be its root. The incidence number ci of the vertex i in a tree is the number of distinct lines containing i. The subset Ve ⊂ V , containing the vertices i with ci = 1, excluding the root (if it has c = 1), is called the set of external vertices. All other vertices are called internal vertices. We denote by T s the set of all trees such that |Ve | = s − 1, s ≥ 2. Subsequently we will consider trees where the set of vertices is identified10 with a set of points in the manifold M. For a tree T s ∈ T s we will call x1 ∈ M its root vertex, and Y = {y2 , . . . , ys } the set of points in M to be identified with its external vertices. Likewise we call Z = {z 1 , . . . , zr } with r ≥ 0 the set of internal vertices of T s . ii) For yi ∈ Y there exists exactly one p ∈ P such that yi ∈ p. For x1 there exist p1 , . . . , pc1 ∈ P with 1 ≤ c1 ≤ s − 1 such that x1 ∈ p1 , . . . , x1 ∈ pc1 . For z j ∈ Z (z )

(z )

(z )

there exist p1 j , . . . , pc j j ∈ P with 2 ≤ c j ≤ s such that z j ∈ p1 j , . . . , z j ∈ (z )

pc j j . We call c1 = c(x1 ) the incidence number of the root vertex and c(z j ) the incidence number of the internal vertex z j of the tree. We call a line p ∈ P an external line of the tree if there exists yi such that yi ∈ p. The set of external lines is denoted J . The remaining lines are called internal lines of the tree and are denoted by I, hence P = J ∪ I. iii) Denoting by vc thenumber of vertices having incidence number c, it follows from the definition that c≥2 (c − 2) vc = s − 3 + δc1 ,1 . By Tls we denote a tree T s ∈ T s satisfying v2 +δc1 ,1 ≤ 3l−2+s/2 for l ≥ 1 and satisfying v2 = 0 for l = 0. Then Tl s denotes the set of all trees Tls . We indicate the external vertices and internal vertices of the tree by writing Tls (x1 , y2, s , z ) with y2, s = (y2 , . . . , ys ), z = (z 1 , . . . , zr ). s,(12) . The trees iv) We also define for i ≤ s the set of twice rooted trees denoted as Tl s,(12) s,(12) s ∈ Tl are defined exactly as the trees Tl apart from the fact that they Tl have two root vertices x1 , x2 with the properties of ii) above, and s − 2 external vertices. Definition 4. For a tree Tls+2 (x1 , y2, s+2 , z ) we define the reduced tree s Tl,y (x1 , y2 , . . . , yi−1 , yi+1 , . . . , y j−1 , y j+1 , . . . , ys+2 , z i j ) to be the unique tree to be i ,y j obtained from Tls+2 (x1 , y2, s+2 , z ) through the following procedure : i) By taking off the two external vertices yi , y j together with the external lines attached to them. ii) By taking off the internal vertices - if any - which have acquired incidence number c = 1 through the previous process, and by also taking off the lines attached to them. iii) If the process ii) has produced a new vertex of incidence number 1 go back to ii). In the sequel we shall bound the CAS folded with test functions. Here we restrict to test functions of the following form : Let 1 ≤ s ≤ n and τ = τ2,s = (τ2 , . . . , τs ) with 0 < τi , 10 In mathematically straight notation a vertex should be viewed as the image of an element of a discrete set under a mapping from this set into M.

342

C. Kopper, V. F. Müller

ϕτ,y2,s (x2,n ) :=

s 

K (τi , xi , yi )

i=2

n 

1(xi ).

(42)

i=s+1

Here 1(x) = 1 ∀x ∈ M. These test functions are factorized11 . The nonconstant functions are smooth, globally defined and rapidly decreasing on M. The pair τ j , y j determines the width and the center of localisation of the test function. This definition can be generalized by choosing any other subset of s coordinates among x2 , . . . , xn . We also define12 for 2 ≤ j ≤ s, s 

( j)

ϕτ,y2,s (x2,n ) := K (1) (τ j , x j , x1 ; y j )

K (τi , xi , yi )

i=2,i= j

n 

1(xi )

(43)

i=s+1

with K (1) (τ j , x j , x1 ; y j ) = K (τ j , x j , y j ) − K (τ j , x1 , y j ).

(44)

Definition 5. Given τ , y2,s , δ > 0, and a set of internal vertices z = (z 1 , . . . , zr ), and attributing positive parameters tI = {t I |I ∈ I} to the internal lines, the weight factor F(tI , τ ; Tls (x1 , y2,s , z )) of a tree Tls (x1 , y2,s , z ) at scales tI is defined as a product of heat kernels associated with the internal and external lines of the tree. We set   Ct I,δ (I ) K (τ J,δ , J ). (45) F(tI , τ ; Tls (x1 , y2,s , z )) := I ∈I

J ∈J

Here we denote by τ J the entry τi in τ carrying the index of the external coordinate yi in which the external line J ends. For I = {a, b} the notation Ct I (I ) stands for Ct I (a, b). We then also define the integrated weight factor of a tree by
s F(t, τ ; Tl ; x1 , y2,s ) := sup F(tI , τ ; Tls (x1 , y2,s , z )). (46) {t I |I ∈I , ε≤t I ≤t} z

It depends on ε, but note that its limit for ε → 0 exists, and that typically the sup is expected to be taken for the maximal values of t admitted. Therefore we suppress the dependence on ε in the notation. Finally we introduce the shorthand notation for the global weight factor Fs,l (t, τ ; x1 , y2,s ) or more shortly Fs,l (t, τ ) which is defined as follows:

F(t, τ ; Tls ; x1 , y2,s ). (47) Fs,l (t, τ ) ≡ Fs,l (t, τ ; x1 , y2,s ) := Tls ∈Tl s

In complete analogy we define the weight factors and global weight factors for twice s,(12) (12) rooted trees which we denote as F(t, τ ; Tl ; x1 , x2 , y3,s ) resp. Fs,l (t, τ ; x1 , x2 , y3,s ) (12)

or Fs,l (t, τ ). Following the definitions (45)–(47) we also define for t ≥ 1, 
t  t s  (Ct I,δ (I )+ sup Ct  (I ) dt ) K (τ J,δ , J ), F (τ ; Tl (x1 , y2,s , z )) := {t I |I ∈I ,ε≤t I ≤1} I ∈I

J ∈J

(48)


F t (τ ; Tls ; x1 , y2,s ) :=

1

z

F t (τ ; Tls (x1 , y2,s , z )),

(49)

11 The function 1(x) is obtained on integrating K (τ, x, y) over y. This could be used to unify the notation. 12 Note that ϕ ( j) τ,y2,s depends on x 1 which is not indicated.

Renormalization Proof for Massive ϕ44 Theory on Riemannian Manifolds

343

and t Fs,l (τ ) :=

F t (τ ; Tls ; x1 , y2,s ).

(50)

Tls ∈Tl s

For s = 1 we set F1,l (t, τ ) ≡ 1. We give more explicitly the form of F2,l (t, τ ; x, y). It is by definition given through F2,l (t, τ ; x, y) =

F2,l (t, τ ; Tl2 ; x, y) = K (τδ , x, y)

Tl2

+

3l−2

sup

⎡ ⎣




⎤ ⎦ Ct I ,δ (x, z 1 ) . . . Ct I ,δ (z n−1 , z n ) K (τδ , z n , y). n 1

n=1 {t Ii |ε ≤t Ii ≤ t,i=1.··· ,n} 1≤i≤n z i

Using (5) we get F2,l (t, τ ; x, y) =

3l−2

sup

n=0 {t Ii |ε ≤ t Ii ≤ t, i=1.··· ,n}

Cτδ +n1 t I ,δ (x, y) em i

2τ δ

.

(51)

Let us shortly comment on why we are led to introduce tree structures and weight factors in our context. In fact we will establish bounds for the CAS (35) inductively by concluding from the CAS appearing on the r.h.s. of the FE (32) on the CAS appearing on the l.h.s. Assume we have bounds in terms of weight factors of trees for the L’s on the r.h.s. The second contribution on the r.h.s. of (32) then lends itself immediately to reproduce such a bound if the factor, associated in our bound with a line of the tree, is a bound on Ct (x, y), and if the vertices x, y appearing in the bound are integrated over. This is the case for our definition of weight factors since in particular internal vertices are integrated over. For the first contribution on the r.h.s. of (32) we would like to pass from a tree associated with Ln+2 to a tree associated with Ln . This requires the bound to be such that its integration over x, y against the factor Ct (x, y) finally leads to an expression bounded by a tree bound on Ln . This is at the origin of the notion of reduced tree introduced above, where two external points have disappeared. For simplicity we choose the test functions appearing in the weight factors to be heat kernels themselves. These form a sufficiently large set. However, to get inductive control of the local counterterms, we also have to admit the situation where some of the external coordinates are just integrated over all of M. This leads to the general form of the admitted test functions (42).

5. Boundary and Renormalization Conditions From the mathematical point of view the renormalization problem in the FE framework appears as a mixed boundary value problem. The relevant terms are fixed by renormalization conditions at a large value t R of the flow parameter t, all other boundary terms are fixed at the short-distance cutoff t = ε. ε,t To extract the relevant terms - contained in Lε,t 2,l (x 1 , ϕ) and Ll,4 (x 1 , ϕ)- a covariant Taylor expansion with remainder term (A.28), (A.29) of the test function ϕ is used,

344

C. Kopper, V. F. Müller

ε ≤ t: µ,ε,t

ε,t Lε,t 2,l (x 1 , ϕ) = al (x 1 ) ϕ(x 1 ) − fl

Lε,t 4,l (x 1 , ϕ)

=

(x1 ) (∇µ ϕ)(x1 ) µν,ε,t −bl (x1 )(∇µ ∇ν ϕ)(x1 ) + ε,t 2,l (x 1 , ϕ), ε,t cl (x1 ) ϕ2 (x1 ) ϕ3 (x1 )ϕ4 (x1 ) + ε,t 4,l (x 1 , ϕ).

Then the relevant terms appear as

µ,ε,t ε,t ε,t L2,l (x1 , x2 ), fl (x1 ) = σ (x2 , x1 )µ Lε,t al (x1 ) = 2,l (x 1 , x 2 ), x2 x2
1 µν,ε,t (x1 ) = − σ (x2 , x1 )µ σ (x2 , x1 )ν Lε,t bl 2,l (x 1 , x 2 ), 2 x2
Lε,t clε,t (x1 ) = 4,l (x 1 , . . . , x 4 ), x2 ,x3 ,x4

(52) (53)

(54) (55)

ε,t and the ‘remainders’ ε,t 2,l and 4,l have the respective forms

ε,t 2,l (x 1 , ϕ) =





s (s − r )2 ν3 ν2 ν1 x˙12 (r ) x˙12 Lε,t (x , x ) dr (r ) x˙12 (r ) 1 2 2,l 2! x2 0   × ∇ν3 ∇ν2 ∇ν1 ϕ (x12 (r )),

(56)

where s = d(x1 , x2 ) and x12 (r ) is the point on the geodesic segment from x1 to x2 at arc length r ; and ε,t 4,l (x 1 , ϕ) =


s12   ν Lε,t (x , . . . , x ) dr x˙12 (r ) ∇ν ϕ2 (x12 (r )) ϕ3 (x3 )ϕ4 (x4 ) 4 4,l 1 x2 ,x3 ,x4 0
s13   ν dr x˙13 (r ) ∇ν ϕ3 (x13 (r )) ϕ4 (x4 ) +ϕ2 (x1 ) 0 
s14   ν +ϕ2 (x1 ) ϕ3 (x1 ) dr x˙14 (r ) ∇ν ϕ4 (x14 (r )) . (57)




0

Reparametrizing the geodesic segment x12 (r ) = X (ρ), r = d(x1 , x2 )ρ, 0 ≤ ρ ≤ 1, we can rewrite the remainder (56) employing (A.30), ε,t 2,l (x 1 , ϕ)



1 (1 − ρ) 2 ˙ ν3 ε,t 3 = d (x1 , x2 )L2,l (x1 , x2 ) dρ X (ρ) X˙ ν2 (ρ) X˙ ν1 (ρ) 2! d 3 (x1 , x2 ) x2 0   (X (ρ)), where ων(3) (x) = ∇ν3 ∇ν2 ∇ν1 ϕ (x). (58) × ων(3) 3 ν2 ν1 3 ν2 ν1

1) Boundary conditions at t = ε. The bare interaction (25) implies that at t = ε - with Lε ≡ Lε,ε Lεn,l (x1 , . . . xn ) ≡ 0 for n > 4, Lε2,0 ≡ 0,

(59)

˜ 2 , x1 ), b = b µν, ε (x2 ), ˜ 2 , x1 ) − (b) δ(x Lε2,l (x1 , x2 ) = a˜ l (x1 ) δ(x 2 l

(60)

Lε4,l (x1 , . . . x4 )

= (δl,0 λ(x1 ) +

(1 − δl,0 ) clε (x1 ))

˜ 2 , x1 ) δ(x ˜ 3 , x1 ) δ(x ˜ 4 , x1 ). (61) δ(x

Renormalization Proof for Massive ϕ44 Theory on Riemannian Manifolds

345

To cope with the relevant part of the expansion (52) we consider a corresponding bare part µ, ε

ε ε L2, l (x 1 , ϕ) = al (x 1 ) ϕ(x 1 ) − fl

µν, ε

(x1 ) (∇µ ϕ)(x1 ) − bl

(x1 )(∇µ ∇ν ϕ)(x1 ). (62)

The identity   µν µν µν − bl (x)(∇µ ∇ν ϕ)(x) = ∇ν bl (x) · ∇µ ϕ (x) − (b) ϕ(x), b = bl (x)

(63)

suggests to decompose the bare vector coefficient appearing in (62) as µ, ε

fl

µ, ε µν, ε (x1 ) = f˜l (x1 ) + ∇ν bl (x1 ).

(64)

By folding (62) with a test function ϕ we obtain after partial integration13

  ε  1 µ, ε µν,ε ε al (x) + ∇µ f˜l (x) ϕ 2 (x) + bl ϕ(x) L2, (x, ϕ) = (x) ∂µ ϕ(x) · ∂ν ϕ(x) . l 2 x x (65) This agrees in form with the corresponding content of the bare action (25). From the boundary conditions (59)–(61) we deduce ε,ε ε,ε 2,l (x 1 , ϕ) = 0, 4,l (x 1 , ϕ) = 0.

(66)

The renormalization problem is related to the behaviour of the heat kernel at small values of t. Therefore this problem is essentially solved if we can integrate the flow equations up to some finite value t R of t. For shortness we choose units such that t R = 1. We will come to the limit t → ∞ later, see Proposition 3. The positive mass m > 0 only plays a role when this limit is taken. We pose 2) Renormalization conditions at t = t R := 1. 14 alε,1 (x1 ) := alR (x1 ),

µ,ε,1

fl

clε,1 (x1 ) µν,R

µ,R

(x1 ) := fl :=

µν,ε,1

(x1 ), bl

µν,R

(x1 ) := bl

(x1 ), (67)

clR (x1 ),

(68) µ,R

where bl (x) is a smooth symmetric tensor of type (2, 0), fl (x) is a smooth vector and alR (x), clR (x) are smooth scalars on M, all uniformly bounded in the norm (A.32). Typically the renormalization conditions are assumed to be cutoff-independent. To be able to analyse the relation between the bare (inter)action and the renormalization conditions in more detail later on, we shall be more general in also admitting weakly ε-dependent renormalization functions satisfying |∂ε alR (x)| < O(ε−η ), |alR (x)| < const + O(ε1−η ), η ≤ 1/2,

(69)

with analogous expressions for the other renormalization functions. In the particular case of M having constant curvature, i.e. where all sectional curvatures of M have a constant value ρ, a transitive isometry group G acts on M. There 13 Here ϕ is assumed to be smooth, and to decrease sufficiently rapidly if M is noncompact. Apart from the present consideration and from the general analysis of the effective action after (20), we do not introduce test functions against which the first (root) vertex is integrated. 14 The scale t is related to the scale T appearing in the bounds on the heat kernel (13), (14), (15) . R

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are three types of such manifolds: The sphere S 4 with ρ = k 2 and G = S O(5), the flat space R4 with ρ = 0 and G = S O(4) ⊗s R4 , the hyperbolic space H4 with ρ = −k 2 and G = S O0 (4, 1), the subscript denoting the component connected to the identity. Requiring the Schwinger functions to show this symmetry G, results in the following restrictions on the relevant terms: i) alε,t (x), clε,t (x) do not depend on x ∈ M, µ,ε,t µν,ε,t µν,ε,t ii) fl (x) ≡ 0, bl (x) = g µν (x) blε,t , hence ∇ν bl (x) ≡ 0. However, there is a further (dimensionless) parameter alε,t , blε,t , clε,t may depend, in general.

ζ = k 2 /m 2 on which

6. Renormalizability The subsequent proposition is proven for test functions of the form ϕτ2,s ,y2,s (x2,n ), (42). In the end of this section we join some remarks on possible extensions of the class of test functions. By Bose symmetry the bounds stay unaltered if any permuted subset of external coordinates (and not x2 , . . . , xs ) is folded with test functions. Proposition 1. We consider 0 < ε ≤ t ≤ 1 and ε < τi , furthermore 1 ≤ s ≤ n, 2 ≤ i ≤ n, 2 ≤ j ≤ s and 0 ≤ r ≤ 3. We consider test functions either of the form ( j) ( j) ϕτ2,s ,y2,s (x2,n ) or ϕτ2,s ,y2,s (x2,n ), which are also denoted in shorthand as ϕs resp. ϕs . In all subsequent bounds we understand Pl to denote a polynomial of degree ≤ sup(l, 0) - each time it appears possibly a new one - with nonnegative coefficients which may depend on l, n, δ 15 , on supM |λ(x)|, as well as on k 2 , κ 2 and the bounds on the first and second covariant derivatives of the curvature tensor (see (13)–(15)), but not on ε, t, m and τ . Constants O(1) in the subsequent proof are to be understood in the same way. By (t, τ ) we denote inf{τ2 , . . . , τs , t}, by (t, τ )i we denote inf{τ2 , . . . , τ i , . . . , τs , t}. Then we claim the following bounds - using the shorthand (47): | Lε,t n,l (x 1 , ϕτ,y2,s )| ≤ t

n−4 2

Pl log(t, τ )−1 Fs,l (t, τ ), n ≥ 4,

(r ) |Lε,t n,l (x 1 , E (i) ϕτ,y2,s )| ≤ | ω (x 1 )| t

n+r −4 2

(r ) (r ) | Lε,t n,l (x 1 , E (i) ϕτ,y2,s )| ≤ | ω (x 1 )| t

n+r −4 2

(r )

Pl log(t, τ )i−1 Fs,l (t, τ ), n > 4, r > 0, (71) Pl−1 log(t, τ )i−1 Fs,l (t, τ ),

(72)

n = 2, r = 3 or n = 4, r > 0, ≤ (t, τ )−1 Pl−1 log(t, τ )−1 F2,l (t, τ ),

(73)

−1/2

(74)

| Lε,t 2,l (x 1 , ϕτ,y2 )| ε,t | L2,l (x1 , E (2) ϕτ,y2 )| (2) | Lε,t 2,l (x 1 , E (2) ϕτ,y2 )| ( j)

(70)

≤ | ω(x1 )| (t, τ ) ≤ |ω

| Lε,t n,l (x 1 , ϕτ,y2,s )| ≤ t

(2)

n−4 2

Pl−1 log(t, τ )

−1

F2,l (t, τ ),

−1

(x1 )| Pl−1 log(t, τ ) F2,l (t, τ ), 1/2 t Pl log(t, τ )−1 Fs,l (t, τ ), n > 2, τj

t 1/2 (2) (t, τ )−1 Pl−1 log(t, τ )−1 F2,l (t, τ ), | Lε,t 2,l (x 1 , ϕτ,y2 )| ≤ ( ) τ

(r ) ε,t Ln,l (x1 , x2 , ϕ)| ≤ | ω(r ) (x1 )| t | F(12)

n−1 2

(12) Pl−1 log(t, τ )−1 Fs,l (t, τ ).

r = 0, 1, 2 and | ω

(0)

(75) (76) (77) (78)

(x1 )| ≡ 1.

15 We suppose that δ > 0 may be chosen arbitrarily small in the definition of F . The constants in P then l

depend on the choice of δ.

Renormalization Proof for Massive ϕ44 Theory on Riemannian Manifolds

347

Remark. The full series of the previous bounds is needed to close the inductive argument in the subsequent proof. The reader who only wants to know what the bounds are can restrict to (70), (73). Proof. The bounds stated in the proposition are proven inductively using the (standard) inductive scheme which proceeds upwards in n + 2l, and for given n + 2l upwards in l. Thus the induction starts with the pair (4, 0). For this term the r.h.s. of the FE vanishes ˜ ˜ ˜ so that Lε,t 4,0 (x 1 , . . . , x 4 ) = λ(x 1 ) δ(x 2 , x 1 ) δ(x 3 , x 1 ) δ(x 4 , x 1 ) which is compatible with our bounds (after folding with suitable ϕ). Generally it is important to note that the boundary conditions are compatible with the bounds of the proposition. (r ) (0) We will first derive bounds for the derivatives ∂t Lε,t n,l (x 1 , E (i) ϕs ), where E (i) ≡ 1, ( j)

and ∂t Lε,t n,l (x 1 , ϕs ). Afterwards these bounds are integrated over w.r.t. t.

A) We start considering the case r = 0 and test functions ϕs . A1) Here we first treat the first term on the r.h.s. of (32),
R1 := ϕs (x2,n ) Ct (x, y) Lε,t xn , x, y). n+2,l−1 ( x2,n ,x,y

We may rewrite this expression as

R1 = ϕs (x2,n ) Ct/2 (x, v) Ct/2 (v, y) Lε,t xn , x, y) n+2,l−1 ( v x2 ,...,xn ,x,y
= Lε,t n+2,l−1 (x 1 , ϕs × C t/2 (·, v) × C t/2 (v, ·)). v

Applying the induction hypothesis to Lε,t n+2,l−1 (x 1 , ϕs × C t/2 (·, v) × C t/2 (v, ·)) we thus obtain the bound



n+2−4 t t s+2 |R1 | ≤ t 2 Pl−1 log(t, τ )−1 F(t, τ, , ; Tl−1 (x1 , y2,s , v, v, z )). 2 2 v z s+2 Tl−1 (x1 ,y2,s ,v,v)

(79) For any contribution to (79) we denote by z  , z  the vertices in the respective tree s+2 (x , y , v, v,  Tl−1 z ) to which the test functions Ct/2 (·, v), Ct/2 (v, ·) are attached. 1 2,s Interchanging z (see (46)) and v and performing the integral over v using (5), (19), we get a contribution
Ct/2 (z  , v) Ct/2 (v, z  ) = Ct (z  , z  ) ≤ O(1) t −2 . (80) v

Using this bound we can majorize

v

s+2 ; x , y , v, v) by F(t, τ, 2t , 2t ; Tl−1 1 2,s

O(1) t −2 F(t, τ ; Tls ; x1 , y2,s ), s+2 by taking away the two external where the tree Tls is the reduced tree obtained from Tl−1 lines ending in v, see Definition 4. The reduction process for each tree fixes uniquely s+2 . Note that the elimination the set of internal vertices of Tls in terms of those of Tl−1 of vertices of incidence number 1 together with their adjacent line is justified by the fact that z  Ct I ,δl (z, z  ) ≤ 1. Note also that in the tree Tls the number v2 of vertices of

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C. Kopper, V. F. Müller

s+2 , so that T s incidence number 2 may have increased by at most 2, as compared to Tl−1 l s is indeed an element from Tl . We keep track of this lower index l in the tree basically to show that the number of vertices always stays finite (in fact does not grow faster than linearly in l for n fixed). The final bound for the first term on the r.h.s. of the FE is thus

n−6 |R1 | ≤ t 2 Pl−1 log(t, τ )−1 F(t, τ ; Tls ; x1 , y2,s ), (81) Tls (x1 ,y2,s )

where constants have been absorbed in Pl−1 log. A2) We now consider the second term in (32) for i) n > 4. Picking a generic term from the symmetrized sum and arguing as in A1) we have to bound
ε,t ϕs (x2,n ) Ct (x, y) Lε,t R2 := n 1 +1,l1 (x 1 , . . . , x n 1 , x) Ln 2 +1,l2 (y, x n 1 +1 , . . . , x n ) x2,n ,x,y

which we rewrite similarly as in A1),

ϕs (x2,n ) Ct/2 (x, v) Ct/2 (v, y) Lε,t R2 = n 1 +1,l1 (x 1 , . . . , x) v

x2 ,...,xn ,x,y ε,t ×Ln 2 +1,l2 (y, . . . , xn ).

(82)

Denoting ϕs 1 (x2,n 1 )

=

n1  r =2

we identify the two terms
x2 ,...,xn 1 ,x

and


xn 1 +1 ,...,xn ,y

ϕr (xr ), ϕs2 (xn 1 +1,n−1 ) =

n−1 

ϕr (xr )

r =n 1 +1

ϕs 1 (x2,n 1 ) Ct/2 (x, v) Lε,t n 1 +1,l1 (x 1 , . . . , x)

ϕs2 (xn 1 +1,n−1 ) Ct/2 (v, y) Lε,t n 2 +1,l2 (y, . . . , x n )

and thus write (82) as

ε,t   R2 = Lε,t n 1 +1,l1 (x 1 , ϕs1 × C t/2 (·, v))Ln 2 +1,l2 (x n , C t/2 (v, ·) × ϕs2 )ϕn (x n ). (83) xn

v

On applying the induction hypothesis to both terms in (83), restricting first to n 1 , n 2 > 1, we obtain the bound



n+2−8 F(t, τ  , t/2; Tls11 +1 ; x1 , y2 , . . . , ys1 , v) · |R2 | ≤ t 2 Pl1 log(t, τ )−1 xn

v

s +1 1

s +1 2

Tl 1 ,Tl 2

·Pl2 log(t, τ )−1 F(t, τ  , t/2; Tls22 +1 ; xn , v, ys1 +1 , . . . . . . , ys(n) )ϕn (xn ).

(84)

Renormalization Proof for Massive ϕ44 Theory on Riemannian Manifolds

349

Here s(n) = s if s < n, and s(n) = s − 1 if s = n. Interchanging the integral over v with the sum over trees we obtain


n−6 |R2 | ≤ t 2 Pl log(t, τ )−1 F(t, τ2,s ; Tls ; x1 , y2 . . . , ys ) (85) s +1 1

s +1 2

Tls (Tl 1 ,Tl 2 )(x1 ,y2 ...,ys )

xn

with the following explanations : Any contribution in the sum over trees Tls (Tls11 +1 , Tls22 +1 ) 

(x1 , y2 , . . . , ys , z ) is obtained from Tls11 +1 (x1 , y2 . . . , ys1 , v, z ) and Tls22 +1 (xn , v, ys1 +1 , 

. . . , ys , z ) by joining these two trees via the lines going from the vertices z  and z  to v, where z  and z  are the vertices attached to v in the two trees. These two lines have parameters t/2. We use the equality
Ct/2 (z  , v) Ct/2 (v, z  ) = Ct (z  , z  ) (86) v

so that the new internal line has a t-parameter in the interval [ε, t] over which the sup is taken in the definition of F.16 When performing the integral over xn in (84) we remember that xn is the root vertex of Tls22 +1 (xn , v, ys1 +1 , . . . . . . , ys , z ). If s = n we have ϕn (xn ) = Cτn (xn , yn ), and xn becomes an internal vertex, and yn an external vertex, of Tls . If s < n, then ϕn (xn ) ≡ 1, and the vertex xn becomes an internal vertex of Tls unless c(xn ) = 1. In this last case integration over xn together with (4) permits to take away the vertex xn and the internal line joining it to an internal vertex z j of the tree Tls22 +1 . 17 If (originally) c(z j ) = 2 this elimination process continues. Thus the final bound for R2 , and hence for the second term in (32) is the same as (81), apart from changing Pl−1 log(t, τ )−1 → Pl log(t, τ )−1 . Note that this bound is established in the same way ε,t if Lε,t n 1 +1,l1 (x 1 , . . . , x) or Ln 2 +1,l2 (y, . . . , x n ) are two-point functions: In this case the parameter τ appearing in (73) equals t/2 so that (t, τ )−1 can be replaced by 2/t. Taking both contributions from the r.h.s. of the FE together and summing over all trees we have established the bounds |∂t Lε,t n,l (x 1 , ϕs )| ≤ t

n−6 2

Pl log(t, τ )−1 Fs,l (t, τ ), n > 4.

(87)

ii) n ≤ 4. In this case we have n 1 + 1 = 2 and/or n 2 + 1 = 2. Thus at least one of the polynomials Pli log(t, τ )−1 appearing in the bounds (84) can by induction be replaced by Pli −1 log t −1 . Therefore proceeding exactly as in the previous case we obtain the bounds |∂t Lε,t n,l (x 1 , ϕs )| ≤ t

n−6 2

Pl−1 log(t, τ )−1 Fs,l (t, τ ), n ≤ 4.

(88)

B) r > 0, cf.(40). For the first term on the r.h.s. of the flow equation resulting from (32) the bounds for 1 ≤ r ≤ 3 are proven exactly as in A1). For the second term we proceed similarly as in A2). We pick a generic term on the r.h.s.,
ϕs (x2,n )Ct (x, y) E(xk , x1 ; ω(r ) ) x2,n ,x,y ε,t ×Lε,t n 1 +1,l1 (x 1 , . . . , x n 1 , x)Ln 2 +1,l2 (y, x n 1 +1 , . . . , x n ).

16 We can of course majorize C (z  , z  ) ≤ O(1) C (z  , z  ). t tδ 17 If x has turned into a vertex of incidence number 2 for T s , the bound v + δ n 2 c1 ,1 ≤ 3l − 2 + s/2 is easily l

verified.

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In the case where k ≤ n 1 the proof is the same as for r = 0, up to inserting the modified induction hypothesis for
x2,n 1 ,x

ϕs1 (x2,n 1 ) Ct/2 (x, v) E(xk , x1 ; ω(r ) )Lε,t n 1 +1,l1 (x 1 , . . . , x n 1 , x).

If k > n 1 we assume without restriction k = n and proceed again as in A2) to obtain the bound



n−6 t 2 Pl1 log(t, τ )−1 |E(xn , x1 ; ω(r ) )| F(t, τ, t/2; Tls11 +1 ; x1 , y2 , n xn

v

s +1 1

s +1 2

Tl 1 ,Tl 2

s2 +1 . . . , ys1 , v) · Pl2 log(t, τ )−1 ; xn , v, ys1 +1 , . . . . . . , ys(n) )ϕn (xn ). n F(t, τ, t/2; Tl2 (89)

Observing the inequality (39) together with d(xn , x1 ) ≤

q

d(va , va−1 ),

(90)

a=1

where {va } are the positions of the internal vertices in the tree Tls (Tls11 +1 , Tls22 +1 ) defined as in A2), on the path joining x1 = v0 and xn = vq , we then use the bound (14). The cases s = n and s < n are treated as in A2), using once more the bound (14). (r ) The previous reasoning holds a fortiori for ∂t F(12) Lε,t n,l (x 1 , x 2 , ϕ), since in these cases (r )

we have | F(12) | ≤ d 3 (x1 , x2 )| ω(r ) (x1 )|, | ω(0) (x1 ) | ≡ 1. Here then x2 takes the role of xn . Proceeding as before we thus obtain for r = 0 (after absorbing again all constants in Pl ) (r )

n+r −6 2

(r )

(r )

(r ) |∂t Lε,t n,l (x 1 , E (i) ϕs )| ≤ |ω (x 1 )|t

Pl log(t, τ )i−1 Fs,l (t, τ ), n > 4,

(91)

r −2 2

Pl−1 log(t, τ )i−1 Fs,l (t, τ ),

(92)

(r ) |∂t Lε,t 2,l (x 1 , E (2) ϕ2 )| ≤ | ω (x 1 )| t

r −4 2

Pl−2 log t −1 F2,l (t, τ ),

(93)

(r ) ε,t | ∂t F(12) Ln,l (x1 , x2 , ϕ)| ≤ | ω(r ) (x1 )| t

n−3 2

(12) Pl−1 log(t, τ )−1 Fs,l (t, τ ).

(94)

(r ) |∂t Lε,t 4,l (x 1 , E (i) ϕs )| ≤ |ω (x 1 )| t

In (94) τ stands for (τ3 , . . . , τs ), furthermore r = 0, 1, 2 and | ω(0) (x1 )| ≡ 1. The bounds for (54)–(55), 1 1 1 Pl−1 log , | ∂t alε,t (x1 ) | ≤ t −2 Pl−1 log , t t t 1 µ,ε,t −3/2 | ∂t fl (x1 ) ωµ (x1 )| ≤ | ω(x1 )| t Pl−2 log , t 1 1 µν,ε,t (2) | ∂t bl (x1 ) ωµν (x1 ) | ≤ | ω(2) (x1 )| Pl−2 log t t | ∂t clε,t (x1 ) | ≤

(95) (96) (97)

Renormalization Proof for Massive ϕ44 Theory on Riemannian Manifolds

351

are obtained on restricting the previous considerations to the case s = 1, in which all external coordinates are integrated over, e.g.


1 2

∂t alε,t (x1 ) =

x2 ,x,y

 Ct (x, y) Lε,t 4,l−1 (x 1 , x 2 , x, y)



 ε,t L2,l1 (x1 , x) Lε,t (y, x ) − 2 2,l2

sym

l1 +l2 =l

 .

The polynomials appearing in (96), (97) are of degree ≤ l − 2, corresponding to the fact ε,t and Llε,t Lε,t with that on the r.h.s. of the FE (32) for these terms, there appear Ll−1,4 1 ,2 l2 ,2 (r )

insertions E (2) , r = 1, 2. Both are bounded inductively by polynomials of total degree ≤ sup(l − 2, 0). ( j) C) We come to the bound on ∂t Lε,t n,l (x 1 , ϕs ), cf. (76). As compared to B) the only case which requires new analysis is the bound on the second term from the r.h.s. of the FE (32), in the case j > s1 . Then we assume without restriction, similarly as in B), that j = s. The term to be bounded corresponding to (84) is then
n−6 −1 2 t Pl1 log(t, τ ) F(t, τ  , t/2; Tls11 +1 ; x1 , y2 , . . . , ys1 , v) · Pl2 log(t, τ )−1 v s +1 s +1 Tl 1 ,Tl 2




1

×

F(t, τ xs



2

; xs , v, ys1 +1 , . . . . . . , ys−1 )|K (1) (τs , xs , x1 ; ys )|.

, t/2; Tls22 +1

(98)

To bound this expression we telescope the difference K (1) (τs , xs , x1 ; ys ), cf.(44), along the tree Tls (Tls11 +1 , Tls22 +1 ) obtained from the two initial trees by joining them via v as in A2) and proceeding similarly as in (90). We then have to bound expressions of the type Ct I,δ (va−1 , va ) |K (τs , va , ys ) − K (τs , va−1 , ys )|, where va−1 , va are adjacent internal vertices in Tls (Tls11 +1 , Tls22 +1 ) on the unique path from x1 to ys . Making use of the covariant Schlömilch formula (A.28)–(A.31) for the difference K (1) (τs , va , va−1 ; ys ), we obtain
s |K (τs , va , ys ) − K (τs , va−1 , ys )| ≤ dr | ∇(1) K (τs , z(r ), ys ) | 0




1

= d(va−1 , va ) 0

dρ | ∇(1) K (τs , va−1,a (ρ), ys ) |

d(va−1 , va ) ≤ O(1) √ τs




1

dρ K (τs,δ  , va−1,a (ρ), ys ),

(99)

0

where z(r ) = va−1,a (ρ) lies at distance r = ρ d(va−1 , va ), 0 ≤ ρ ≤ 1, from va−1 on the reparametrized geodesic segment from va−1 to va . The last inequality results from (15). Introducing for 3δ < 1 :

b=2

1 + 3δ , 1 − 3δ

(100)

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we then bound, with δ  > 0 to be fixed later,  ≤

Ct I,δ (va−1 , va ) | K (τs , va , ys ) − K (τs , va−1 , ys )| ≤ Ct I,δ (va−1 , va ) K (τs , va , ys ) + Ct I,δ (va−1 , va ) K (τs , va−1 , ys ), b t ≥ δ  τs 1 O(1) Ct I,2δ (va−1 , va ) ( τt Is )1/2 0 dρ K (τs,δ  , va−1,a (ρ), ys ) , b t < δ  τs . (101)

The last line follows using (99) and absorbing the factor d(va−1 , va ) in Ct I,δ (va−1 , va ) with the aid of (14), by changing δ to 2δ . The last line in (101) has to be bounded in such a way as to reproduce a contribution compatible with the induction hypothesis. To this end we use the (upper) bound (13),

d 2 (v1,2 (ρ), y) 1 1 d 2 (v1 , v2 ) . − Ct2δ (v1 , v2 ) K (τδ  , v1,2 (ρ), y) ≤ O(1) 2 2 exp − t τ 4t (1 + 3δ) 4τ (1 + δ  )2 Noting that d(v1 , v2 ) = d(v1 , v1,2 (ρ)) + d(v1,2 (ρ), v2 ) we deduce 1 1 2 d (v1 , v2 ) + d 2 (v1,2 (ρ), y) ≥  d 2 (v1 , v1,2 (ρ)) + d 2 (v1,2 (ρ), y) ≥ δ δ 2 1  1 d(v1 , v1,2 (ρ)) + d(v1,2 (ρ), y) ≥ d 2 (v1 , y). 1 + δ 1 + δ Hence, observing (100), we find for b t < δ  τ , d 2 (v1 , v2 ) d 2 (v1 , v2 ) d 2 (v1,2 (ρ), y) d 2 (v1 , v2 ) d 2 (v1,2 (ρ), y) + + + = 4t (1 + 3δ) 4τ (1 + δ  )2 8t 4bt 4τ (1 + δ  )2 2 2 d (v1 , y) d (v1 , v2 ) + ≥ . 8t 4τ (1 + δ  )3 With the aid of the lower bound (13) we then arrive at
1 Ct I,2δ (v1 , v2 ) dρ K (τs,δ  , v1,2(ρ), ys ) ≤ O(1) C2t I ,δ (v1 , v2 )K (τs (1 + δ  )4 , v1 , y). 0

(102) Taking into account (5) and choosing δ  such that (1 + δ  )4 = 1 + δ, i.e. δ  = δ/4 + O(δ 2 ), we may thus bound the last line in (101) by
t 1/2 O(1) ( ) K (τs,δ , va−1 , ys ) Ct I , δ (va−1 , v) Ct I , δ (v, va ), b t < δ  τ. (103) τs v Note that the addition of a new internal vertex v of incidence number 2 in (103) is compatible with the inequality v2 + δc1 ,1 ≤ 3l − 2 + s/2. Using these bounds and going back to (98) we realize that the two terms in the first line of (101) - case b t ≥ δ  τs - correspond to two new trees of type Tl s , where in comparison to Tls (Tls11 +1 , Tls22 +1 ) the incidence number of the vertex va−1 or va has increased by one unit. Similarly (103) - case b t < δ  τs - corresponds to a new tree where the incidence number of the vertex va−1 has increased by one unit. In (98) an integral over xs is performed. If in the new tree

Renormalization Proof for Massive ϕ44 Theory on Riemannian Manifolds

353

i) xs has c(xs ) > 1,18 then xs takes the role of an internal vertex of the new tree, ii) xs has c(xs ) = 1 we integrate over xs using (4) so that the vertex xs disappears. As a consequence of the previous bounds, on replacing again s → j in (101),(103), we thus obtain for n > 4 - see also A2) ii) for (105), (106)  n−6 t 2 Pl log(t, τ )−1 Fs,l (t, τ ), b t ≥ δ  τ j ( j) ε,t , (104) |∂t Ln,l (x1 , ϕτ2,s ,y2,s )| ≤ n−6 t 1/2 t 2 ( τ j ) Pl log(t, τ )−1 Fs,l (t, τ ), b t < δ  τ j  t −1 Pl−1 log(t, τ )−1 Fs,l (t, τ ), b t ≥ δ  τ j ( j) ε,t |∂t L4,l (x1 , ϕτ2,s ,y2,s )| ≤ t −1 ( t )1/2 P log(t, τ )−1 F (t, τ ), b t < δ  τ , (105) l−1 s,l j τj  −2 −1  t Pl−1 log t F2,l (t, τ ), b t ≥ δ τ2 (2) |∂t Lε,t (106) −1 F (t, τ ), b t < δ  τ . 2,l (x 1 , ϕτ,y2 )| ≤ t −2 ( t )1/2 P l−1 log t 2,l 2 τ D) From the bounds on the derivatives ∂t Lε,t n,l we then verify the induction hypothesis on integrating over t. In all cases we need the bound Fs,l (t  , τ ) ≤ Fs,l (t, τ ) for t  ≤ t,

(107)

which follows directly from the definition (46). a) In the cases n + r > 4 we have, due to the boundary conditions encoded in the form of (25),
t  ε,t Ln,l (x1 , ϕ) = dt  ∂t  Lε,t n,l (x 1 , ϕ). ε

Then we get from (87), (91)–(93), due to (107), | Lε,t n,l (x 1 , ϕs )| ≤ t

n−4 2

Pl log(t, τ )−1 Fs,l (t, τ ),

(r )

(r ) |Lε,t n,l (x 1 , E (k) ϕs )| ≤ | ω (x 1 )| t

n+r −4 2

|Lε,t 4,l (x 1 ,

(r ) E (k) ϕs )|

≤ | ω(r ) (x1 )| t

r 2

|Lε,t 2,l (x 1 ,

(3) E (2) ϕs )|

(3)

1 2

≤ |ω

(108)

Pl log(t, τ )−1 k Fs,l (t, τ ), n > 4,

Pl−1 log(t, τ )−1 k

(x1 )| t Pl−1 log t

−1

Fs,l (t, τ ), r > 0,

Fs,l (t, τ ),

(109) (110) (111)

which proves the proposition for these cases. b) Similarly, for n ≥ 4 the boundary conditions (25) imply that
t  ( j) ( j) ε,t dt  ∂t  Lε,t Ln,l (x1 , ϕs ) = n,l (x 1 , ϕs ), ε

and we then obtain from (104), (105) together with (107), 1/2 n−4 t ( j) 2 (x , ϕ )| ≤ t Pl log(t, τ )−1 Fs,l (t, τ ). |Lε,t 1 s n,l τj

(112)

t δ  τ /b t We note that for b t ≥ δ  τ j the integral ε dt  has to be split into ε j dt  + δ  τ j /b dt  , and that in the case n = 4 the polynomial in logarithms may increase in degree by 18 Remember that the vertex x in T s2 +1 ) is a root vertex by construction. s l2

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C. Kopper, V. F. Müller

one unit due to the logarithmically divergent t-integral, see (115)–(117) below for more details. c) In the case n = 4, r = 0 we start from the decomposition (53), ε,t ε,t Lε,t 4,l (x 1 , ϕ) = cl (x 1 ) ϕ(x 1 , x 1 , x 1 ) + 4,l (x 1 , ϕ).

On integrating the bound for clε,t (x1 ), (95), from t to 1 and using the boundary condition (68) we get | clε,t (x1 ) | ≤ Pl log t −1 .

(113)

Taking together (95) and (88) we verify −1 |∂t ε,t Pl−1 log(t, τ )−1 Fs,l (t, τ ). 4,l (x 1 , ϕs )| ≤ t

A sharper bound for ∂t ε,t 4,l (x 1 , ϕs ), which when integrated over t ≥ ε stays uniformly bounded in ε, is obtained as follows. In the case s = 419 we decompose the test function ϕ4 (x2 , x3 , x4 ) :=

4 

K (τi , xi , yi ) = ϕ4 (x1 , x1 , x1 ) + ψ(x2 , x3 , x4 ),

i=2

ψ(x2 , x3 , x4 ) =

4  i−1

4 

K (τ f , x1 , y f ) K (1) (τi , xi , x1 ; yi )

i=2 f =2

K (τ j , x j , y j ).

j=i+1

ε,t Then ε,t 4,l (x 1 , ϕ4 ) = L4,l (x 1 , ψ), and hence the FE (32) provides

∂t ε,t 4,l (x 1 , ϕ4 ) =

1 2 −




 ψ(x2 , x3 , x4 )Ct (x, y) Lε,t 6,l−1 (x 1 , . . . , x 4 , x, y)

x2 ,x3 ,x4 ,x,y

 ε,t L4,l1 (x1 , x2 , x3 , x) Lε,t 2,l2 (y, x 4 )

l1 +l2 =l

ε,t +Lε,t 2,l1 (x 1 , x) L4,l2 (y, x 2 , x 3 , x 4 )

 sym

 .

(114) ( j)

The r.h.s. is a sum over expressions of the same form as the one for ∂t Lε,t 4,l (x 1 , ϕτ2,s ,y2,s ) in part C. Setting τ = inf j {τ j } we obtain, in the same way as there, the bound  −1 t Pl−1 log(t, τ )−1 Fs,l (t, τ ), b t ≥ δ  τ, (115) |∂t ε,t (x , ϕ )| ≤ 4,l 1 s t −1 ( τt )1/2 Pl−1 log(t, τ )−1 Fs,l (t, τ ), b t < δ  τ. Using the boundary condition (66) we integrate (115) over t. This gives for b t < δ  τ (and ε sufficiently small)  
t  t 1/2   dt  ∂t  ε,t (x , ϕ ) Pl−1 log t −1 Fs,l (t, τ ), (116)  1 s  ≤ ( ) 4,l τ ε 19 In this case ϕ (x ) = K (τ , x , y ), 1 ≤ i ≤ 4. If ϕ = 1 for some i, the corresponding contribution to i i i i i i the subsequent sum over i vanishes.

Renormalization Proof for Massive ϕ44 Theory on Riemannian Manifolds

355

and for b t > δ  τ , in which case we may have t > τ or t < τ , 
δ  τ/b 
t   
t           ε,t    ε,t  ≤ + dt  ∂t  ε,t (x , ϕ ) dt ∂  (x , ϕ ) dt ∂  (x , ϕ )       1 s 1 s 1 s t t 4,l 4,l 4,l ε ε δ  τ/b   

δ 1/2 b b ≤ Fs,l (t, τ ). (117) Pl−1 log  + Pl log  b δτ δτ Hence, absorbing powers of log(δ  /b) in the coefficients of Pl log as usual, −1 |ε,t Fs,l (t, τ ). 4,l (x 1 , ϕs )| ≤ Pl log τ

(118)

From (53), (113), (116) and (118) we obtain −1 |Lε,t Fs,l (t, τ ). 4,l (x 1 , ϕs )| ≤ Pl log(t, τ )

d) In the case n = 2 we have the decomposition (52). In addition to the bounds (95)–(97) to be integrated from 1 to t ≤ 1, we need for ∂t ε,t 2,l (x 1 , ϕ), (56), a bound, which upon integration from ε to t becomes a uniformly bounded function on ε ≥ 0. To this end we use the form (58) and choose the test function ϕ(x2 ) = K (τ, x2 , y2 ). Taking into account the bound (94) for n = 2, r = 0 together with (A.31) yields | ∂t ε,t 2,l (x 1 , ϕ)| ≤




(12)

1

x2 1

t − 2 Pl−1 log t −1 F2,l (t) 3

≤ t − 2 τ − 2 Pl−1 log t −1







1

dρ 0

(12)

x2




F2,l (t)

(1 − ρ)2 3 | ∇(1) K (τ, X (ρ), y2 )| 2! 1

0

dρ K (τδ  , X (ρ), y2 ),

where (15) has been used. By definition we have (12) F2,l (t) = F2,l (t; x1 , x2 ) =

F2,l (t; Tl2,(12) ; x1 , x2 )

2,(12)

Tl

= =

3l−2




sup

[

sup

Cn1 t Iν ,δ (x1 , x2 ),

n=1 {t Iν |ε ≤ t Ii ≤ t, i=1.··· ,n} 1≤ν≤n z ν 3l−2

n=1 {t Iν |ε ≤ t Iν ≤ t, ν=1.··· ,n}

] Ct I1 ,δ (x1 , z 1 ) . . . Ct In ,δ (z n , x2 )

where we used (5) and (51). We then proceed similarly as in and after (101). Setting N = 3l − 2 we bound for N b t < δ  τ and for n ≤ N as in (102),
Cn1 t Iν ,δ (x1 , x2 )

0

1

dρ K (τδ  , X (ρ), y2 ) ≤ O(1)C2 n1 t Iν ,δ (x1 , x2 )K (τδ , x1 , y2 ) (119)

so that, observing (4), −3/2 −1/2 | ∂t ε,t t Pl−1 log t −1 K (τδ , x1 , y2 ), 2,l (x 1 , ϕ)| ≤ τ

N b t < δ  τ. (120)

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C. Kopper, V. F. Müller

To verify the induction hypothesis (73) we resort to the decomposition (52) and denote the sum of the first, second and third terms there by Lε,t 2,l (x 1 , ϕ)r el . Integrating the corresponding bounds (95)–(97) from 1 to t, and using again (15) gives | Lε,t 2,l (x 1 , ϕ)r el | <

1 t

Pl−1 log t −1 +

1 1

(tτ ) 2

Pl−2 log t −1

 1 + Pl−1 log t −1 K (τδ , x1 , y2 ). τ

(121)

Integrating the remainder (120) from (small) ε with vanishing initial condition (66) to t < δ  τ/(bN ) leads to −3/2 1/2 | ε,t t Pl−1 log t −1 K (τδ , x1 , y2 ). 2,l (x 1 , ϕ)| ≤ τ

(122)

By way of (52) we obtain from the bounds (88) and (95)–(97) the bound for N b t ≥ δ  τ −2 | ∂t ε,t Pl−1 log(t, τ )−1 F2,l (t, τ ) 2,l (x 1 , ϕ)| ≤ t 2 

+ t −2 Pl−1 log t −1 + t

j−4 2

 τ − j/2 Pl−2 log t −1 K (τδ , x1 , y2 ).

j=1

(123) Hence, integration and majorization, again observing both τ > t and t < τ , gives for t > δ  τ/(bN ), | ε,t 2,l

1 bN bN −1 Pl−1 log  + θ (t − τ ) Pl−1 log τ F2,l (t, τ ) (x1 , ϕ)| ≤ δτ δτ τ



bN bN 1/2 1 bN bN bN + P + + P log P log log l−1 l−2 l−1 δτ δτ δτ 2 δτ τ δτ (124) ×K (τδ , x1 , y2 ).

From (121), (122) and (124), absorbing constants as usual in P log, we then get20 −1 | Lε,t Pl−1 log(t, τ )−1 F2,l (t, τ ) 2,l (x 1 , ϕ) | ≤ (t, τ )

(125)

in accord with (73). 20 The bound (124) diverges linearly with δ  , whereas in (117) the divergence was only logarithmic. This indicates rapid growth since the bounds then behave as (δ  )−l , a factor of (δ  )−1 being produced per loop order. Without trying at all to optimize constants, we still note that it is possible to choose for this case δ = 2 in K (τδ , x1 , y2 ) and bound the two point function inductively by F2,l (t, 3τ ) without changing the bounds on the other functions. The only place in the proof where there is a modification due to this factor is in part A2). But here the value of τ appearing is t/2, see (86), and 3t/2 can be accommodated for in the proof by introducing a new vertex of incidence number 2 while respecting the bound on the number of those vertices. A value δ = 2 then gives for suitable choice of b the value b/δ   6.

Renormalization Proof for Massive ϕ44 Theory on Riemannian Manifolds

357

(r )

To establish the bounds on Lε,t 2,l (x 1 , E (2) ϕ), r = 1, 2, we expand the respective test functions as follows, employing (A.30) and using the notations (54), (41), µ,ε,t

µν,ε,t

(1) Lε,t (x1 ) ωµ (x1 ) + 2 bl (x1 ) ωµ (x1 ) (∇ν ϕ)(x1 ) 2,l (x 1 , E (2) ϕ) = ϕ(x 1 ) f l

1 (1 − ρ) ˙ µ (1) + F(12) Lε,t X (ρ) X˙ ν (ρ)(∇µ ∇ν ϕ)(X (ρ)), 2,l (x 1 , x 2 ) dρ 2 d (x1 , x2 ) 0 x2 (126) µν,ε,t

(2)

(2) (x1 ) ωµν (x1 ) Lε,t 2,l (x 1 , E (2) ϕ) = −2 ϕ(x 1 ) bl

1 1 (2) + F(12) Lε,t dρ X˙ µ (ρ) (∇µ ϕ)(X (ρ)). 2,l (x 1 , x 2 ) d(x1 , x2 ) 0 x2

(127)

The local, i.e. relevant terms have already been dealt with in (96), (97), and the remainders are treated as ε,t 2,l (x 1 , ϕ); one obtains (1)

(1) −1/2 Pl−1 log(t, τ )−1 F2,l (t, τ ), | Lε,t 2,l (x 1 , E (2) ϕ2 )| ≤ | ω (x 1 )| (t, τ ) (2)

(2) −1 F2,l (t, τ ). | Lε,t 2,l (x 1 , E (2) ϕ2 )| ≤ |ω (x 1 )| Pl−1 log(t, τ ) (2)

Finally, we realize that Lε,t 2,l (x 1 , ϕ2 ) equals the r.h.s. of (52) without its first term. Proceeding again similarly as before - see (121), (122) and (124) - provides 1/2   t (2) ε,t |L2,l x1 , ϕ2 | ≤ (t, τ )−1 Pl−1 log(t, τ )−1 F2,l (t, τ ). τ This ends the proof of Proposition 1.

 

The behaviour of the CAS upon removing the UV cutoff, i.e. ε  0, follows from Proposition 2. Let ε be (sufficiently) small. With the notations, conventions and the same class of renormalization conditions as in Proposition 1 we have the bounds −2 | ∂ε Lε,t Pl log ε−1 t n,l (x 1 , ϕτ2,s ,y2,s )| ≤ ε 1

| ∂ε Lε,t n,l (x 1 ,

(r ) E (i) ϕτ2,s ,y2,s )|

≤ε

( j)

− 21

Pl log ε

−1

n−5 2

Fs,l (t, τ ),

(r )

| ω (x1 )| t

−2 Pl log ε−1 t | ∂ε Lε,t n,l (x 1 , ϕτ2,s ,y2,s )| ≤ ε

− 12

n+r −5 2

Fs,l (t, τ ),

1

n−4 2

τj

| ∂ε F(12) Lε,t n,l (x 1 , x 2 , ϕτ2,s ,y2,s )| ≤ ε

− 21

n−2 2

Fs,l (t, τ ),

| ∂ε Lε,t 2,l (x 1 , ϕτ,y )| ≤ ε

− 21

Pl−1 log ε−1 (t, τ )

− 21

(12) Pl−1 log ε−1 F2,l (t).

(0)

(0) | ∂ε F(12) Lε,t 2,l (x 1 , x 2 )|

≤ε

Pl log ε−1 t

Fs,l (t, τ ),

(12) − 23

F2,l (t, τ ),

(128) (129) (130) (131) (132) (133)

Proof. We apply the method developed in the previous proof. The bound (128) obviously holds in the starting case n = 4, l = 0. Because of the bare interaction (25) the FE (33) is used if n +r > 4, where the difference test function in (130) and the modified insertion in (131),(133) count as r = 1 and r = 3, respectively. Regarding the r.h.s. of (33) we note that the first and second term do not contribute to the cases considered, and the third one only if n = 2, 4, 6.

358

C. Kopper, V. F. Müller

Proceeding inductively as in A), B) and C) of the previous proof, and using the bounds of Proposition 1, reproduces (128) for n > 4 and (129)–(131) and (133). The FE (34) provides bounds on the relevant parts of the cases n + r ≤ 4. As the renormalization conditions (67), (68), (69) depend at most weakly on ε, we obtain inductively | ∂ε clε,t (x1 )| ≤ ε− 2 Pl−1 log ε−1 · t − 2 , | ∂ε alε,t (x1 )| ≤ ε− 2 Pl−1 log ε−1·t − 2 , 1

1

1

3

1 µ,ε,t |∂ε fl (x1 ) ωµ (x1 )| ≤ | ω(x1 )| ε− 2 Pl−1 log ε−1 · t −1 , 1 1 µν,ε,t (2) | ∂ε bl (x1 ) ωµν (x1 ) | ≤ | ω(2) (x1 ) | ε− 2 Pl−1 log ε−1 · t − 2 .

(134) (135) (136)

With the aid of the decomposition (53), the bound (128) for n = 4 follows from (134) and (130). It remains to show (132). We use the decomposition (52) and perform similar steps as in D)d). From (58) and (133) we obtain

1 (1 − ρ)2 (0) ε,t | ∂ε ε,t (x , ϕ )| ≤ | ∂ F L (x , x ) | dρ | (∇ 3 ϕτ,y )(X (ρ)) | 1 τ,y ε 1 2 2,l (12) 2,l 2! x2 0
1
(12) − 12 −1 − 23 Pl−1 log ε τ F2,l (t) dρ K (τδ  , X (ρ), y) ≤ε x2

0

and herefrom, cf. (51), (119) for N b t < δ  τ (N = 3l − 2), −2 | ∂ε ε,t Pl−1 log ε−1 τ −3/2 K (τδ , x1 , y). 2,l (x 1 , ϕτ,y )| ≤ ε 1

(137)

From (134)–(136) follows −2 | ∂ε Lε,t Pl−1 log ε−1 · 2,l (x 1 , ϕτ,y )r el | < ε 1

1 1

t2



1 1 1 + + 1 t (tτ ) 2 τ

K (τδ , x1 , y).

(138)

On account of (52) the bounds (137), (138) establish (132) for N b t < δ  τ . To obtain an extension of the bound (137) to N b t ≥ δ  τ we again resort to the decomposition (52), yielding ε,t ε,t ∂t ∂ε ε,t 2, l (x 1 , ϕ) = ∂t ∂ε L2, l (x 1 , ϕ) − ∂t ∂ε al (x 1 )ϕ(x 1 ) µ,ε,t

+ ∂t ∂ε fl

µν,ε,t

(x1 ) ωµ (x1 ) + ∂t ∂ε bl

(2) (x1 ) ωµν (x1 ), (139)

(2)

with ωµ (x) = ∇µ ϕ(x), ωµν (x) = ∇µ ∇ν ϕ(x), ϕ(x) = K (τ, x, y). Employing on the r.h.s. of (139) in the various terms the corresponding FE (32) derived w.r.t. ε and then making use of bounds of Proposition 1 and of Proposition 2 already established inductively, leads with now familiar steps to  5 − 12 | ∂t ∂ε ε,t Pl−1 log ε−1 · (t, τ )− 2 F2, l (t, τ ) 2, l (x 1 , ϕ)| ≤ ε   5  1 3 + t − 2 + τ − 2 t −2 + τ −1 t − 2 K (τδ , x1 , y) . (140) On integrating ∂t ∂ε ε,t 2,l (x 1 , ϕ) from t = ε (small) with vanishing initial condition up to t ≥ δ  τ/bN the integral has to be split at t = δ  τ/bN . A bound on the lower part of the integral is given by (137). The upper part of the integral can be bounded using

Renormalization Proof for Massive ϕ44 Theory on Riemannian Manifolds

359

(140) observing both τ > t and τ < t, and majorizing constants. Combining both contributions yields for t ≥ δ  τ/bN ,

3 bN 2 δτ

   δ  1 δ 2 K (τδ , x1 , y) . · F2, l (t, τ ) + 1 + + bN bN

−2 Pl−1 log ε−1 · | ∂ε ε,t 2, l (x 1 , ϕ)| ≤ ε 1



(141)

Taking into account once more the decomposition (52), the bound (138) on the relevant part together with the bounds (137), (141) on the remainder reproduce (132). Thus the proof of Proposition 2 is complete.   From (128), (132) follows the integrability at ε = 0 and hence the existence of finite limits lim Lε,t n, l (x 1 , ϕτ2,s ,y2,s ), n ≥ 2.

ε0

Proposition 3. With the notations, conventions and the same class of renormalization conditions as in Proposition 1 - up to the fact that the constants in Pl log may now also depend on the mass m - we claim the following bounds for the CAS in the interval 1 ≤ t ≤ ∞: −1 t Fs,l (τ ), n ≥ 4, | Lε,t n,l (x 1 , ϕτ,y2,s )| ≤ Pl log τ

| Lε,t 2,l (x 1 , ϕτ,y )|

t ≤ (1, τ )−1 Pl−1 log(1, τ )−1 F2,l (τ ).

(142) (143)

t (τ ) is given in (50). The definition of Fs,l

Proof. The bounds stated in the proposition are proven inductively using again the standard scheme. The boundary conditions are the bounds from Proposition 1 taken at t = 1. They obviously satisfy the bounds (142), (143). The FE is treated in the same way as in parts A1) and A2) of the proof of Proposition 1. The integration w.r.t t is performed t (τ ) is montonically increasing with t. As regards part A1) we now using the fact that Fs,l use for t ≥ 1 instead of (80) now Ct (z, z  ) ≤ O(1) exp(−(m 2 − δ) t ), which results from the upper bounds (6), (8) on the heat kernel, and obtain upon integration


t 1



t dt  Fs,l (τ ) e−(m

2 −δ) t 

t ≤ O(1/m 2 ) Fs,l (τ ).

As regards A2) the internal line generated, which connects the two (partial) trees, see (84), (85), has the weight (86). Integrating, we majorize the weights of the other internal lines by their values at t and use for (86),


t 1

dt  Ct  (z  , z  ) ≤ Ct (z  , z  ) +




t

dt  Ct  (z  , z  )

1

t (τ ), (50), in this case, valid for any 0 < t ≤ 1, thus reproducing the weight factor Fs,l too.  

360

C. Kopper, V. F. Müller

Note that the renormalization conditions at t = 1 are in one to one relation with the values of the corresponding relevant terms at t = ∞, which have been shown to be finite for m 2 > 0 according to Proposition 3. Therefore renormalization conditions at t = 1 are tantamount to renormalization conditions at t = ∞. We want to close this section with some comments on the test functions considered and on possible extensions of the class of test functions. We stay with some informal remarks here, we did not rigorously analyse the problem of what is a "natural large" class of test functions. First note that our test functions can be arbitrarily well localized around any point of the manifold. This is an essential criterion for their viability from the physical point of view. Secondly the class of test functions can be extended by linearity (36). Since our bounds are in terms of the weight factors decaying with the tree distance between the points x1 , y2 , . . . , ys it is quite evident that the functionals Lε,t continuously by bounded convergence to test functions n,l (x 1 , ϕ) can also be extended   which are infinite sums i λi ϕi,τ (i) ,y (i) with |λi | < ∞. To go further one could either 2,s

2,s

prove (in a more functional analysis type of approach) that our test functions are dense e.g. in the set of smooth rapidly decaying functions on M w.r.t. a suitable norm, and that the Lε,t n,l (x 1 , ϕ) are continuous w.r.t. this norm. Or one could try to directly extend the previous proof to more general test functions in a second step. In this case the crucial part would be to maintain the line of argument presented in part C), (99) to (103), of the previous proof. 7. Scaling Transformations and the Minimal Form of the Bare Action In this section we want to show that the theory can be renormalized starting from a bare (inter)action of the form (21). This requires that we do not introduce any position dependent quantity in the theory which is not intrinsic to (M, g). Thus we only consider position independent coupling λ, and renormalization conditions in terms of intrinsic geometric quantities. We then introduce scaling tranformations of the following kind: For a four-dimensional Riemannian manifold (M, g) we scale its metric by a constant conformal factor, [NePa], ρ ∈ R+ :

gµν (x) → ρ 2 gµν (x), shortly g → ρ 2 g.

(144)

This leads to corresponding changes of geometrical quantities ˜ g µν → ρ −2 g µν ,  → ρ −2 , | g|1/2 → ρ 4 | g|1/2 , δ˜ → ρ −4 δ, µ µ d(x, y) → ρ d(x, y), σ (x, y) → σ (x, y) , (145) λ λ λ λ −2 µν → µν , ∇µ → ∇µ , R µνσ → R µνσ , Rµν → Rµν , R → ρ R. Moreover, the heat kernel K (t, x, y ; g) satisfies the scaling relation K (t, x, y ; g) = ρ 4 K ( ρ 2 t, x, y ; ρ 2 g),

(146)

which follows from its evolution equation (∂t − g )K (t, x, y ; g) = 0 together with stochastic completeness (4). As a consequence the regularized free propagator (18), 0 < ε < t ≤ ∞,
t 2  dt  e −m t K (t  , x, y; g), C ε, t (x, y; m 2 , g) = ε

Renormalization Proof for Massive ϕ44 Theory on Riemannian Manifolds

361

satisfies C ε, t (x, y; m 2 , g) = ρ 2 C ρ

2 ε, ρ 2 t

(x, y;

m2 2 , ρ g). ρ2

Regarding for a moment the action of the classical scalar field theory,
 1 λ  ϕ(−)ϕ + m 2 ϕ 2 + ξ R(x) ϕ 2 + 2 ϕ 4 , S(ϕ, m 2 , ξ, λ; g) = 2 x 4!

(147)

(148)

we observe that it is invariant if we supplement the scaling (144) of the metric by the transformations ϕ(x) → ρ −1 ϕ(x), m 2 → ρ −2 m 2 , ξ → ξ, λ → λ.

(149)

λφ 4 -

theory without We now consider the perturbative expansion of a regularized counterterms, i.e. in (20) we have L ε,ε (φ) = λ d V (x) φ 4 (x). A Feynman diagram contributing to an n-point CAS having v four-vertices and I internal lines obeys the topological relation 4v = n + 2I. This together with the scaling property (147) of the propagator implies for an n-point function folded with a test function ϕ = ϕ(x2 , . . . , xn ), ρ 2 ε, ρ 2 t

ε, t 2 4−n Ln, Ln, l l (x 1 , ϕ ; m , λ, g) = ρ

(x1 , ϕ ;

m2 , λ, ρ 2 g). ρ2

(150)

In the renormalization proof the CAS were constructed by imposing renormalization conditions for the relevant terms, see (67), (68), and by requiring the irrelevant terms to vanish at scale ε, see (59)–(61). As noted the renormalization conditions will now be supposed to be expressed in terms of intrinsic quantities, and they will be supposed to satisfy scaling (both statements are true for vanishing renormalization conditions). Because of the behaviour of σ (x, y)µ under scaling, (145), this means ρ 2 ε, ∞

alε, ∞ (x; m 2 , g) = ρ 2 al fl

µ, ε, ∞

µν, ε, ∞

bl

(x; m , g) = ρ fl 2

2

(x; ρ −2 m 2 , ρ 2 g),

µ, ρ 2 ε, ∞

(x; ρ

µν, ρ 2 ε, ∞

(x; m 2 , g) = ρ 2 bl

ρ 2 ε, ∞

clε, ∞ (x; m 2 , g) = cl

−2

(151)

m , ρ g),

(152)

(x; ρ −2 m 2 , ρ 2 g),

(153)

2

2

(x; ρ −2 m 2 , ρ 2 g).

(154)

For the standard case of ε-independent renormalization conditions the scaling of ε can of course be ignored. At the tree level the relation (150) holds as shown above. Using the FE with the standard inductive scheme it then follows that (150) holds in the case of renormalization conditions satisfying (151)–(154). Renormalization conditions imposed at some scale t R < ∞ are in one to one relation to those imposed at t = ∞, and the local terms alε, t R , etc. can be viewed either as renormalization conditions imposed at this scale or as resulting from integrating the FE over [t R , ∞) with renormalization conditions imposed at ∞. From this fact and (150) one deduces that the relations corresponding to (151)–(154) for renormalization conditions imposed at finite t R are ρ 2 ε, ρ 2 t R

alε, t R (x; m 2 , g) = ρ 2 al

(x; ρ −2 m 2 , ρ 2 g), etc.

(155)

In the subsequent analysis of the counterterms it will be helpful to first analyse the massless theory for t in the interval [ε, T ] to eliminate one of the parameters subject

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C. Kopper, V. F. Müller

to scaling. While restricting to [ε, T ], the less singular corrections stemming from the massiveness (see (170) below) can be dealt with afterwards. The same can then be done (trivially) for the finite contributions coming from integrating the FE of the massive theory over [T, ∞). For the massless theory we introduce the following notation: we denote ε,t alε,t (x; g) → al,t (x; g), R

etc.

to explicitly introduce all parameters subject to scaling, including the scale of the renormalization point t R . Furthermore we will introduce the sequence of scales tn := κ −n t R , κ > 1, 1 ≤ n ≤ N , such that ε = t N . Then we use the shorthands ε,tn n al,t (x; g) := al,t (x; g), R R

etc.,

(156)

etc.

(157)

and for the renormalization constants at t = t R , ε,t R alt R (x; g) := al,t (x; g), R

n (x; g), etc. are As a consequence of the properties of the heat kernel, the terms al,t R smooth scalars on the manifold. For the manifolds considered (of sectional curvature bounded above and below, as defined in Sect. 2), we have proven bounds which are uniform in the curvature since our bounds on the heat kernel are uniform in this case. n (x; g), etc., since we obtain The same holds for their (low order) derivatives (t)s al,t R the same bounds for these derivatives due to (15). We can therefore decompose these terms according to their tensorial character into individual contributions from curvature, respecting the scaling property, such that in this decomposition there will only appear terms depending smoothly on the geometric quantities. This gives n n n n al,t (x; g) = αl,t + R(x) ξl,t + δal,t (x; g), R R R R

µ, n fl,t R (x; g) µν, n bl,t R (x; g) n cl,t (x; g) R

= = =

µ, n 0 + δ fl,t R (x; g), µν, n n g µν (x) βl,t + δbl,t R (x; g), R n n γl,t + δcl,t (x; g). R R

(158) (159) (160) (161)

The zero written in (159) reminds us that this term vanishes identically in the case of constant curvature. The remainder terms in this decomposition may be analysed further  (1,n) (1 ,n) (1 ,n) n δal,t R(x) h l (x; g) = t + R 2 (x) h l + R µν (x)R µν (x) h l R R  (1 ,n) + ···, (162) + R µνλσ (x)R µνλσ (x) h l µ, n

(2,n)

δ fl,t R (x; g) = t R g µν (x) R, ν (x)h l + ··· ,   µν, n (3,n) (3 ,n) + ···, + g µν (x)R(x)h l δbl,t R (x; g) = t R R µν (x) h l (4,n)

n δcl,t (x; g) = t R R(x) h l R

+ ··· .

(163) (164) (165)

All the h-functions in this decomposition have mass dimension zero and are therefore independent of t R which is the only scale. The dots indicate terms of higher scaling

Renormalization Proof for Massive ϕ44 Theory on Riemannian Manifolds

363

dimension in the expansion w.r.t. curvature terms. We then assume that these expansions are asymptotic21 , in the sense that the remainders satisfy µ, n

µν,n

(2)

n (x; ρ 2 g)|, |ω (x)δ f 2 2 −4 |δal,t µ l,t R (x; ρ g)|, |ωµν (x)δbl,t R (x; ρ g)| ≤ O(ρ ), R n (x; ρ 2 g)| |δcl,t R

≤ O(ρ −2 ).

(166) (167)

Here n and t R are (of course) kept fixed and furthermore, the rank 1, resp. rank 2, cotensor (2) fields ωµ (x), ωµν (x) are assumed to stay invariant under scaling g → ρ 2 g. The bounds are in agreement with the leading terms written in (162)–(165). This assumption appears plausible and is often taken for granted, see e.g. [HoWa3]. Its proof requires a more thorough analysis of the heat kernel and its convolutions than is given here. Proposition 4. Assuming (166),(167), then for position independent coupling λ there exist renormalization conditions of the form (67, 68) such that the bare action takes the simple form (21); this means that for l ≥ 1,
1 2 ε 4 L lε (ϕ) = c ϕ (x)} (168) {( αlε + ξlε R(x)) ϕ 2 (x) − blε ϕ(x)ϕ(x) + 2 x 4! l with the following bounds | αlε | ≤

1 1 1 1 1 Pl−1 log , | ξlε | ≤ Pl log , | blε | ≤ Pl−1 log , | clε | ≤ Pl log . ε ε ε ε ε (169)

Proof. We first note that Proposition 1 can be proven in complete analogy when imposing renormalization conditions of the form (67), (68) at scale t R = T > 0 for the massless theory. The scale T is the one up to which we have precise control on the heat kernel, cf. (13), and it is thus related to the geometry of M. Furthermore we can expand for ε ≤ t ≤ T, ε,t ε,t 2 2 Lε,t n,l (m ; x 1 , ϕτ,y2,s ) = Ln,l (0; x 1 , ϕτ,y2,s ) + m ∂m 2 Ln,l (0; x 1 , ϕτ,y2,s )
1 2 + m4 dλ (1 − λ) ∂m2 2 Lε,t n,l (λm ; x 1 , ϕτ,y2,s ). (170) 0

We first analyse the massless theory and then comment on the derivative terms. We use the notation (156), (157). The theory is specified through renormalization conditions of the form (67), (68) imposed at scale t R = T : µ,T

alT (x; g) = 0, fl

µν,T

(x; g) = 0, bl

(x; g) = 0, clT (x; g) = 0,

(171)

together with boundary conditions of the type (59)–(61) at scale ε = κ −N T for l  ≤ l. Our aim is to analyse the bare action. From Proposition 1 we obtain for l > 0 the bounds n | al,T (x; g)| ≤ O(1) κ n nl−1 , µ, n | fl,T (x; g) ωµ (x)| µν, n (2) | bl,T (x; g) ωµν (x)| n | cl,T (x; g)|

≤ O(1) | ω(x)| κ

(172) n 2

n

l−1

,

(173)

≤ O(1) | ω(2) (x)| nl−1 ,

(174)

≤ O(1) n .

(175)

l

21 Asymptoticity is obviously required up to second order in ρ 2 only.

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C. Kopper, V. F. Müller

In the sequel we present the detailed argument for the relevant term a(x; g), whereas the analogous treatment of the other ones is stated in shortened form. In view of the decomposition (158) we want to prove inductively in n 22 n |αl,T | ≤ O(1)

n



κ n n

l−1

n , |ξl,T | ≤ O(1)

n  =1

n

n

l−1

,

n  =1

n |δal,T (x; g)| ≤ O(1)

n



κ −n n 

l−1

.

(176)

n  =1

First note that the uniqueness of the solutions of the FE implies that the relevant term n+1 (x, g) satisfies al,T n+1 1 (x; g) = aˆ l,κ al,T −n T (x; g),

(177)

1 −(n+1) T for where aˆ l,κ −n T (x; g) is defined to be the corresponding relevant term at scale κ −n the theory renormalized at scale κ T , with renormalization conditions of the following form

aˆ lκ

−n T

n (x; g) = al,T (x; g) (analogously for the f, b, c-terms).

(178)

This just means that we take renormalization conditions at scale T , integrate down to κ −n T , and take the values we arrive at for the local terms, as renormalization conditions at the scale κ −n T . By the uniqueness statement we obtain the same Schwinger functions as when imposing alT (x; g), etc. at scale T . From the scaling relations, cf. (155), we have n n n n T al,T (x; g) = κ n al,κ ˆ l (x; κ n g), n T (x; κ g) = κ a

(179)

n+1 al,T (x; g)

(180)

=κ

n

n+1 n al,κ n T (x; κ g)

= κ

n

1 aˆ l,T (x; κ n g).

n (x; g) such relations hold without the external factor κ n . Moreover, In the case of cl,T µν, n µν,T (2) (2) (x) = κ n bˆl (x; κ n g) ωµν (x), bl,T (x; g) ωµν µν, n+1 (2) bl,T (x; g) ωµν (x)

= κ

n

µν,1 (2) (x), bˆl,T (x; κ n g) ωµν

(181) (182)

(2) (x) by ωµ (x). Using and the analogue for f µ is obtained replacing bµν by f µ and ωµν (172)–(175) and (179)–(182), we then obtain

|aˆ lT (x; κ n g)| ≤ O(1) nl−1 , | fˆl

µ,T

(x; κ n g)ωµ (x)| ≤ O(1)|ω(x)| κ n g nl−1 ,

µν,T (2) (x; κ n g)ωµν (x)| ≤ O(1)|ω(2) (x)| κ n g nl−1 , | bˆl

| cˆlT (x; κ n g)| ≤ O(1)nl ,

where we denoted by | · | κ n g the norm (A.32) generated by

κ n g.

(183) (184)

−1

We now consider more general massless Schwinger functions Lˆ κp,l T,t (x1 , ϕτ,y2,s ; g) ˜ 23 resulting from a metric g˜ of the class defined in Sect. 2 and satisfying renormalization conditions of the form (183), (184) at loop orders l  < l. At loop order l we first 22 More precisely induction is in (l, n) in the order (l, 1), (l, 2), . . . , (l, N ), (l + 1, 1), . . ., but the step (l, N ) → (l + 1, 1) is trivial. 23 g˜ = κ n g certainly belongs to this class if g does.

Renormalization Proof for Massive ϕ44 Theory on Riemannian Manifolds

365

assume vanishing renormalization conditions. Afterwards the contribution coming from renormalization conditions at loop order l, bounded as in (183),(184) will be added to the result obtained. Integrating the flow equations for these Schwinger functions within the interval [κ −1 T, T ], one verifies with the aid of the usual inductive scheme and analogously as in Proposition 1, for t ∈ [κ −1 T, T ], the bounds −1 T,t

| Lˆ κp,l

κ −1 T,t

| Lˆ p,l

(x1 , ϕτ,y2,s ; g)| ˜ ≤ O(1) nl Fs,l (t, τ ),

p ≥ 6,

(x1 , ϕτ,y2,s ; g)| ˜ ≤ O(1) nl−1 Fs,l (t, τ ),

(185)

p ≤ 4.

(186)

These bounds are dictated by the size of the boundary conditions for l  < l which enter  on the r.h.s. of the FE. In fact one realizes that the factors of nl appearing in the bound on the r.h.s. can be factored out and majorized by nl resp. nl−1 . The remainder is then inductively bounded (uniformly in n) by the Fs,l (t, τ )-factors times a ( p, l)-dependent constant. For the relevant terms these bounds imply µ,1,0 1,0 | aˆ l,T (x; g)| ˜ ≤ O(1) nl−1 , | fˆl,T (x; g) ˜ ωµ (x)| ≤ O(1) | ω(x)| g˜ nl−1 , µν,1,0 1,0 (2) | bˆl,T (x; g) ˜ ωµν (x)| ≤ O(1) | ω (2) (x)| g˜ nl−1 , | cˆl,T (x; g)| ˜ ≤ O(1) nl−1 . (187)

Here the upper index 0 indicates that we were calculating with vanishing renormalization conditions at loop order l. On decomposing as in (158), 1,0 1,0 1,0 1,0 ˜ aˆ l,T (x; g) ˜ = αˆ l,T + R(x) ξˆl,T + δ aˆ l,T (x; g), ˜

(188)

we then obtain from (187) by linear independence 1,0 1,0 1,0 |, | ξˆl,T |, | δ aˆ l,T (x; g) ˜ | ≤ O(1) nl−1 . | αˆ l,T

(189)

Specializing to g˜ = κ n g in (188) yields 1,0 1,0 1,0 1,0 (x; κ n g) = αˆ l,T + κ −n R(x) ξˆl,T + δ aˆ l,T (x; κ n g), aˆ l,T

(190)

1,0 where our smoothness assumption (166) on δ aˆ l,T (x; κ n g) implies 1,0 | δ aˆ l,T (x; κ n g)| ≤ O(1) κ −2n nl−1 .

(191)

Upon scaling according to (180) we then obtain n+1,0 n+1,0 n+1,0 n+1,0 (x; g) = αl,T + R(x) ξl,T + δal,T (x; g) al,T

(192)

with the bounds n+1,0 n+1,0 n+1,0 | ≤ O(1)κ n nl−1 , |ξl,T | ≤ O(1)nl−1 , |δa l,T (x; g)| ≤ O(1) |αl,T

nl−1 . κn

(193)

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C. Kopper, V. F. Müller

Adding the contributions from the renormalization condition obeying the inductive bounds from (176), then yields n+1 | αl,T | ≤ O(1)

n+1 | ξl,T | ≤ O(1)

n+1 | δa l,T (x; g) | ≤ O(1)

n+1

n  =1 n+1

n  =1 n+1



κ n n n

l−1



l−1

≤ O(1) κ n+1 (n + 1) l−1 ,

≤ O(1) (n + 1)l ,

κ −n n 

l−1

≤ O(1),

n  =1

thus establishing the bounds (176) by induction. The statement for n + 1 = N implies Proposition 4, noting in particular that the last inequality allows for eliminating the term N (x; g) by a finite change of the corresponding renormalization condition at scale T . δal,T The other relevant terms are dealt with analogously. Regarding cl we obtain in place of (192), (193), n+1,0 n+1,0 n+1,0 cl,T (x; g) = γl,T + δcl,T (x; g),

(194)

n+1,0 n+1,0 | ≤ O(1) nl−1 , |δcl,T (x; g)| ≤ O(1) κ −n nl−1 . |γl,T

(195)

µν

As for bl , decomposing as in (160), µν, 1,0 µν, 1,0 1,0 (x; g) ˜ = g˜ µν (x) βˆl,T + δ bˆl,T (x; g), ˜ bˆl,T

(196)

we get from (187), µν, 1,0 1,0 (2) (2) | g˜ µν (x) ωµν (x) βˆl,T |, | ωµν (x) δ bˆl,T (x; g)| ˜ ≤ O(1) | ω (2) (x)|g˜ nl−1 .

(197)

The second bound implies for g˜ = g, µν, 1,0

(2) | ωµν (x) δ bˆl,T

(x; g)| ≤ O(1) | ω (2) (x)|g nl−1 ,

and using (166) then provides µν, 1,0

(2) | ωµν (x) δ bˆl,T

(x; κ n g)| ≤ O(1) | ω (2) (x)|g κ −2n nl−1 .

(198)

Upon scaling, (182), and observing κ n | ω (2) (x)| g˜ = | ω (2) (x)|g , we obtain from (196)– (198), n+1,0 (2) | g µν (x) ωµν (x) βl,T | ≤ O(1) | ω (2) (x)|g nl−1 , µν, n+1,0

(2) | ωµν (x) δbl,T

(x; g)| ≤ O(1) | ω (2) (x)|g κ −n nl−1 .

(199) (200)

Finally, proceeding similarly we find µ, n+1,0

| ωµ (x) δ fl,T

(x; g)| ≤ O(1) | ω(x)|g κ −n nl−1 .

(201)

Since (200), (201) hold with general ω (2) and ω, respectively, the bounds extend to the individual tensorial components. The proof of Proposition 4 is finished through the following remarks:

Renormalization Proof for Massive ϕ44 Theory on Riemannian Manifolds

367

i) To go back to the massive theory we have to add the two derivative terms from (170). An m 2 -derivative acting on the propagator produces an additional factor of t. As a consequence of this we get the bounds |∂ms 2 Lε,t n,l (x 1 , ϕs )| ≤ t

n+2s−4 2

Pl log(t, τ )−1 Fs,l (t, τ ).

(202)

This implies that for s ≥ 1 there is only one relevant term
∂m 2 Lε,t 2,l (x 1 , x 2 ), x2

which by the previous statement is logarithmically bounded. Applying the expansion (158) to this term, all terms produced can be absorbed-respecting the bounds- in the terms already present in the massless theory. So the previous result is maintained. ii) We restore the massive theory at scale T by adding the contributions from the last two terms on the r.h.s. of (170). According to Proposition 3, renormalization conditions at scale T can then be translated into renormalization conditions at scale t → ∞ for the massive theory.   Acknowledgement. The authors are indebted to the referee for careful study of the paper and for demanding clarification of two items.

A. Some Notions from Riemannian Geometry Here, we briefly recall some basic properties of Riemannian manifolds pertinent to the main text and thereby introduce the definitions and conventions used. For a detailed exposition we refer to [Wil]. We consider a connected four-dimensional smooth manifold M. A Riemannian metric on M is a tensor field g of type (0,2) (more technically: a section of ⊗2 T ∗ M, where T ∗ M is the cotangent bundle ) which associates to each point p ∈ M a positive-definite inner product on T p M, the tangent space to M at p. Given a chart with local coordinates x = (x 1 , x 2 , x 3 , x 4 ) ∈ R4 , and denoting by ∂µ := ∂/∂ x µ and by d x µ , µ = 1, 2, 3, 4, the corresponding coordinate vector and covector fields, respectively, the Riemannian metric tensor has the form g = gµν (x) d x µ ⊗ d x ν ,

gµν (x) = g(∂µ , ∂ν ).

(A.1)

At each point x the components gµν (x) form the entries of a symmetric positive-definite matrix. In (A.1) and henceforth the summation convention is implied. Moreover, with   |g(x)| ≡ det gµν (x) (A.2) g λµ (x)gµν (x) := δνλ , the Riemannian volume element reads 1

d V (x) = |g(x)| 2 d x 1 d x 2 d x 3 d x 4 ,

(A.3)

and the Laplace-Beltrami operator acting on a scalar field is defined by 1

1

φ(x) = |g(x)|− 2 ∂µ g µν (x)|g(x)| 2 ∂ν φ(x).

(A.4)

The Levi-Civita connection ∇ of the Riemannian metric g leads to the covariant derivative of the coordinate vector fields λ (x) ∂λ ∇∂ν ∂µ = µν

(A.5)

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C. Kopper, V. F. Müller

with the Christoffel symbols λ µν (x) =

 1 λ  λ . g ∂µ gν + ∂ν gµ − ∂ gµν = νµ 2

(A.6)

The Riemannian curvature tensor R of the connection ∇ maps the triple of vector fields X, Y, Z to the vector field24   (A.7) R(X, Y )Z = ∇ X ∇Y − ∇Y ∇ X − ∇[X,Y ] Z . In local coordinates with X = X µ (x) ∂µ and similarly for Y, Z the curvature tensor has the form R(X, Y )Z = R σ µ ν Z σ X µ Y ν ∂

(A.8)

with components 



R σ µ ν (x) = ∂µ σ ν − ∂ν σ µ + λµ σλ ν − λν σλ µ .

(A.9)

The components of the Ricci tensor follow by internal contraction as Rσ ν (x) := R µσ µν (x),

(A.10)

and the Ricci curvature at the point p with local coordinates x in the direction of the tangent vector v ∈ T p M is defined by Rσ ν (x) v σ v ν . gσ ν (x) v σ v ν

Ric p (v) :=

(A.11)

Moreover, the scalar curvature is given by R(x) := g σ ν (x) Rσ ν (x).

(A.12)

Let v, w ∈ T p M span the two-dimensional subspace S. Then the sectional curvature of M at the point p along the section S is defined as Sec p (v, w) := −

g p (R p (v, w)v, w) . g p (v, v)g p (w, w) − g p (v, w) 2

(A.13)

It depends only on the section S, not on the spanning vectors v, w. Given in T p M an µ orthonormal basis ξ(r ) , r = 1, .., 4, with components {ξ(r ) } implies g µν (x) =

4

r =1

µ

ξ(r ) ξ(rν )

(A.14)

and leads to sectional curvatures, r = s, µ

α ν Sec p (ξ(r ) , ξ(s) ) = Rσ αµν (x) ξ(rσ ) ξ(s) ξ(r ) ξ(s) .

(A.15)

24 There is obviously a freedom in choosing an overall sign, which has to be observed, similarly in the case of the Ricci tensor.

Renormalization Proof for Massive ϕ44 Theory on Riemannian Manifolds

Herefrom it follows that Ric p (ξ(s) ) =

369

Sec p (ξ(r ) , ξ(s) ),

(A.16)

Sec p (ξ(r ) , ξ(s) ).

(A.17)

r, r =s

R(x) = 2

r< s

The geodesics passing through a point p ∈ M can in general only be defined for values of the (affine) parameter confined to a finite interval. They generate a map from an open domain of the tangent space into the manifold, called the exponential map, exp :  ⊂ T p M → M. Its inverse are the Riemannian normal coordinates. A manifold is geodesically complete, if this parameter interval everywhere extends to R, and hence  = T p M, for all p ∈ M. For points p, q ∈ M the distance function d( p, q) = d(q, p) is defined by d( p, q) = inf α L(α), where α runs over all C 1 curve segments joining p to q, i.e. α : [a, b ] → M, α(a) = p, α(b) = q, and its arc length given by
b    21 L(α) = ˙ α(t) ˙ dt gα(t) α(t), . (A.18) a

If p is sufficiently close to q there is always a unique geodesic determining d( p, q). Regarding a geodesic ball in M with center p and with radius r , B( p, r ) = {q ∈ M| d( p, q) < r },

(A.19)

its Riemannian volume is denoted by
|B( p, r )| =

B

d V.

(A.20)

For x, y ∈ M we introduce the bi-tensor of type scalar-vector σ (x, y)µ :=

1 µν ∂ g (y) ν d 2 (x, y) 2 ∂y

(A.21)

which satisfies σ (x, y)µ σ (x, y)ν gµν (y) = d 2 (x, y).

(A.22)

In the renormalization proof we need covariant Taylor expansion formulae in the Schlömilch form, i.e. with integrated whichare obtained as follows:25 Given  remainders,   a complete Riemannian manifold M, g , and a chart U, x with local coordinates x, a geodesic x(s) parametrized by its arc length s satisfies λ (x(s)) x˙ µ (s) x˙ ν (s) = 0, x¨ λ (s) + µν

(A.23)

gµν (x(s)) x˙ (s) x˙ (s) = 1.

(A.24)

µ

ν

Let f ∈ C ∞ (M) and F(s) := f (x(s)), then  d n   F(s) = ∇νn · · · ∇ν1 f (x(s)) x˙ ν1 (s) · · · x˙ νn (s). ds 25 We give the complete argument since we only found part of it in the literature [BaVi].

(A.25)

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C. Kopper, V. F. Müller

The proof is by induction, using (A.23). We consider the geodesic segment with initial point x0 = x(0) and end point x = x(s), hence d(x, x0 ) = s. With (A.21) we then have the relation, see e.g. [Wil, Sect. 6.3], σ (x, x0 )ν = −s x˙ ν (0).

(A.26)

From the Taylor formula with remainder F(s) = F(0) +

n

sl

l!

l=1




(l)

F (0) + Rn ,

s

Rn =

dr 0

(s − r )n (n+1) F (r ), n!

(A.27)

we obtain, using (A.25), (A.26), f (x) = f (x0 )+

n

(−1)l

l=1 d(x,x0 )


Rn (x, x0 ) =

dr 0

l!

  σ (x, x0 )νl · · ·σ (x, x0 )ν1 ∇νl · · · ∇ν1 f (x0 )+ Rn , (A.28)

  (d(x, x0 ) − r )n νn+1 x˙ (r ) · · ·x˙ ν1 (r ) ∇νn+1 · · · ∇ν1 f (x(r )). n! (A.29)

Between fixed x, x0 we can reparametrize the geodesic segment x(r ) = X (ρ), with r = d(x0 , x)ρ, 0 ≤ ρ ≤ 1, implying gµν (X (ρ)) X˙ µ (ρ) X˙ ν (ρ) = d 2 (x0 , x). Then


1

Rn (x, x0 ) =

dρ 0

  (1 − ρ) n ˙ νn+1 (ρ) · · · X˙ ν1 (ρ) ∇νn+1 · · · ∇ν1 f (X (ρ)). X n!

(A.30)

In the remainder Rn the contraction of a tensor of type (n + 1, 0) with a tensor of type (0, n + 1) can be viewed via the (inverse) Riemannian metric as the scalar product of two tensors of type (0, n + 1). To bound |Rn (x, x0 )|, Cauchy’s inequality is used observing (A.24),
|Rn (x, x0 )| ≤

d(x,x0 )

(d(x, x0 ) − r )n |(∇ n+1 f )(x(r ))| = d n+1 (x0 , x) n! 0
1 (1 − ρ) n |(∇ n+1 f )(X (ρ))|, × dρ (A.31) n! 0 dr

where the norm square is given by     |(∇ n+1 f )(x)| 2 = ∇µn+1 · · · ∇µ1 f (x) g µn+1 νn+1 (x) · · · g µ1 ν1 (x) ∇νn+1 · · · ∇ν1 f (x). (A.32) Majorising in (A.31) the norm on the geodesic segment γ between x0 and x yields the bound | Rn (x, x0 )| ≤

d n+1 (x, x0 ) sup |(∇ n+1 f )(y)|. (n + 1)! y∈γ

(A.33)

Renormalization Proof for Massive ϕ44 Theory on Riemannian Manifolds

371

References [BaVi]

Barvinsky, A.O., Vilkovisky, G.A.: The generalized Schwinger-De Witt technique in gauge theories and quantum gravity. Phys. Rep. 119, 1–74 (1985) [BEM] Bros, J., Epstein, H., Moschella, U.: Towards a general theory of quantized fields on the Anti-de Sitter space-time. Commun. Math. Phys. 231, 481–528 (2002) [BFV] Brunetti, R., Fredenhagen, K., Verch, R.: The generally covariant locality principle - a new paradigm for local quantum field theory. Commun. Math. Phys. 237, 31–68 (2003) [BiDa] Birrell, N.D., Davies, P.C.W.: Quantum Fields in Curved Space. Cambridge, Cambridge University Press, 1982 [BiFr] Birke, L., Fröhlich, J.: KMS, etc. Rev. Math. Phys. 14, 829–873 (2002) [Bir] Birrell, N.D.: Momentum space renormalization of λ φ 4 in curved space-time. J. Phys. A 13, 569–584 (1980) [BPP] Bunch, T.S., Panangaden, P., Parker, L.: On renormalization of λ φ 4 field theory in curved spacetime: I. J. Phys. A 13, 901–918 (1980) [BrFr] Brunetti, R., Fredenhagen, K.: Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds. Commun. Math. Phys. 208, 623–661 (2000) [Bun1] Bunch, T.S.: Local momentum space and two-loop renormalization of λ φ 4 field theory in curved space-time. Gen. Rel. Grav. 13, 711–723 (1981) [Bun2] Bunch, T.S.: BPHZ renormalization of λ φ 4 field theory in curved space-time. Ann. Phys. (N.Y.) 131, 118–148 (1981) [BuPn] Bunch, T.S., Panangaden, P.: On renormalization of λ φ 4 field theory in curved space-time: II. J. Phys. A 13, 919–932 (1980) [BuPr] Bunch, T.S., Parker, L.: Feynman propagator in curved space-time: a momentum-space representation. Phys. Rev. D 20, 2499–2510 (1979) [Cha] Chavel, I.: Riemannian Geometry: A Modern Introduction. Cambridge, Cambridge University Press, 1993 [CLY] Cheng, S.Y., Li, P., Yau, S.-T.: On the upper estimate of the heat kernel of a complete riemannian manifold. Am. J. Math. 103, 1021–1063 (1981) [Dav1] Davies, E.B.: Heat kernels and spectral theory. Cambridge University Press, Cambridge, 1989 [Dav2] Davies, E.B.: Gaussian upper bounds for the heat kernels of some second order operators on riemannian manifolds. J. Funct. Anal. 80, 16–32 (1988) [Dav3] Davies, E.B.: Pointwise bounds on the space and time derivatives of heat kernels. J. Operator Theory 21, 367–378 (1989) [Gri] Grigor’yan, A.: Estimates of heat kernels on Riemannian manifolds. In: Davies, E.B., Safarov, Yu. (eds.) Spectral Theory and Geometry, London Math. Soc. Lecture Notes 273, pp. 140–225. Cambridge, Cambridge Univ. Press, 1999 [HoWa1] Hollands, S., Wald, R.M.: Local Wick polynomials and time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 223, 289–326 (2001) [HoWa2] Hollands, S., Wald, R.M.: Existence of local covariant time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 231, 309–345 (2002) [HoWa3] Hollands, S., Wald, R.M.: On the renormalization group in curved spacetime. Commun. Math. Phys. 237, 123–160 (2003) [KKS] Keller, G., Kopper, Ch., Salmhofer, M.: Perturbative renormalization and effective lagrangians in 44 . Helv. Phys. Acta 156, 32–52 (1992) [Kop1] Kopper, Ch.: Renormierungstheorie mit Flussgleichungen Aachen, Shaker Verlag, 1998 [Kop2] Kopper, Ch.: Renormalization Theory based on Flow equations. Lecture in honour of Jacques Bros. In: Rigorous Quantum Field Theory, Progress in Mathematics, Basel, Birkhäuser, 2006 [LiYa] Li, P., Yau, S.-T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 153–201 (1986) [Lü] Lüscher, M.: Dimensional regularization in the presence of large background fields. Ann. Phys. (N.Y.) 142, 359–392 (1982) [Mü] Müller, V.F.: Perturbative renormalization by flow equations. Rev. Math. Phys. 15, 491–557 (2003) [NePa] Nelson, B.L., Panangaden, P.: Scaling behavior of interacting quantum fields in curved spacetime. Phys. Rev. D 25, 1019–1027 (1982) [Pol] Polchinski, J.: Renormalization and effective lagrangians. Nucl. Phys. B 231, 269–295 (1984) [Sal] Salmhofer, M.: Renormalization - An Introduction. Berlin-Heidelberg-New York, Springer-Verlag, 1998 [SoZh] Souplet, P., Zhang, Q.: Sharp gradient estimate and Yau’s liouville theorem for the heat equation on noncompact manifolds. Bull. London Math. Soc. 38(6), 14045–1053 (2006) [Tay] Taylor, M.E.: Partial Differential Equations I. AMS 115. Springer-Verlag, 1996

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Varopoulos, N.Th.: Small time gaussian estimates of heat diffusion kernel. I. The semigroup technique. Bull. Sc. Math., 2 Série 113, 253–277 (1989) Willmore, T.J.: Riemannian Geometry. Oxford, Oxford University Press, 1996

Communicated by J.Z. Imbrie

Commun. Math. Phys. 275, 373–400 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0300-9

Communications in

Mathematical Physics

Toric G 2 and Spi n(7) Holonomy Spaces from Gravitational Instantons and Other Examples Gastón E. Giribet, Osvaldo P. Santillán Physics Department, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, 1428 Buenos Aires, Argentina. E-mail: [email protected] Received: 16 September 2006 / Accepted: 16 March 2007 Published online: 3 August 2007 – © Springer-Verlag 2007

Abstract: Non-compact G 2 holonomy metrics that arise from a T 2 bundle over a hyper-Kähler space are constructed. These are one parameter deformations of certain metrics studied by Gibbons, Lü, Pope and Stelle in [1]. Seven-dimensional spaces with G 2 holonomy fibered over the Taub-Nut and the Eguchi-Hanson gravitational instantons are found, together with other examples. By using the Apostolov-Salamon theorem [2], we construct a new example that, still being a T 2 bundle over hyper-Kähler, represents a non-trivial two parameter deformation of the metrics studied in [1]. We then review the Spin(7) metrics arising from a T 3 bundle over a hyper-Kähler and we find a two parameter deformation of such spaces as well. We show that if the hyper-Kähler base satisfies certain properties, a non-trivial three parameter deformation is also possible. The relation between these spaces with half-flat and almost G 2 holonomy structures is briefly discussed. 1. Introduction The spaces of special holonomy have received remarkable attention within the context of high energy physics; the spaces of G 2 holonomy are one of the most important examples. This is due to the fact that these geometries, in the presence of singularities, give rise to a natural framework for reducing eleven-dimensional M-theory to N = 1 fourdimensional realistic models [3–5]; see also [6, 7]. Actually, it is a well known fact that the geometries of this sort admit at least one globally defined covariantly constant spinor [8] and, when singularities are present, non-abelian gauge fields and chiral matter can also appear. All these features make evident that the study of spaces of G 2 holonomy deserved the attention of theoretical physicists. However, this is actually a hard tool to be studied, mainly because the explicit examples of compact spaces of G 2 holonomy have so far eluded discovery. In fact, even though such spaces are known to exist [21], the metrics which are explicitly known [8–20] turn out to be non-compact. Nevertheless, it was shown that the M-theory dynamics near the singularity is not strongly dependent

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on the global properties of the space [3]. On the other hand, applications for which the presence of singularities plays no role also exist. Furthermore, still in the case of non-compact spaces, it turns out that finding and classifying metrics of holonomy G 2 represents a highly non-trivial problem. In this work we discuss non-compact examples of toric geometries of such special holonomy, which are based on four-dimensional hyper-Kähler gravitational instantons. The problem of classifying spaces of G 2 holonomy possessing one isometry whose Killing vector orbits form a Kähler six-dimensional space was analyzed both by physicists and mathematicians. In Ref. [23], it was concluded that such geometries are described by a sort of holomorphic monopole equation together with a condition related to the integrability of the complex structure. Such a condition turns out to be stronger than the one required by supersymmetry. On the other hand, Apostolov and Salamon have proven in [2] that the Kähler condition yields the existence of a new Killing vector that commutes with the first, so that these metrics are toric. Besides, it was shown that such a G 2 metric yields a four-dimensional manifold equipped with a complex symplectic structure and a one-parameter family of functions and 2-forms linked by second order equations (henceforth called the Apostolov-Salamon equation). The inverse problem, i.e. the one of constructing a torsion-free G 2 structure starting from such a four-dimensional space was also discussed in [2]. Then, a natural question arises as to whether both descriptions of this classification problem are equivalent. In Ref [24], it was argued that this is indeed the case. Moreover, in Ref. [1, 2, 24] such a construction was employed to generate new G 2 -metrics. In the present work, the solution generating technique will be analyzed in detail and a wider family of G 2 -metrics will be written down. In particular, the Eguchi-Hanson and the Taub-Nut metrics will be dimensionally extended to new examples of G 2 holonomy following the construction proposed in [1]. In Sect. 2 we discuss the Apostolov-Salamon theorem [2], which formalizes a method for systematically constructing spaces with special holonomy G 2 by starting with a hyper-Kähler space in four-dimensions. This construction is actually the one previously employed by Gibbons, Lü, Pope and Stelle in Ref. [1] and here we discuss it within the framework of [2]. Then, we describe some explicit examples in order to illustrate the procedure. In particular, we show how some of the G 2 metrics obtained in such a way are one-parameter deformations of those examples discussed in the literature. In Sect. 3, we present the examples that are based on non-trivial Gibbons-Hawking solutions. We discuss the G 2 spaces obtained by starting with the four-dimensional gravitational instantons and we write down the corresponding metrics explicitly. Other examples are also discussed. In particular, we find a two parameter deformation of the T 3 bundles with Spin(7) holonomy over hyper-Kähler studied in [1]. Besides, in the strictly almost Kähler case, we obtain Spin(7) metrics which correspond to non-trivial T 3 bundles over hyper-Kähler metrics. To our knowledge, such metrics were not considered before in the literature. We show that all the presented metric spaces are foliated by equidistant hypersurfaces. In the G 2 holonomy case, these surfaces are half-flat manifolds, while for the Spin(7) metrics they are almost G 2 holonomy spaces, so that we implicitly find a family of half-flat T 2 bundles and a family of almost G 2 holonomy T 3 bundles over hyper-Kähler spaces. 2. The General Setup Let us begin by explaining how the Apostolov-Salamon scheme works; detailed proofs can be found in [2]. Let us consider a four dimensional complex manifold M with a metric g4 = δab ea ⊗ eb . It is also assumed that this metric is a function of a certain

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parameter µ, i.e. g4 = g4 (µ). This parameter should not be confused with a coordinate of g4 . The property of M as being “complex” means the following: Consider the manifold M, for which there exists a (1, 1)-tensor J1 with the property J1 · J1 = −I , such that g4 (J1 ·, ·) = −g4 (·, J1 ·). A metric for which the last property holds is called hermitian with respect to J1 , and it follows that J
1 = g4 (J1 ·, ·) is an antisymmetric tensor, so that it is a µ-dependent 2-form. In the cases in which there is no dependence on µ we will denote this tensor as J 1 . The tensor J1 is called an almost complex structure and is not uniquely defined. Indeed, any S O(4) rotation of the frame ea induces a new almost complex structure for which the metric turns out to be again hermitian. If at least one element of such a family of almost complex structures is covariantly constant with respect to the Levi-Civita connection  of g4 (that is, ∇ X J1 = 0, X being an arbitrary element of T M) then the tensor J1 will be called a complex structure, and then the manifold M will be complex. Conversely, for any complex manifold it is possible to find at least one complex structure. The condition ∇ X J1 = 0 is equivalent to the condition d J 1 = 0 (or d M J
1 = 0 if there is a dependence on µ, d M being the differentiation over M which does not involve derivatives with respect to µ) together with the integrability of J1 that is, the vanishing of the Nijenhuis tensor associated to J1 . It will be assumed also that the metric g4 admits a complex symplectic form  = J 2 + i J 3 , where being “symplectic” means that it is closed, d = 0. On the other hand, being “complex” implies that J 2 ∧ J 2 = J 3 ∧ J 3,

J 2 ∧ J 3 = 0,

and that J 2 (J1 ·, ·) = J 3 (·, ·).

(2.1)

Now let us introduce a function u depending on the coordinates of M and on the parameter µ, and satisfying 2µ J
1 (µ) ∧ J
1 (µ) = u ∧ .

(2.2)

This function always exists because the wedge products appearing in (2.2) are proportional to the volume form of M. With the help of the quantities defined above a seven-dimensional metric [2, 1]   (dα + H2 )2 (dβ + H1 )2 2 g7 = + g4 (µ) , + µ u dµ + (2.3) µ2 u can be constructed. Here β and α are two new coordinates while H1 and H2 are certain 1-forms independent on β and α. The parameter µ of g4 is now a coordinate of g7 and the metric tensor g7 is also independent on the coordinates α and β. This means that the vectors ∂α and ∂β are obviously Killing and commuting and therefore the metrics (2.3) are to be called “toric”. The Apostolov-Salamon construction states that if the quantities appearing in (2.3) are related by the evolution equation ∂ 2 J
1 c = −d M d M u, ∂ 2µ

(2.4)

and the forms H1 and H2 are defined on M × Rµ and M respectively by the equations c d H1 = (d M u) ∧ dµ +

∂ J
1 , ∂µ

d H2 = −J 2 ,

(2.5)

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c = J d , then the seven-dimensional metric (2.3) has holonomy in G . with d M 1 M 2 Moreover, the six-dimensional metric

g6 = u dµ2 +

(dβ + H1 )2 + g4 (µ), u

(2.6)

is Kähler with Kähler form K = (dβ + H1 ) ∧ dµ + J
1 .

(2.7)

This condition is usually referred to as the “strong supersymmetry condition” in the physical literature [23]. The calibration 3-form corresponding to the metrics (2.3) is  = J
1 (µ) ∧ (dα + H2 ) + dµ ∧ (dβ + H1 ) ∧ (dα + H2 )   + µ J 2 ∧ (dβ + H1 ) + u J 3 ∧ dµ ,

(2.8)

and by means of (2.5), (2.4) and (2.2) it can be seen directly that d = d ∗  = 0. This is an standard feature of the reduction of the holonomy from S O(7) to G 2 [8]. Moreover, the Killing field ∂α preserves  and ∗. The converse of all these statements are also true. That is, for a given G 2 holonomy manifold Y with a metric g7 possessing a Killing vector that preserves the calibration forms  and ∗ and such that the six-dimensional metric g6 obtained from the orbits of the Killing vector is Kähler, then there exists a coordinate system in which g7 takes the form (2.3) g4 (µ) being a one-parameter four-dimensional metric admitting a complex symplectic structure , and a complex structure J1 being the quantities appearing in this expression related by (2.1) and the conditions (2.5), (2.4) and (2.2). Details of these assertions can be found in the original reference [2]. 2.1. G 2 holonomy metrics fibered over hyper-Kähler manifolds. It is worth mentioning that the integrability conditions   ∂ J
1 c = 0, d J2 = 0 d (d M u) ∧ dµ + ∂µ for (2.5) are identically satisfied. The second is the closure of J 2 while the first is a direct consequence of (2.4) together with the closure of J
1 . Equations (2.5) are considerably c u = 0. In this case, the first equation in (2.5) and the condition simplified when d M 2 d H1 = 0 imply that ∂ J
1 /∂µ should be µ-independent and closed. If this is so then (2.4) is trivially satisfied. One possible solution is to choose J
1 (µ) = (M + Qµ)J 1 , J 1 being µ-independent and closed, and where M and Q are certain real parameters. The metric g4 for which J
1 is a Kähler form is g4 , g4 (µ) = (M + Qµ)


(2.9)


g4 being a µ-independent metric and J 1 its Kähler form. Relation (2.2) gives a simple algebraic equation for u, yielding u = µ(M + Qµ)2 .

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In conclusion, the metric
g4 is quaternion hermitian with respect to the tensors Ji , and from (2.1) it follows that Ji · J j = −δi j +i jk Jk . Besides, the two forms J 1 , J 2 and J 3 are closed. In four dimensions the closure of the hyper-Kähler triplet implies that the Ji are integrable. This means that
g4 is hyper-Kähler. The G 2 holonomy metric corresponding to this case is then given by g7 =

(dβ + Q H1 )2 (dα + H2 )2 + + µ2 (M + Qµ)2 dµ2 + µ (M + Qµ)
g4 , (M + Qµ)2 µ2

(2.10)

and the 1-forms H1 and H2 obey d H1 = J 1 and d H2 = −J 2 , J 1 and J 2 being any pair of the three Kähler forms on M. It is important to remark that (2.10) gives three G 2 holonomy metrics for a given hyper-Kähler metric. This is because one pair of 2-forms, ∗ selected among those that form the hyper-Kähler triplet J i , is necessary. I.e. although there are six possible choices, the order of the selection has no relevance and there are essentially three pairs. If the parameter M is set to zero, then the resulting metrics correspond to those appearing in Sect. 6.2 of reference [1]. Although (2.10) are just a subfamily of the Apostolov-Salamon metrics, the extension (2.10) gives rise to a powerful method to construct new G 2 examples. In principle, these have to be distinguished from other G 2 metrics presented in the literature [9–13], which are of the Bryant-Salamon type [8]. Regarding this, let us briefly comment on the amount of effective parameters appearing in both constructions. For Apostolov-Salamon metrics (2.10) there appear two parameters M and Q but only one of them is an effective one. It is not difficult to see that if M = 0, then it can be set at 1 by simply rescaling as
g4 → M
g4 ,

H1 → M H1 ,

H2 → M H2 , Q . Q → Q = α → Mα, β → M 2 β, M Then, the number of effective parameters will be 1 + n, n being the numbers of those belonging to the base 4-space. On the other hand, if M = 0, then we can also make the following redefinitions:
g4 → Q
g4 , H1 → Q H1 , α → Qα, β → Q 2 β,

H2 → Q H2 ,

and set Q = 1. In this case, the only parameters appearing in the metric will be those n of the hyper-Kähler base. There is another way to check that g7 is given by (2.10). Let us define the tetrad 1-forms dα + H2 dβ + Q H1 e5 = µ (M + Qµ) dµ, , e7 = . (2.11) e6 = µ M + Qµ Then the calibration form (2.8) is expressed as

   = µ (M + Qµ) J 1 ∧ e6 + e5 ∧ e6 ∧ e7 + µ (M + Qµ) J 2 ∧ e7 + J 3 ∧ e5 . (2.12)

It is convenient to choose a tetrad basis
ei for which the hyper-Kähler metric is diagonal, a b i.e. g4 = δab
e ⊗
e , and for which the hyper-Kähler triplet takes the form J1 =
e1 ∧
e2 +
e3 ∧
e4 ,

J2 =
e1 ∧
e3 +
e4 ∧
e2 ,

J3 =
e1 ∧
e4 +
e2 ∧
e3 . (2.13)

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Then, by making the redefinition ea ea = µ1/2 (M + Qµ)1/2
we see that (2.12) takes the familiar octonionic form  = cabc ea ∧ eb ∧ ec , where cabc is the octonion constants. The form  is G 2 invariant as a consequence of the fact that G 2 is the automorphism group of the octonion algebra. The G 2 holonomy metric corresponding to  is simply g7 = δab ea ⊗ eb and it can be checked that g7 is indeed given by (2.10). This follows from the expressions for ea given above. It is convenient to remark that, in (2.13), we were assuming that the hyper-Kähler triplet is positive oriented. In the negative oriented case it can be analogously shown that the holonomy will be also G 2 . 2.2. The simplest example. An interesting example to illustrate this construction comes from considering the simplest hyper-Kähler 4-manifold, namely R4 provided with its flat metric E4 = g4 = d x 2 + dy 2 + dz 2 + dt 2 . A closed hyper-Kähler triplet for R4 is given by J 1 = dt ∧ dy − dz ∧ d x,

J 2 = dt ∧ d x − dy ∧ dz,

J 3 = dt ∧ dz − d x ∧ dy. (2.14)

Now, let us extend this example to a seven-dimensional metric by means of ( 2.10). This innocent looking case is indeed rather rich and instructive. As a will be explained below, it gives rise to metric with holonomy exactly G 2 . This example was already discussed in Ref. [9] and we will extend it here to less simple cases. As it has been mentioned, there are three possible G 2 metrics that can be constructed, depending on which pair of forms we select from (2.14). But in the flat case, this choice just corresponds to a permutation of coordinates and the resulting metrics will be actually the same. By selecting the first two J i among those in (2.14) we obtain the potential forms H1 = −xdz − ydt, H2 = −ydz − xdt. Hence, the corresponding G 2 holonomy metrics read (dβ − Q(xdz + ydt))2 (dα − ydz − xdt)2 + + µ2 (M + Qµ)2 dµ2 (M + Qµ)2 µ2

g7 =

+ µ (M + Qµ)(d x 2 + dy 2 + dz 2 + dt 2 ).

(2.15)

If we select M = 0 and Q = 1 the metric tensors (2.15) reduce to g7 =

(dβ − xdz + ydt)2 µ2 (dα − ydz − xdt)2 + + µ4 dµ2 + µ2 (d x 2 + dy 2 + dz 2 + dt 2 ). µ2

(2.16)

Toric G 2 and Spin(7) Holonomy Spaces from Gravitational Instantons

379

Actually, metrics (2.16) have been already obtained in the literature [1]. They have been constructed within the context of eleven-dimensional supergravity, by starting with a domain wall solution of the form g5 = H 4/3 (d x 2 + dy 2 + dz 2 + dt 2 ) + H 16/3 dµ2 , with a = 1, . . . , 4 and H a warp function (see [1] for the details). By making use of the often cited oxidation rules, and by starting from the above space in five-dimensions, it is feasible to get a eleven-dimensional background of the form g11 = g(3,1) + g7 with the seven-dimensional metric being g7 =

(dβ − xdz + ydt)2 (dα − ydz − xdt)2 + H2 H2 4 2 2 2 2 + H dµ + H (d x + dy + dz 2 + dt 2 ).

(2.17)

By selecting the homothetic case H = µ the last metric reduces to (2.16). It can be shown by explicit calculation of the curvature tensor that (2.17) is irreducible and has holonomy exactly G 2 , and this happens even in the particular case H = µ, i.e. the one in (2.16 ), cf. [1]. The isometry corresponding to (2.17) and (2.15) is the same SU (2) group that acts linearly on the coordinates (x, y, z, t) on M and which simultaneously preserves the two forms J 1 and J 2 . The SU (2) action on M supplies H1 and H2 with total differential terms that can be absorbed by a redefinition of the coordinates α and β. For instance, a general translation of the form x → x + α1 ,

y → y + α2 ,

z → z + α3 ,

t → t + α4 ,

(2.18)

does preserve J 1 and J 2 but does not preserve the 1-forms H1 and H2 . Nevertheless, the effect of the translations (2.18) can be compensated by a coordinate transformation of the form β → β + α5 + α1 z − α2 t,

α → α + α6 + α2 z + α1 t.

(2.19)

Besides, more general SU (2) transformations on M preserving J 1 and J 2 can be also absorbed by a redefinition of the coordinates α and β. For the metric (2.16) we also have the scale invariance under transformation x → λx, y → λy, z → λz, t → λt, 4 4 α → λ α, β → λ β, µ → λ2 µ, for a real parameter λ, which is generated by the homothetic Killing vector, D = 2x∂x + 2y∂ y + 2z∂z + 2t∂t + µ∂µ + 4α∂α + 4β∂β .

(2.20)

Our task now is to construct more elaborate examples by means of a similar procedure. We do that in the next section.

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3. G 2 Metrics with Three Commuting Killing Vectors By construction, the metrics (2.10) possess at least two Killing vectors. If a larger isometry group is desired, an inspection of the formula (2.10) shows that the hyperKähler basis
g4 should already possess Killing vectors and that the action of the isometry group on H1 and H2 should be induced by gauge transforming H1 → H1 + d f 1 and H2 → H2 + d f 2 . The effect of this transformation can be compensated by a redefinition of the coordinates α → α + f 1 and β → β + f 2 , so that the local form of the metric would be unaltered. Considering d H1 ∼ J 1 and d H2 ∼ J 2 we see that J 2 and J 3 will be actually preserved by the isometry group. An obvious example is an hyper–kähler metric with a Killing vector which is tri-holomorphic, namely, one satisfying L K J 1 = L K J 2 = L K J 3 = 0. For any metric admitting a tri-holomorphic Killing vector ∂t there exists a coordinate system in which it takes generically the Gibbons-Hawking form [32] g = V −1 (dt + A)2 + V d xi d x j δ i j ,

(3.21)

with a 1-form A and a function V satisfying the linear system of equations ∇V = ∇ × A.

(3.22)

These metrics are hyper-Kähler with respect to the hyper-Kähler triplet J 1 = (dt + A) ∧ d x − V dy ∧ dz, J 2 = (dt + A) ∧ dy − V dz ∧ d x, J 3 = (dt + A) ∧ dz − V d x ∧ dy,

(3.23)

which is actually t-independent. The isometry group of the total G 2 space will be then enlarged to T 3 . Actually, one could naively suggest another possibility: namely, to choose a Killing vector which is not actually tri-holomorphic, but still preserves two of the three Kähler forms. However, as it will be shown below, an isometry preserving J 1 and J 2 is necessarily tri-holomorphic. If, on the other hand, a hyper-Kähler metric possesses an isometry that is not tri-holomorphic, then there always exists a coordinate system (x, y, z, t) for which the distance element takes the form [31]    2 gh = u z eu (d x 2 + dy 2 ) + dz 2 + u −1 dt + (u x dy − u y d x) , (3.24) z with u a function of (x, y, z) satisfying the SU (∞) Toda equation; namely (eu )zz + u yy + u x x = 0,

(3.25)

where we denote f x i = ∂x i f . It is evident that the vector field ∂t is a Killing vector of (3.24). Metric ( 3.24) is thus hyper-Kähler with respect to the t-dependent hyper-Kähler triplet,   J 1 = eu u z d x ∧ dy + dz ∧ dt + (u x dy − u y d x) , (3.26) J = eu/2 cos(t/2)
J + eu/2 sin(t/2)
J , (3.27) 2

J3 = e

2

u/2

3

sin(t/2)
J 2 − eu/2 cos(t/2)
J 3;

(3.28)

Toric G 2 and Spin(7) Holonomy Spaces from Gravitational Instantons

where we defined the 2-forms  
J 2 = −u z dz ∧ dy + dt + u y d x ∧ dy,

381


J 3 = u z dz ∧ d x + (dt + u x dy) ∧ d x.

These should not be confused with the hyper-Kähler forms J i in (3.28). From (3.28) it is clear that ∂t preserve J 1 , but the other two Kähler forms turn out to be dependent on t. It means that in the non-tri-holomorphic case it is impossible to preserve two of the three J i without preserving the third one. The local form (3.24) is general enough, and therefore, in four dimensions, a U (1) isometry that preserves two of the closed Kähler forms of an hyper-Kähler metric is automatically tri-holomorphic.1 Actually, there exists a shorter argument in order to find the same conclusion. All the complex structures form a two-sphere, since u 1 J1 + u 2 J2 + u 3 J3 is a complex structure so long as u 21 + u 22 + u 23 = 1. Now any U(1) action on a two sphere either is trivial and keeps all the complex structures inert or keeps only two points (i.e. only one complex structure and its conjugate) on the sphere fixed.2 3.1. On G 2 metrics and asymptotic behavior of ALG spaces. Let us consider now the Gibbons-Hawking metrics (3.21) for which the potential V and a vector potential A are independent on certain coordinates, say x. It is simple to check that Eqs. (3.22) reduce, up to a gauge transformation, to the Cauchy-Riemann equations A y = A z = 0,

∂ y A x = −∂z V,

∂z A x = ∂ y V.

This means that we can write A x + i V = τ1 + iτ2 = τ (w) with w = z + i y; namely, A x and V turn out to be the real and the imaginary part of an holomorphic function τ of one complex variable w. The corresponding hyper-Kähler metric can be written as g4 =

|dt + τ d x|2 + τ2 dwdw. τ2

(3.29)

The “local aspect” of the isometry group corresponds to R2 , R × U (1) or T 2 = U (1) × U (1), depending on the range of the values of the coordinates x and t. Notice that the Cauchy-Riemann equations imply that the 1-forms B1 = τ1 dz − τ2 dy,

B2 = τ1 dy + τ2 dz

(3.30)

are closed. Therefore, two functions (x, y) and ω(x, y) such that d = B1 and dω = B2 can be “locally” defined. It is easy to see that the Bi are the real and imaginary parts of the complex 1-form B1 + i B2 = τ (w)dw = dT (w), T (w) being the primitive of the function τ . Therefore  + iω = T (w). With the help of these functions the hyper-Kähler triplet for the family (3.29) is expressed as   1 ∗ J 1 = dt ∧ d x + dw ∧ dT , 2 (3.31) ∗ ∗ J 2 =  (ϒ), J 3 = (ϒ), 1 This discussion concerns only U (1)n isometry groups, in other cases it does not apply. 2 We thank S. Cherkis for this argument and for other important suggestions.

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where we introduced the complex two 2-form ϒ = dt ∧ dw + d x ∧ dT, and where the symbols and  denote the real and the imaginary part of the quantity between parentheses. It is clear that both ∂t and ∂x preserve (3.31). Therefore ∂x is also a tri-holomorphic Killing vector, which clearly commutes with ∂t . The 1-forms Hi∗ ∗ satisfying d Hi∗ = J i are easily found from (3.31), the result is   1 H1∗ = − xdt + T dw , H3∗ + i H2∗ = −wdt − T d x, 2 up to total differential terms. By selecting H1 = H1∗ and H2 = H2∗ in (2.10) the following G 2 holonomy metric is obtained  2 dβ − Q (xdt + 21 T dw) (dα − (wdt + T d x))2 g7 = + + µ2 (1 + Qµ)2 dµ2 (1 + Qµ)2 µ2   |dt + τ d x|2 + µ (1 + Qµ) + τ2 dwdw . (3.32) τ2 Metrics (3.32) constitute a family of G 2 holonomy metrics constructed essentially from a single holomorphic function τ and its primitive T . There are two more G 2 holonomy metrics obtained by selecting H1 = H1∗ and H2 = H3∗ , and also H1 = H2∗ and H2 = H3∗ in (3.32). For conciseness, we will not write them explicitly here, but the procedure to find them follows straightforwardly. In all the cases there will be four commuting Killing vectors namely, ∂α , ∂β , ∂t and ∂x . Metrics of the form (3.29) have been considered in the physical literature. For instance, the asymptotic form of any “ALG” instanton is (3.29) if the two coordinates x and t are periodically identified. Several examples of such geometries have been considered in [36, 37]. Let us recall that the term ALG usually stands for a complete elliptically fibered hyper-Kähler manifold. The metric (3.29) is not an ALG metric and in general is not complete; however, it is what an ALG metric approaches at infinity. Another context in which (3.29) has been considered is the context of stringy cosmic strings [38]. Besides, a particular case of such a class of metrics was shown to describe the single matter hypermultiplet target space for type IIA superstrings compactified on a CalabiYau threefold when supergravity and D-instanton effects are switched off [39]. In this case we have τ = log(w) with T (w) = w(log(w) − 1), which, from (3.32), yields the following explicit G 2 holonomy metric: g7 =

(dα −  (wdt + w(log(w) − 1)d x))2 µ2  2 dβ − Q (xdt + w2 (log(w) − 1)dw) + (1 + Qµ)2   | dt + log(w)d x |2 + log |w| dwdw . + µ2 (1 + Qµ)2 dµ2 + µ (1 + Qµ) log |w| (3.33)

The base hyper-Kähler metric possesses a rotational Killing vector ∂arg(w) . However, such vectors do not preserve the hyper-Kähler triplet (3.31) and, for this reason, they will not be a Killing vector of the seven-metric (3.33).

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It is interesting to analyze the transformation properties of the generic metric (3.29) under the S L(2, R) action τ→

aτ + b , cτ + c

t → at − bx,

x → d x − ct,

where the parameters a, b, c, d satisfy ad − bc = 1. Then, we obtain that τ2 →

adτ2 , |cτ + d|2

and the metric (3.29) transforms as g4 =

dwdw |dt + τ d x|2 + τ2 . τ2 |cτ + d|2

By defining new complex coordinates ξ, by dξ =

dw , cτ + d

the metric becomes g4 =

|dt + τ d x|2 + τ2 dξ dξ , τ2

which is of the form (3.29) and therefore hyper-Kähler. However, notice the transformed metric is different than the former one, because τ will be a different function after the coordinate transformation w → ξ . Therefore, in principle, the S L(2, R) action is not a symmetry of (3.29), but instead it maps any element of the family of toric hyper-Kähler metric (3.29) into another one; namely, it is closed among such a family and can be regarded as a subgroup of the asymptotic ALG symmetries. If the coordinates x and t are periodic, the transverse space to the w-plane will be T 2 = U (1) × U (1). However, again, let us emphasize that only one of the U (1) isometries of those which would correspond to the referred T 2 is globally defined for the metric (3.29). In this case τ can be interpreted as the modulus of the tori by restricting it to the fundamental S L(2, Z) domain. Under this domain we have the action of the modular transformations τ → τ + 1 and τ → −1/τ . Function τ will have certain magnetic source singularities around the w-plane. By going around such singularities one comes up against a jump τ → τ +1, so that the singularities behave like 2π τ ∼ −i log(w −wi ). By changing coordinate in the complex plane the metric can be expressed as g4 =

|dt + τ d x|2 + τ2 |h(ξ )|2 dξ dξ , τ2

h(ξ ) being an holomorphic function. If we require this metric to be modular invariant we obtain certain restrictions over τ and h(ξ ) [38]. This requirement implies that τ is given by the rational function j (τ ) =

P(ξ ) , Q(ξ )

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P and Q being polynomials in the complex variable ξ . The function h is related to τ by h(w) = η(τ )η(τ )

N 

(ξ − ξi )−1/12 ,

k=1

where η(τ ) is the Dedekind function  η(τ ) = q 1/24 (1 − q n ),

q = exp(2πiτ ),

n

and N is the number of singularities ξi of the function τ . This corresponds to the seven brane solution in [38]. The corresponding 4-metric g4 is hyper-Kähler with respect to the triplet   1 ∗ 2 J 1 = dt ∧ d x + τ |h(ξ )| dξ ∧ dξ , 2 ∗

∗

J 2 =  (ϒ), J 3 = (ϒ), ϒ = h(ξ )dt ∧ dξ + d x ∧ dT.

(3.34)

We see that only the first 2-form in (3.34) is modular invariant, therefore when modular invariant hyper-Kähler metrics are extended to a G 2 holonomy one, the modular invariance is generically lost through the extension. 4. G 2 Holonomy Metrics Fibered over Gravitational Instantons 4.1. The Taub-Nut case. The family of G 2 metrics presented in the previous subsection was constructed with a toric hyper-Kähler basis whose corresponding Killing vectors were tri-holomorphic, i.e. they preserved the hyper-Kähler triplet (3.23). For this reason, they turned out to be isometries of the full G 2 metric. The following examples we consider do deal with toric hyper-Kähler metrics, but possessing only one tri-holomorphic Killing vector. Therefore only this vector will extend the isometry of the resulting sevendimensional metric, and the isometry group will be U (1) × R2 . As the first example of a less simple family, let us consider the Gibbons-Hawking metrics (3.21), and select V as the potential for the electric field of certain configuration of charges. Then it follows from (3.22) that A will be a Wu-Yang potential describing a configuration of Dirac monopoles located at the same position where the electric charges are. For a single monopole located at the origin the potentials will take the form V =1+

a , r

A=

a(yd x − xdy) r (r + z)

z > 0,


= a(yd x − xdy) A r (r − z)

z ≤ 0, (4.35)

where we have defined here the radius r 2 = x 2 + y 2 + z 2 . The vector potential is not globally defined in R3 due to the string singularities in the z axis. In the overlapping
differ one to each other by a gauge transformaregion, the potential A and potential A
tion of the form A = A − 2a d arctan(y/x). Besides, if a further gauge transformation
→ A
− a d arctan(y/x) is performed, the vector A → A + a d arctan(y/x) and A potential will be given by a single expression, namely A=

az d arctan(y/x); r

(4.36)

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385

nevertheless, the potential (4.36) is clearly discontinuous at the origin, as it can be seen by evaluating the limit δ A = lim z→0+ A − lim z→0− A = 2a d arctan(y/x). As usual, we are assuming that this limit is taken by crossing the origin along the z axis. The upper and lower limits are different but related by a gauge transformation, as expected. The Gibbons-Hawking metric corresponding to this single monopole configuration is the Taub-Nut self-dual instanton. The local form of its metric, written in Cartesian coordinates, reads  r   2  r + a   az g= dτ + d arctan(y/x) + d x 2 + dy 2 + dz 2 . (4.37) r +a r r Although the function z/r is discontinuous, the metric tensors corresponding to the regions z > 0 and z ≤ 0 can be joined to form a globally defined regular metric by defining a new variable
τ = τ + 2a arctan(y/x) in the region z ≤ 0. This implies that the variable τ turns out to be periodic. To see this clearly it is convenient to introduce cylindrical coordinates x = ρ cos ϕ,

y = ρ sin ϕ,

z = η,

d arctan(y/x) = dϕ;

and then it follows that
τ = τ +2aϕ. The angle ϕ is periodic with period 2π and therefore the coordinate τ should be actually periodic with period 4aπ . Then τ can be interpreted as an angular coordinate if the parameter a is an integer number, a ∈ Z=0 . The explicit form for the hyper-Kähler triplet for the Taub-Nut metric ( 4.37) is given by    az a dy ∧ dz, J 1 = dτ + d arctan(y/x) ∧ d x − 1 + r r     a az dz ∧ d x, (4.38) J 2 = dτ − d arctan(y/x) ∧ dy − 1 + r r    a az d x ∧ dy. J 3 = dτ + d arctan(y/x) ∧ dz − 1 + r r Then it is elementary to find the integral forms Hi . These are H1 = −x dτ + (a log(r + z) + z) dy − a x d arctan(y/x), H2 = −y dτ − (a log(r + z) + z) d x − a y d arctan(y/x), H3 = −z dτ − a r d arctan(y/x) − x dy,

(4.39)

up to a total differential term. Let us show that the one monopole solution actually corresponds to the Taub-Nut hyper-Kähler metric. In spherical coordinates x = r sin θ cos ϕ,

y = r sin θ sin ϕ,

z = r cos θ,

d arctan(y/x) = dϕ,

the metric tensor acquires the form  r  r + a  (dr 2 + r 2 (dθ 2 + sin2 θ dϕ 2 )). (4.40) g= (dτ + a cos θ dϕ)2 + r +a r Next, by defining a new radial coordinate R = 2r + a, it becomes     1 R+a 1 R−a (dτ +a cos θ dϕ)2 + d R 2 + (R 2 − a 2 )(dθ 2 + sin2 θ dϕ 2 ), g= R+a 4 R−a 4 (4.41)

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which turns out to be the most familiar expression for the Taub-Nut instanton. Nevertheless, this metric is defined only in the domain R > a and therefore it has no well defined flat limit a 2 → ∞. With the help of Eq. (4.39), it is not difficult to find the G 2 holonomy metrics (2.10) fibered over the Taub-Nut instanton. The first one is thus obtained by selecting H1∗ = H1 and H2∗ = H2 , and the result is  r  r + a  (dτ + A(x, y, z))2 + µ (M + Qµ) r +a r ×(d x 2 + dy 2 + dz 2 ) + µ−2 (dα + y G(x, y) + F(r, z) d x)2 + µ2 (M + Qµ)2 dµ2 1 + (dβ − Qx dτ + Q F(r, z) dy − Qa x d arctan(y/x))2 , (4.42) (M + Qµ)2

g7 = µ (M + Qµ)

where the function F and the 1-forms G and A are given by F(r, z) = a log(r + z) + z, G(x, y) = dτ + a d arctan(y/x) az A(x, y, z) = d arctan(y/x), r and where Q and M are two positive real parameters. Again, for M > 0, it can be set to 1 without loss of generality. The case M = 0, on the other hand, is the one considered in Ref. [1], cf. Eq. (126) there. Besides, a second G 2 holonomy metric is obtained by selecting H1∗ = H3 and H2∗ = H2 , leading to the form  r  r + a  (dτ + A(x, y, z))2 + µ (M + Qµ) r +a r 2 2 2 × (d x + dy + dz ) + µ−2 (dα + y G(x, y) + F(r, z) d x)2 + µ2 (M + Qµ)2 dµ2 1 + (dβ − Qz dτ − Q a r d arctan(y/x) − Q x dy)2 . (4.43) (M + Qµ)2

g7 = µ (M + Qµ)

Notice that the metric holding for the case H1∗ = H2 and H2∗ = H3 just corresponds to a permutation of the coordinates in (4.43) and then gives no new geometry. The Killing vectors corresponding to both metrics above are ∂τ , ∂α and ∂β . We have actually worked out these metrics in other coordinate systems, but the corresponding expressions are too cumbersome to write here. The curvature tensor corresponding to both cases is irreducible and the holonomy is exactly G 2 .

4.2. The Eguchi-Hanson case. Now, let us discuss the case of two monopoles on the z axis. Without losing generality, it can be considered that the monopoles are located in the positions (0, 0, ±c). The potentials for these configurations are 1 1 V = + , r+ r−

 A = A+ + A− =

2 r± = x 2 + y 2 + (z ± c)2 .

z+ z− + r+ r−

 d arctan(y/x),

Toric G 2 and Spin(7) Holonomy Spaces from Gravitational Instantons

This case corresponds to coordinates, reads  1 g= + r+  1 + r+

387

the Eguchi-Hanson instanton, whose metric, in Cartesian −1    2 z+ z− dτ + d arctan(y/x) + r+ r−  1 (d x 2 + dy 2 + dz 2 ), + r−

1 r−

(4.44)

where z ± = z ±c. In order to recognize the Eguchi-Hanson metric in its standard form it is convenient to introduce a new parameter a 2 = 8c, and the elliptic coordinates defined by [44] x=

r2 8

1 − (a/r )4 sin ϕ cos θ,

y=

r2 8



1 − (a/r )4 sin ϕ sin θ, z =

r2 cos ϕ. 8

In this coordinate system it can be checked that r± =

  r2  r2  1 ± (a/r )2 cos ϕ , cos ϕ ± (a/r )2 , z± = 8 8 −1 16  4 2 V = 2 1 − (a/r ) cos ϕ , r   −1  A = 2 1 − (a/r )4 cos2 ϕ 1 − (a/r )4 cos ϕ dθ,

and, with the help of these expressions, we find g=

 −1  r2  1 − (a/r )4 (dθ + cos ϕdτ )2 + 1 − (a/r )4 4 r2 (dϕ 2 + sin2 ϕdτ ). × dr 2 + 4

(4.45)

This is actually a more familiar expression for the Eguchi-Hanson instanton, indeed. Its isometry group is U (2) = U (1) × SU (2)/Z2 , the same as the Taub-Nut one. Actually, the Eguchi-Hanson is a limit form of the Taub-Nut instanton [45]. The holomorphic Killing vector is ∂τ . This space is asymptotically locally Euclidean (ALE), which means that it asymptotically approaches the Euclidean metric, and therefore the boundary at infinity is locally S 3. However, the situation is rather different regarding its global properties. This can be seen by defining the new coordinate   u 2 = r 2 1 − (a/r )4 for which the metric is rewritten as −2  u2 r2 g= (dθ + cos ϕdτ )2 + 1 + (a/r )4 (dϕ 2 + sin2 ϕdτ ). du 2 + 4 4

(4.46)

The apparent singularity at r = a has been moved now to u = 0. Near the singularity, the metric looks like g

u2 1 a2 (dθ + cos ϕdτ )2 + du 2 + (dϕ 2 + sin2 ϕdτ ), 4 4 4

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and, at fixed τ and ϕ, it becomes g

u2 1 dθ 2 + du 2 . 4 4

This expression “locally” looks like the removable singularity of R2 that appears in polar coordinates. However, for actual polar coordinates, the range of θ covers from 0 to 2π , while in spherical coordinates in R3 , 0 ≤ θ < π . This means that the opposite points on the geometry turn out to be identified and thus the boundary at infinity is the lens space S 3 /Z2 . For an arbitrary ALE space, the boundary will be S 3 / ,  being a finite subgroup that induces the identifications. In general, the multi Taub-Nut metrics, corresponding to V with no constant term, will be ALE spaces [30]. In particular, any ALE space admits an unique self-dual metric [47]. The expressions for the integral forms corresponding to the Eguchi-Hanson metrics is a bit longer than those of the Taub-Nut case, yielding H1 = −x dτ + (log(r+ + z + ) + log(r− + z − )) dy − 2a x d arctan(y/x), H2 = +y dτ + (log(r+ + z + ) + log(r− + z − )) d x + 2a y d arctan(y/x), H3 = −zdτ − a (r+ + r− ) d arctan(y/x).

(4.47) (4.48) (4.49)

By using the formulas (4.48) and (2.10) the following G 2 holonomy metric is found

  1 1 −1 g7 = µ (M + Qµ) µ (M + Qµ) dµ2 + + r+ r−  2      z+ z− 1 1 d arctan(y/x) × dτ + + + + µ(M + Qµ) r+ r− r+ r− × (d x 2 + dy 2 + dz 2 ) +

(dβ + Q H1∗ )2 (dα + H2∗ )2 + . (M + Qµ)2 µ2

(4.50)

As before, H1∗ and H2∗ are any pair of 1-forms selected among those in (4.47)–(4.49), and there are essentially two different metrics coming from these possible choices. The Killing vectors of (4.50) turn out to be the same as those for the Taub-Nut case, namely, ∂t , ∂α and ∂β . 4.3. Relation to the Ward metrics. In the previous subsections we considered an array of one and two monopoles along one axis. In this one we consider an arbitrary array along one axis. This case also corresponds to axially symmetric hyper-Kähler metrics, such as the Eguchi-Hanson and Taub-Nut instantons. Thus, it is convenient to write the flat three dimensional metric in cylindrical coordinates d x 2 + dy 2 + dz 2 = dρ 2 + dη2 + ρ 2 dϕ 2 . Then the potentials V and A, and consequently the hyper-Kähler metrics corresponding to such an array, will be ϕ-independent. This means that ∂ϕ is a Killing vector but, unlike ∂t , it is not tri-holomorphic. For this reason, ∂ϕ will not be necessarily a Killing vector of the full G 2 metric. In this sense, there is no advantage in considering these toric 4-metrics if one wants to enlarge the isometry group of the 7-metrics. One can consider a configuration with at least three monopoles that are not aligned [34, 35], the resulting

Toric G 2 and Spin(7) Holonomy Spaces from Gravitational Instantons

389

metrics will not be toric, but the isometry group of the G 2 holonomy space will be the same, that is, ∂t , ∂α and ∂β . The interest in considering the toric hyper-Kähler examples is mainly that they already encode many well known hyper-Kähler examples. Such spaces corresponding to aligned monopoles are known as the Ward spaces [33]. The solution of the Gibbons-Hawking equation in this case reads A = ρUρ dϕ and V = Uη , U being a solution of the Ward monopole equation (ρUρ )ρ +(ρUη )η = 0. The local form for such metrics, when written in cylindrical coordinates, is g=

(dt + ρUρ dϕ)2 + Uη (dρ 2 + dη2 + ρ 2 dϕ 2 ). Uη

(4.51)

For instance, the Taub-Nut metric (4.37) would look like 2

ρ 2 + η2 a + ρ 2 + η2 η g= dϕ + (dρ 2 + dη2 + ρ 2 dϕ 2 ), dt + a + ρ 2 + η2 ρ 2 + η2 ρ 2 + η2 (4.52) where, for instance, we recognize that the expression (4.52) is actually of the Ward form (4.51) for the function U = η + a log(η + η2 + ρ 2 ), which can be checked to satisfy (ρUρ )ρ + (ρUη )η = 0. The form (4.51) is characteristic of a hyper-Kähler metric with two commuting isometries, one of which is self-dual while the other is not. Just for completeness, let us point out that the integral forms (4.39) for the Taub-Nut metric would be expressed in cylindrical coordinates as H1 = −ρ cos ϕ (dτ + a dϕ) + U (sin ϕdρ + ρ cos ϕdϕ), H2 = +ρ sin ϕ (dτ + a dϕ) + U (cos ϕdρ − ρ sin ϕdϕ), H3 = −η dτ − a ρ 2 + η2 dϕ − ρ cos ϕ (sin ϕdρ + ρ cos ϕdϕ).

(4.53)

On the other hand, the Eguchi-Hanson solution corresponds to     2 2 2 2 U = log η − c + (η − c) + ρ + log η + c + (η + c) + ρ with c2 > 0. That is, we have two point sources on the η-axis (z -axis). Also, the case c2 < 0 corresponds to the potential for an axially symmetric circle of charge, called Eguchi-Hanson metric of the type I, and which is always incomplete. Further, let us consider the fundamental solution of the Ward equation, namely Ui = ai log(η − ηi + (η − ηi )2 + ρ 2 ). This corresponds to a monopole of charge ai located in the position ηi . It is not difficult to find the explicit expressions for the forms Hi in an analogous way to that in the case analyzed before. If we consider an array of aligned monopoles, the forms Hi will be given by H1 = −ρ cos ϕ (dt + Adϕ) + U (sin ϕdρ + ρ cos ϕdϕi ) + bη(sin ϕdρ + ρ cos ϕdϕ), H2 = −ρ sin ϕ (dt + Adϕ) − U (cos ϕdρ − ρ sin ϕdϕ) + bη(cos ϕdρ − ρ sin ϕdϕ), (4.54)

ai ρ 2 + (η − ηi )2 dϕ − bρ cos ϕ (sin ϕdρ + ρ cos ϕdϕ), H3 = −η dτ − i

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  where now U would be given by U = i Ui and A = i ai . The parameter b adds a constant to V ; for the Eguchi-Hanson metric we have b = 0 and for Taub-Nut b = 1. The case of infinite array of monopoles periodically distributed over the axis has been also considered in the literature; and this was within the context of D-instantons in type IIA superstring theory [39]. All these examples concern the Ward-type spaces in fourdimensions. From (2.10), one can find the corresponding G 2 holonomy metrics, which again take the form g7 =

2 µ (M + Qµ)  dt + ρUρ dϕ + µ (M + Qµ) Uη (dρ 2 + dη2 + ρ 2 dϕ 2 ) Uη +

(dβ + Q H1∗ )2 (dα + H2∗ )2 + + µ2 (M + Qµ)2 dµ2 , (M + Qµ)2 µ2

(4.55)

being H1∗ and H2∗ any pair of forms selected among those in the hyper-Kähler triplet. Again, notice that expression (4.55) gives essentially a pair of G 2 metrics for a given Ward space ( 4.51). 5. Non-trivial T 2 Bundle over Hyper-Kähler All the G 2 metrics considered in the previous sections are solutions of the Apostolovc u = 0. As we Salamon system (2.4), (2.5) and (2.2) together with the condition d M have seen, Eq. (2.5) together with the integrability condition for H1 implies that ∂ J
1 /∂µ should be µ-independent and closed. We have selected the solution J
1 = (M + Qµ)J 1 , being J 1 a closed two-form. The resulting base space was found to be hyper-Kähler with respect to certain triplet of 2-forms J i . Nevertheless, there exist in the literature examples of hyper-Kähler structures (g4 , Ji , J i ) which also admit a strictly almost Kähler structure (g4 , J0 , J 0 ) compatible with the opposite orientation defined by J 1 , J 2 , J 3 [40–43]. This means that J i ∧ J i = −J 0 ∧ J 0 . Being “strictly almost hyper-Kähler” means that the 2-form J 0 is closed though the corresponding almost complex structure J0 defined by J 0 = g4 (J0 ·, ·) is not integrable. Thus J0 is not a complex structure. If this is the case, then we can consider the 2-form J
1 = (M + Qµ)J 1 + (M  + Q  µ)J 0 ,

(5.56)

for which the integrability condition d 2 H1 = 0 is also satisfied. From (2.2) an algebraic equation for u is obtained with solution   u = µ (M + Qµ)2 − (M  + Q  µ)2 , (5.57) M  and Q  being two additional parameters. This solution again defines a G 2 holonomy space, which is in principle different from the one with Q  = M  = 0 of the previous sections. The task of finding the corresponding G 2 holonomy metric is a little bit more complicated because the four dimensional base metric
g4 (µ) will not be simply given by a µ-dependent scaling of the hyper-Kähler metric g4 , as it was before. It is better to illustrate how to construct the G 2 metric with an example. Let us consider the distance element g4 = x(d x 2 + dy 2 + dz 2 ) +

1 1 1 (dt + zdy − ydz)2 . x 2 2

(5.58)

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It is not hard to see that such a metric tensor is of the Gibbons-Hawking type (3.21) and is therefore hyper-Kähler. Let us define the positive and negative oriented triplets 1 ± J 1 = (dt + zdy) ∧ dz ± xd x ∧ dy, 2 1 1 ± J 2 = (dt + zdy − ydz) ∧ d x ± xdy ∧ dz, 2 2 1 ± J 3 = (dt − ydz) ∧ dy ± xdz ∧ d x. 2

(5.59)

Being “negative oriented” means that the square of such forms is minus the volume form of M. The metric (5.58 ) is hyper-Kähler with respect to the positive oriented triplet. But − − it is easy to see that also d J 1 = d J 3 = 0. One can also consider any rotated 2-form −

−

−

J θ = cos θ J 1 − sin θ J 3 , where θ runs from 0 to 2π . Such forms will be also closed and we have a whole circle of − negative oriented symplectic forms J θ . Nevertheless the almost complex structures Jθ− − associated to J θ are not integrable, that is, their Nijenhuis tensor is not zero. Thus they are not truly complex structures. This means that the metric (5.58) admits a circle bundle of negative oriented almost Kähler structures which are not Kähler [40–43]. This is the situation that we were talking about. The structures of this kind are known as strictly almost Kähler, for obvious reasons. − Now, let us select θ = 0 for simplicity, and take J 1 as J 0 in (5.56). Then we have + − J
1 = (M + Qµ)J 1 + (M  + Q  µ)J 1 .

(5.60)

It is convenient to introduce the basis
e1 =

(dt + 21 zdy − 21 ydz) , √ x


e2 =

√

xdz,


e3 =

√

xd x,


e4 =

√

xdy, (5.61)

for the metric (5.58). Then the positive and negative oriented triplet will be written as ±

J1 =
e1 ∧
e2 ±
e3 ∧
e4 ,

±

J2 =
e1 ∧
e3 ±
e4 ∧
e2 ,

±

J3 =
e1 ∧
e4 ±
e2 ∧
e3 , (5.62)

and, combining (5.62) with (5.60), we obtain that J
1 = (δ+ M + δ+ Qµ)
e1 ∧
e2 + (δ− M + δ− Qµ)
e3 ∧
e4 .

(5.63)

Here we have denoted δ± M = M ± M  and δ± Q = Q ± Q  . By making the redefinitions e1 = (δ+ M + δ+ Qµ)1/2
e1 , e3 = (δ− M + δ− Qµ)1/2
e3 ,

e2 = (δ+ M + δ+ Qµ)1/2
e2 , e4 = (δ− M + δ− Qµ)1/2
e4 ,

we see that the 2-form (5.63) becomes J
1 = e1 ∧ e2 + e3 ∧ e4 .

(5.64)

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Thus J
1 is the Kähler form of the metric
g4 (µ) = δab ea ⊗ eb . The explicit expression for g4 (µ) can be obtained from (5.64) and (5.61); the result is g4 (µ) = (δ− M + δ− Qµ) x (d x 2 + dy 2 ) + (δ+ M + δ+ Qµ)   1 1 1 2 2 × xdz + (dt + zdy − ydz) . x 2 2

(5.65)

We immediately observe that
g4 (µ) is not proportional to the hyper-Kähler metric (5.58) except if Q  = M  = 0, which corresponds to the cases considered in the previous sections. Now, we can construct a G 2 holonomy metric for (5.65) by means of (2.3); the result is   dα + H2 2 (dβ + H1 )2 + g7 = (M + Qµ)2 − (M  + Q  µ)2 µ   2 2   2 + µ (M + Qµ) − (M + Q µ) dµ2 + µ(δ− M + δ− Qµ) x (d x 2 + dy 2 ) + µ (δ+ M + δ+ Qµ)   1 1 1 × xdz 2 + (dt + zdy − ydz)2 , x 2 2

(5.66)

where the forms H1 and H2 are defined by −

d H1 = ∂µ J
1 = Q J 1 + Q  J 1 , +

+

d H2 = −J 2 .

(5.67)

It is really not difficult to obtain the explicit expressions of Hi from (5.59). Therefore the expression (5.66) is explicit. As before, more metrics can be obtained by selecting another element of the hyper-Kähler triplet in order to solve (5.67). There is another way to check that the expression (5.65) for the G 2 metric is correct. Let us consider the calibration form (2.8). Then from (5.57), (5.56) and (2.11) we find that   + −  = (M + Qµ)J 1 + (M  + Q  µ)J 1 ∧ e6 + e5 ∧ e6 ∧ e7   (5.68) + µ (M + Qµ)2 − (M  + Q  µ)2 J 2 ∧ e7 + J 3 ∧ e5 . By using (5.62) together with the redefinitions (5.64) it can be checked again that  takes the octonionic form  = cabc ea ∧ eb ∧ ec . The corresponding G 2 metric is g7 = δab ea ⊗ eb and after some calculation the expression (5.66) is found, which is what we wanted to show. Although in principle the metric (5.66) contains four parameters (Q , M, Q  , M  ), only two of them are effective ones. In fact, by a convenient scaling in (5.64) we can select δ+ M and δ− M equal to one, which means that M = 1 and M  = 0. Therefore the G 2 extension presented in this subsection add two parameters to the 4-dimensional base space, unlike the extensions considered in previous sections, which added only one. Nevertheless here we are imposing much stronger conditions on the 4-base metric. It should be not only hyper-Kähler, but also should possess a bundle of opposite oriented strictly almost Kähler structures. Only a few such spaces are known in the literature, and they are usually too simple. For instance, the 4-metric that we have presented in this subsection is one of the simplest Gibbons-Hawking ones, and contains no parameters.

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The resulting G 2 metric possesses two effective parameters, the same as the G 2 metrics presented in the previous sections. Investigating the existence of less trivial examples of this kind does deserve attention. 6. Half-flat Associated Metrics Now, let us revisit the discussion of Ref. [2–9] about the half-flat metrics that are related to the G 2 spaces discussed here. For all the G 2 holonomy spaces presented so far, we can consider the six-dimensional hyper-surfaces corresponding to the foliation µ = const. Then it follows from (2.10) that the metrics that have the form g6 = c1 (dβ + Q H1 )2 + c2 (dα + H2 )2 +
g4 ,

(6.69)

are then defined over such hyper-surfaces. Here, c1 and c2 are simply constants. As we will show below, metrics (6.69) are half-flat spaces [2]. These spaces are of interest in physics, especially in heterotic string compactifications [48–51] It is not difficult to see that there exists a coordinate system for which metrics (2.10) take the simple form g7 = dτ 2 + g6 (τ ),

(6.70)

g6 (τ ) being a six-dimensional metric depending on τ as an evolution parameter. In fact, by introducing the new variable τ, defined by µ2 (M + Qµ)2 dµ2 = dτ 2 ,

(6.71)

it can be seen that (2.10) takes the desired form. Therefore these G 2 holonomy metrics are a wrapped product Y = Iτ × N  , I being a real interval. The coordinate τ is just a function of µ and is given by  Mµ2 Qµ3 τ − τ0 = µ (M + Qµ) dµ = + . (6.72) 2 3 An aspect to be emphasized is that g6 (τ ) is a half-flat metric on any hypersurface Yτ for which τ takes a constant value. Indeed, the G 2 structure (2.8) can be decomposed as 3 ,  = J ∧ dτ + ψ 1 3 ∧ dτ + J∧ J, ∗ = ψ 2

(6.73) (6.74)

where we have defined J = z 1/2 J 3 + z −1/2 (dβ + A1 ) ∧ (dα + A2 ), 3 = z −1/2 J
1 ∧ (dα + A2 ) + µJ 2 ∧ (dβ + A1 ), ψ 3 = µ z −1/2 J 2 ∧ (dα + A2 ) − µ2 z 1/2 J
1 ∧ (dβ + A1 ), ψ

(6.75) (6.76) (6.77)

and z = µ2 (M + Qµ)2 . Then the G 2 holonomy conditions d = d ∗  = 0 for (2.8) yield 3 ∂ψ 3 + (d J − ) ∧ dτ = 0, d = d ψ ∂τ ∂ J 3 + J ∧ d ∗  = J ∧ d J + (d ψ ) ∧ dτ = 0. ∂τ

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The last equations are satisfied if and only if 3 = J∧ d J = 0 dψ

(6.78)

for any fixed value of τ , and 3 ∂ψ = d J, ∂τ

∂ J 3 . J∧ = −d ψ ∂τ

(6.79)

These flow equations were considered by Hitchin in a rather different context, concerning a certain Hamiltonian system whose details are not important here; see [26] and [2]. Equations (6.78) imply that, for every constant value τ, the metric g6 , together with 3 and ψ  , form a half-flat or half-integrable structure [25]. A constant value for τ J, ψ 3 implies a constant value for µ, and the generic form of such a half-flat metric corresponds to (6.69). The reasoning presented above can be also applied to the metric (5.66) by defining τ by  τ − τ0 = µ (1 + Qµ)2 − Q  2 µ2 dµ. As an example, let us consider the G 2 metric (2.15). From (6.69), the following half-flat space is obtained g6 = c1 (dυ − Q x dz − Q y dt)2 + c2 (dχ − y dz − x dt)2 + (d x 2 + dy 2 + dz 2 + dt 2 ).

(6.80)

There are no technical difficulties in finding the half-flat metrics corresponding to each G 2 holonomy metric described throughout this work. Therefore a family of half-flat metrics for the stringy cosmic string, the Eguchi-Hanson, the Taub-Nut, the almost Kähler and the Ward cases have been found through this procedure. 7. Toric Spi n(7) Holonomy Metrics 7.1. Spin(7) metrics that are T 3 bundle over hyper-Kähler. Now, we will dedicate the last section to try to extend the construction described here to the case of eightdimensional spaces with special holonomy in Spin(7). In reference [1] a construction of eight-dimensional Spin(7) metrics as T 3 bundles over hyper-Kähler metrics was presented. This construction is actually analogous to (2.10) for the G 2 holonomy case and leads to the following Spin(7) metric: g8 =

(dα + H1 )2 (dβ + H2 )2 (dγ + H3 )2 + + + µ6 dµ2 + µ3 g4 , µ2 µ2 µ2

(7.81)

where, as before, the metric g4 is hyper-Kähler and the 1-forms Hi are given by d Hi = J i . There are no major difficulties in proving that the holonomy of this metric is in Spin(7). Indeed, by defining the tetrad basis e0 = µ3 dµ,

e1 =

dα + H1 , µ

e2 =

dβ + H2 µ

e3 =

dγ + H3 , µ

ei = µ3/2 ei ,

Toric G 2 and Spin(7) Holonomy Spaces from Gravitational Instantons

395

where ei is a tetrad for the hyper-Kähler basis and where the indices run over i = 1, 2, 3, 4 and a = 1, 2, 3. It follows that the 4-form defined by the dual octonion constants cabcd 4 = cabcd ea ∧ eb ∧ ec ∧ ed = e0 ∧ e1 ∧ e2 ∧ e3 + + µ3 (e0 ∧ ea +

abc a a e ∧ eb ) ∧ J , 2

µ6 a a J ∧J 6

turn out to be closed. The presence of such a closed form is characteristic from the reduction from S O(8) from Spin(7). It is possible to deform this metric in order to get a new one, which will be again a T 3 bundle over an hyper-Kähler base, but now containing two more effective parameters. The natural deformation ansatz from ( 7.81) would be g8 = µ2 (M1 + Q 1 µ)2 (M2 + Q 2 µ)2 dµ2 + µ (M1 + Q 1 µ) (M2 + Q 2 µ) g4 (dα + Q 1 H1 )2 (dβ + Q 2 H2 )2 (dγ + H3 )2 + + + , (7.82) (M1 + Q 1 µ)2 (M2 + Q 2 µ)2 µ2 where Mi and Q i are four real parameters. By defining the tetrad basis dα + Q 1 H1 dβ + Q 2 H2 , e2 = , M1 + Q 1 µ M2 + Q 2 µ dγ + H3 , ei = µ (M1 + Q 1 µ) (M2 + Q 2 µ) ei , e3 = µ e1 =

it can be checked that also the 4-form 4 = cabcd ea ∧ eb ∧ ec ∧ ed = e0 ∧ e1 ∧ e2 ∧ e3 µ2 (M1 + Q 1 µ)2 (M2 + Q 2 µ)2 a a + J ∧J 6 abc a a e ∧ eb ) ∧ J , + µ (M1 + Q 1 µ) (M2 + Q 2 µ)(e0 ∧ ea + 2 is closed. Therefore, the deformation (7.82) also defines an Spin(7) holonomy metric. Although there are four parameters in the expression (7.82), only two of them are effective. It is easy to see that, when M1 and M2 are nonzero, we can set M1 = M2 = 1 by rescaling
g4 → M1 M2
g4 ⇒ H1 → M1 M2 H1 , H2 → M1 M2 H2 , α → M12 M2 α, β → M1 M22 β, γ → M1 M2 γ ,

H3 → M1 M2 H3 , Qi Qi → . Mi

It is actually feasible to extend any of the hyper-Kähler basis considered along this work to the case of metrics of Spin(7) holonomy. By construction, the resulting metrics will possess four commuting Killing vectors at least. For instance, for the hyper-Kähler

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metrics (3.29 ) the resulting Spin(7) metrics will be given by  2 dβ − Q 1 (xdt + 21 T dw) (dα − Q 2 (wdt + T d x))2 + g8 = 2 (1 + Q 1 µ) (1 + Q 2 µ)2 (dα − (wdt + T d x))2 + µ2 + µ2 (1 + Q 1 µ)2 (1 + Q 2 µ)2 dµ2 + µ (1 + Q 1 µ) (1 + Q 2 µ)   |dt + τ d x|2 + τ2 dwdw , × τ2

(7.83)

τ being an holomorphic function on the variable w and T its primitive. This metric possesses five commuting Killing vectors ∂α , ∂β , ∂γ , ∂t and ∂x . This procedure can be extended to the Eguchi-Hanson, Taub-Nut and Ward cases straightforwardly. 7.2. Non trivial T 3 bundle over hyper-Kähler. As a second example, let us comment on a case which is a non-trivial example of T 3 bundle over hyper-Kähler. We can generalize our discussion of Sect. 5 in order to find metrics that are not of the Gibbons-Lü -Pope-Stelle type. As before, let us consider an hyper-Kähler structure ( g4 , Ji , J i ) which also admits a strictly almost K ähler structure (g4 , J0 , J 0 ) compatible with the opposite orientation defined by J 1 , J 2 , J 3 . Let us note that the 4-form 4 of an eight-dimensional metric g8 = δab ea ⊗ eb can be expressed as 4 = e 0 ∧ e 1 ∧ e 2 ∧ e 3 +

J
a ∧ J
a abc b + (e0 ∧ ea + e ∧ ec ) ∧ J
a , 6 2

where J
1 = e5 ∧ e6 + e7 ∧ e8 ,

J
2 = e5 ∧ e7 + e8 ∧ e6 ,

J
3 = e5 ∧ e8 + e6 ∧ e7 . (7.84)

For the deformed Gibbons-Lü-Pope-Stelle metrics we have that i J
i = µ (M1 + Q 1 µ) (M2 + Q 2 µ)J ,

(7.85)

i

J being the hyper-Kähler triplet of the hyper-Kähler base metric. But if also (g4 , J0 , J 0 ) defines an strictly almost Kähler structure then Eq. (5.56) suggests that we modify the definition (7.85) and consider   1 0 J
1 = µ (M2 + Q 2 µ) (M1 + Q 1 µ) J + (M1 + Q 1 µ) J , 2 J
2 = µ (M2 + Q 2 µ) (M1 + Q 1 µ)2 − (M1 + Q 1 µ)2 J , (7.86) 3 J
3 = µ (M2 + Q 2 µ) (M1 + Q 1 µ)2 − (M1 + Q 1 µ)2 J , where Q 1 and M1 are new parameters. There always exists an einbein ei for which the hyper-Kähler triplet is written as J 0 = e1 ∧ e2 − e3 ∧ e4 ,

J 1 = e1 ∧ e2 + e3 ∧ e4 ,

J 2 = e1 ∧ e3 + e4 ∧ e2 ,

J 3 = e1 ∧ e4 + e2 ∧ e3 .

(7.87)

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Then the natural generalization of (5.64) for an einbein which “diagonalizes” the tensors J
i is given by e1 = µ1/2 (M2 + Q 2 µ)1/2 (δ+ M1 + δ+ Q 1 µ)1/2 e1 , e2 = µ1/2 (M2 + Q 2 µ)1/2 (δ+ M1 + δ+ Q 1 µ)1/2 e2 , e3 = µ1/2 (M2 + Q 2 µ)1/2 (δ− M1 + δ− Q 1 µ)1/2 e3 , e =µ 4

1/2

(M2 + Q 2 µ)

1/2

(δ− M1 + δ− Q 1 µ)

M1

(7.88)

e ,

1/2 4

Q 1 .

and δ± Q 1 = Q 1 ± In terms of this basis it is easy to where δ± M1 = M1 ± check by using (7.87) and (7.88) that the expressions (7.86) for J
i take the diagonal form (7.84), which is what we need. Also, by analogy with the cases discussed in the previous sections, we define the 1-forms dα + H1 dβ + H2 , e1 = , e2 = M2 + Q 2 µ (M1 + Q 1 µ)2 − (M1 + Q 1 µ)2 dγ + H3 e3 = , e0 = µ (M2 + Q 2 µ) (M1 + Q 1 µ)2 − (M1 + Q 1 µ)2 dµ, (7.89) µ where the forms Hi will be given now by the equations d H1 = Q 1 J 1 + Q 1 J 0 ,

d H2 = Q 3 J 2 ,

d H3 = J 3 .

(7.90)

The form 4 corresponding to (7.89) is 4 = dµ ∧ (dα + H1 ) ∧ (dβ + H2 ) ∧ (dγ + H3 )   µ2 (M2 + Q 2 µ)2 a a f 2 − f 2 J ∧ J + 6   1 0 + (dβ + H2 ) ∧ (dγ + H3 ) ∧ f J + f  J + (M2 + Q 2 µ)(dγ + H3 ) ∧ (dα + H1 ) ∧ J + µ (dα + H1 ) ∧ (dβ + H2 ) ∧ J

3

+ µ (M2 + Q 2 µ) dµ ∧ (dα + H1 ) ∧ 2

2

2



1



f J + f J

0



+ µ (M2 + Q 2 µ)   2 3 ×( f 2 − f 2 ) µ dµ ∧ (dα + H2 ) ∧ J + (M2 + Q 2 µ) dµ ∧ (dγ + H3 ) ∧ J , where we have defined f = M1 + Q 1 µ and f  = M1 + Q 1 µ. By virtue of (7.90) it follows that d4 = 0, therefore 4 defines an Spin(7) holonomy metric. The expression for the metric g8 = δab ea ⊗ eb is given by     (dα + H1 )2 dβ + H2 2 dγ + H3 2 + + g8 = (M1 + Q 1 µ)2 − (M1 + Q 1 µ)2 M2 + Q 2 µ µ + µ (M2 + Q 2 µ) g4 (µ), (7.91) where we have defined the one parameter depending on a four-dimensional metric g4 (µ) = (δ+ M1 + δ+ Q 1 µ)(e1 ⊗ e1 + e2 ⊗ e2 ) + (δ− M1 + δ− Q 1 µ)(e3 ⊗ e3 + e4 ⊗ e4 ).

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Therefore, if we deal with an hyper-Kähler basis which is also strictly almost Kähler with a Kähler form with opposite orientation to the one defined by the hyper-Kähler triplet, then the expression (7.91 ) gives a Spin(7) metric if Eqs. (7.90) are satisfied. This result can be applied for instance to the example (5.58). It is not difficult to check that the number of effective parameters appearing in this expression is three; we can select M1 = M2 = 1 and M1 = 0 by an appropriate rescaling of coordinates. Metric (7.91) is constructed with an einbein ei of an hyper-Kähler metric, and is a non trivial example of T 3 bundle over hyper-Kähler. 7.3. Almost G 2 holonomy hypersurfaces living inside Spin(7) metrics. All the Spin(7) metrics obtained in the previous subsections are of the form g8 = dτ + g7 (τ ), τ being a certain coordinate. This means that all these spaces are foliated by equidistant hypersurfaces and the coordinate τ is the distance to a fixed hypersurface M. For instance for the metrics (7.81) the coordinate τ is defined by τ − τ0 =

µ4 . 4

In such cases we have that e0 = dτ and that the eight-space over which the metric is defined is decomposed as M8 = Iτ × M7 , Iτ being a real interval. The closed 4-form 4 = e 0 ∧ e 1 ∧ e 2 ∧ e 3 +

J
a ∧ J
a abc b + (e0 ∧ ea + e ∧ ec ) ∧ J
a 6 2

can be expressed in this coordinate as 4 = dτ ∧ ω + ∗7 ω, where ∗7 is the Hodge star operation defined on M7 . The closure of 4 implies that d(∗7 ω) = 0, ∂τ (∗7 ω) = d(ω). These equations were considered in [26]. The second is known as the gradient flow equation. The first implies that the 3-form ω is co-closed. The seven-spaces with this property are called almost G 2 holonomy spaces. Technically, there are no difficulties to find the almost G 2 holonomy metrics for the examples presented in this section. The surfaces τ are constant as are those for which µ is constant. The expression for these almost G 2 metrics is g4 , g7 = c1 (dα + H1 )2 + c2 (dβ + H2 )2 + c3 (dγ + H3 )2 +


(7.92)

g4 being an hyper-Kähler metric, and where Hi refers to the with ci being constants,
usual one-forms satisfying d Hi = J i . In fact, by making use of (7.92), any of the hyper-Kähler metrics presented throughout this work can be extended to an almost G 2 holonomy metric straightforwardly. Acknowledgement. We are grateful to Sergey Cherkis and José Edelstein for reading the manuscript and for important suggestions; we thank them for their interest in our work. We thank the Commun. Math. Phys. referee for the corrections and very important remarks. We also thank Jorge Russo for pointing out interesting references. This work was partially supported by CONICET and Universidad de Buenos Aires.

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50. Gurrieri, S., Lukas, A., Micu, A.: Phys. Rev. D 70, 126009 (2004) 51. Gurrieri, S., Micu, A.: Class. Quant. Grav. 20, 2181 (2003) Communicated by G.W. Gibbons

G. E. Giribet, O. P. Santillán

Commun. Math. Phys. 275, 401–442 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0303-6

Communications in

Mathematical Physics

Unitarity in “Quantization Commutes with Reduction” Brian C. Hall
, William D. Kirwin University of Notre Dame, Notre Dame, IN 46556-4618, USA. E-mail: [email protected]; [email protected] Received: 25 September 2006 / Accepted: 2 March 2007 Published online: 8 August 2007 – © Springer-Verlag 2007

Abstract: Let M be a compact Kähler manifold equipped with a Hamiltonian action of a compact Lie group G. In this paper, we study the geometric quantization of the symplectic quotient M//G. Guillemin and Sternberg [Invent. Math. 67, 515–538 (1982)] have shown, under suitable regularity assumptions, that there is a natural invertible map between the quantum Hilbert space over M//G and the G-invariant subspace of the quantum Hilbert space over M. Reproducing other recent results in the literature, we prove that in general the natural map of Guillemin and Sternberg is not unitary, even to leading order in Planck’s constant. We then modify the quantization procedure by the “metaplectic correction” and show that in this setting there is still a natural invertible map between the Hilbert space over M//G and the G-invariant subspace of the Hilbert space over M. We then prove that this modified Guillemin–Sternberg map is asymptotically unitary to leading order in Planck’s constant. The analysis also shows a good asymptotic relationship between Toeplitz operators on M and on M//G. 1. Introduction Let M be an integral compact Kähler manifold with symplectic form ω. Following the program of geometric quantization, suppose we are given a Hermitian holomorphic line bundle  with connection over M, chosen in such a way that the curvature of  is equal to −iω. We then consider ⊗k , the k th tensor power of , where in this setting we interpret k as the reciprocal of Planck’s constant
. The Hilbert space of quantum states associated to M is then (for a fixed value of
= 1/k) the space of holomorphic sections of ⊗k . Now suppose that we are given a Hamiltonian action of a connected compact Lie group G on M. Then we can construct the symplectic quotient M//G, which is another
Supported in part by NSF Grant DMS-02000649.

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compact Kähler manifold (under suitable regularity assumptions on the action of G). The line bundle  naturally descends to a bundle ˆ over M//G, and the quantum Hilbert space associated to M//G is then the space of holomorphic sections of ˆ⊗k . This space of sections is what we obtain by performing reduction by G before quantization. Alternatively, we may first perform quantization and then perform reduction at the quantum level, which amounts to restricting to the space of G-invariant sections of ⊗k . A classic result of Guillemin and Sternberg is that there is a natural invertible linear map between the “first reduce and then quantize” space (the space of all holomorphic sections of the bundle ˆ⊗k over M//G) and the “first quantize and then reduce” space (the space of G-invariant holomorphic sections of the bundle ⊗k over M). This result is sometimes described as saying that quantization commutes with reduction. However, from the point of view of quantum mechanics, it is not just the vector space structure of the quantum Hilbert space that is important, but also the inner product. It is natural, then, to investigate the extent to which the Guillemin–Sternberg map is unitary. Some work in this direction has already been done, and in general, it has been found that the obstruction to unitarity is a certain function on M//G (sometimes referred to as the effective potential). This obstruction, in some form, was identified independently in the works [Flu98, Char06b, Pao05 and MZ06]. See also [Got86, Hal02 and Hue06]. We discuss the relation of our work to these in Sect. 1.2. Our first main result is a new proof (similar to that of Charles in [Char06b] for the torus case) that the Guillemin–Sternberg map is not unitary in general, and indeed that the map does not become asymptotically unitary as k → ∞ (i.e., as
→ 0). We show that the obstruction to asymptotic unitarity of the map is the volume of the G-orbits inside the zero-set of the moment map in M. If these G-orbits do not all have the same volume, then the Guillemin–Sternberg map will not be asymptotically unitary. This failure of asymptotic unitarity is troubling from a physical point of view. After all, although one expects different quantization procedures (e.g., performing quantization and reduction in different orders) to give different results, one generally regards these differences as “quantum corrections” that should disappear as
tends to zero. The main contribution of this paper is a solution to this unsatisfactory situation: we include the so-called metaplectic correction, which involves tensoring the original line bundle with the square root of the canonical bundle (assuming that such a square root exists). We first show that one can define a natural map of Guillemin–Sternberg type in the presence of the metaplectic correction, and this natural map is invertible for all sufficiently large values of the tensor power k. We then show that this modified Guillemin–Sternberg map, unlike the original one, is asymptotically unitary in the limit as k tends to infinity. In the rest of this introduction, we describe our results in greater detail and compare them to previous results of other authors. 1.1. Main results. Let M 2n be a compact Kähler manifold with symplectic form ω, complex structure J, and Riemannian metric B = ω(·, J ·). Assume that M is quantizable, that is, that the class [ω/2π ] is integral. Choose a Hermitian line bundle  over M with compatible connection ∇ in such a way that the curvature of ∇ is −iω. (Such a bundle exists because M is quantizable.) We call  a prequantum bundle for M. We denote the Hermitian form on , which we take to be linear in the second factor, by (·, ·) and we denote the pointwise magnitude of a section s of  by |s|2 (x) = (s, s)(x). The connection and Hermitian form on  induce a connection and Hermitian form on ⊗k which we denote by the same symbols.

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The complex structure on M gives ⊗k the structure of a holomorphic line bundle in which the holomorphic sections are those that are covariantly constant in the (0, 1)-directions. We let H(M; ⊗k ) denote the (finite-dimensional) space of holomorphic sections of ⊗k . The symplectic form on M induces a volume form εω :=

ω∧n , n!

(1.1)

which is known as the Liouville volume form. We make H(M; ⊗k ) into a Hilbert space by considering the inner product given by
s1 , s2  := (k/2π )n/2 (s1 , s2 ) εω . (1.2) M

In geometric quantization, we interpret the tensor power k as the reciprocal of Planck’s constant
. Thus the study of holomorphic sections of ⊗k in the limit k → ∞, familiar from algebraic geometry, is in this setting interpreted as the “semiclassical limit” (i.e., the limit
→ 0). The semiclassical limit in geometric quantization has garnered much recent interest. See for example the work of Borthwick and Uribe [BU96] and [BU00] on a symplectic version of the Kodaira embedding theorem and the work of Borthwick, Paul, and Uribe [BPU95] on applications to relative Poincaré series. Suppose now that we are given a Hamiltonian action of a connected compact Lie group G of dimension d, together with an equivariant moment map  for this action. We assume (as in [GS82]) that 0 is in the image of , that 0 is a regular value of , and that G acts freely on −1 (0). The symplectic quotient M//G is then defined to be M//G = −1 (0)/G. The quotient M//G acquires from M the structure of a quantizable Kähler manifold of dimension 2(n − d); in particular the symplectic form ω on M descends to a symplectic form  ω ∈ 2 (M//G). We assume that the action of G lifts to . Then  descends to a holomorphic Hermitian line bundle ˆ with connection over M//G, with curvature −i ω. Let H(M//G; ˆ⊗k ) denote the space of holomorphic sections of ˆ⊗k and use on this space the inner product given by
s1 , s2  := (k/2π )(n−d)/2 (s1 , s2 ) ε (1.3) ω. M//G

We are interested in the relationship between two different Hilbert spaces: first, the space of G-invariant holomorphic sections of ⊗k over M, with the inner product (1.2); and second, the space of all holomorphic sections of ˆ⊗k over M//G, with the inner product (1.3). The first Hilbert space, denoted H(M; ⊗k )G , is the one obtained by first quantizing and then reducing by G; the second one is obtained by first reducing by G and then quantizing. According to Guillemin and Sternberg [GS82], there is (for each k) a natural one-to-one and onto linear map Ak from H(M; ⊗k )G onto H(M//G; ˆ⊗k ). In particular, these two spaces have the same dimension. The map Ak is defined in the only reasonable way: one takes a G-invariant holomorphic section s of ⊗k , restricts it to −1 (0), and then lets s descend from −1 (0) to −1 (0)/G. From the way that the complex structure on M//G is defined, it is easy to see that Ak maps holomorphic sections of ⊗k to holomorphic sections of ˆ⊗k . The hard part is to show that this map is invertible, i.e., that a section of ˆ⊗k , after being

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lifted to −1 (0), can be extended holomorphically to a (G-invariant) section over all of M. In proving this, Guillemin and Sternberg make use of a holomorphic action of the “complexified” group G C on M. We wish to determine the extent to which the map Ak is unitary. We show that (for each k) there exists a function Ik on M//G with the property that for each G-invariant holomorphic section s of ⊗k we have

|s|2 εω = (k/2π )(n−d)/2 |Ak s|2 Ik ε s 2 = (k/2π )n/2 ω. M

M//G

Our first main result is the following. Theorem 1.1. The functions Ik satisfy lim Ik ([x0 ]) = 2−d/2 vol(G · x0 )

k→∞

for all x0 ∈ −1 (0), and the limit is uniform. Here vol(G · x0 ) denotes the volume of the G-orbit of x0 with respect to the Riemannian structure inherited from M. If all of the G-orbits in the zero-set of the moment map have the same volume, then Theorem 5.1 implies that the natural map Ak is asymptotically a constant multiple of a unitary map in the limit as
tends to zero. If, as is usually the case, the G-orbits in the zero-set do not all have the same volume, then using peaked sections, we can show that Ak is not asymptotically a constant multiple of a unitary map, in a sense described in Theorem 5.3. Now, it is probably unreasonable to expect the natural map between the “first reduce and then quantize” space and the “first quantize and then reduce” space to be unitary. Different ways of performing quantization (e.g., before reduction and after reduction) in general give inequivalent results, especially if equivalence is measured by the existence of a geometrically natural unitary (as opposed to merely invertible) map. On the other hand, one expects differences between the procedures to be “quantum corrections” that vanish for small
. The situation reflected in Theorem 1.1, then, is unsatisfactory; the geometrically natural map between H(M; ⊗k )G and H(M//G; ˆ⊗k ) is not unitary even to leading order in
. To remedy this situation, we introduce the “metaplectic” or “half-form” correction. There are various results which seem to indicate the geometric quantization works better with half-forms (indeed, our main results can be seen as further justification of this claim). For example, it is only with half-forms that the quantization of the simple harmonic oscillator has the correct zero-point energy (see [Woo91, Chap. 10] for details and further examples). For this reason, we might expect (and it is indeed the case) that the inclusion of half-forms remedies the unsatisfactory situation described by Theorem 1.1. The metaplectic correction consists of tensoring ⊗k with a square root of the canonical bundle K of M, assuming such a square root exists (equivalently, the vanishing of the second Stiefel–Whitney class), and similarly tensoring ˆ⊗k with a square √ root of the ⊗k  canonical bundle K of M//G. Assuming that the G-action lifts to  ⊗ K , our second main result is an analog of [GS82, Thm 5.2] in the presence of half-forms. Theorem 1.2. For each k, there is a natural √ linear map Bk between the space of ⊗k ⊗ G-invariant holomorphic sections of  K over M and the space of all holo√  over M//G. Furthermore, this map is invertible for all morphic sections of ˆ⊗k ⊗ K sufficiently large k.

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We then show that there exists√a function Jk on M//G such that for each G-invariant holomorphic section r of ⊗k ⊗ K we have

2 n/2 2 (n−d)/2 |r | εω = (k/2π ) |Bk r |2 Jk ε r := (k/2π ) ω. M//G

M

Our last main result is the following, which implies that the maps Bk , unlike the maps Ak , are asymptotically unitary in the limit k → ∞ (i.e.,
→ 0). Theorem 1.3. The functions Jk satisfy lim Jk ([x0 ]) = 1

k→∞

for all x0 ∈ −1 (0), and the limit is uniform. The origin of the unwanted volume factor in Theorem 1.1 is the relationship between the volume measure on −1 (0) and the volume measure on −1 (0)/G (Lemma 4.2). The metaplectic correction introduces a compensating volume factor in the pointwise magnitude of a section over −1 (0) and the pointwise magnitude of the corresponding section over −1 (0)/G (Theorem 3.3). The proofs of our main results make use of a holomorphic action of the “complexified” group G C on M, obtained in [GS82] by analytic continuation of the action of G. We let Ms denote the stable set, that is, the set of points in M that can be moved into the zero-set of the moment map by the action of G C . The stable set is an open set of full measure in M and the symplectic quotient M//G = −1 (0)/G can be identified naturally section (of ⊗k or √ with Ms /G C . We show how the magnitude of a G-invariant ⊗k −1  ⊗ K ) varies in a predictable way as one moves off of  (0) along a G C -orbit; this fact, which is crucial to our analysis, has been noticed before by Guillemin and Sternberg [GS82] and by Donaldson [Don04]. The densities Ik and Jk are obtained by integrating certain quantities related to the moment map over each G C -orbit. The asymptotic behavior of these densities is then determined by Laplace’s method (sometimes referred to as the stationary phase approximation or the method of steepest descent). In proving the invertibility of our Guillemin–Sternberg-type map Bk , we first show √ that a G-invariant section of ⊗k ⊗ K defined over −1 (0) can be extended to a holomorphic, G C -invariant section over the stable set Ms . We then show that for all sufficiently large k, these sections have a removable singularity over the complement of the stable set and thus extend to holomorphic sections over all of M. Here, the presence of half-forms makes the situation slightly more complicated than in [GS82], where the natural map Ak is shown to be invertible for all k. A trivial modification of the arguments described above yields a formula that relates a Toeplitz operator with a G-invariant symbol upstairs, restricted to the space of G-invariant sections, to a certain Toeplitz operator downstairs. Suppose f is a smooth, G-invariant function on M. Then in the case with half-forms, the result is that the Toeplitz operator with symbol f upstairs, restricted to the G-invariant subspace, is asymptotically equivalent to the Toeplitz operator with symbol fˆ downstairs, where fˆ is obtained by restricting f to −1 (0) and letting it descend to −1 (0)/G. For the reader’s convenience, we collect here the various assumptions made in this paper. We begin with a compact Kähler manifold M of real dimension 2n, with symplectic form ω. We assume that [ω/2π ] is an integral cohomology class. We assume that M is equipped with a holomorphic and Hamiltonian action of a compact d -dimensional

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Lie group G with equivariant moment map  : M → g∗ . We assume that 0 is in the image of the moment map, that 0 is a regular value of the moment map, and that G acts freely on the zero-set −1 (0). Since the symplectic form is integral, there exists over M at least one complex Hermitian line bundle with compatible connection with curvature −iω; we fix some choice and denote it by . In a canonical way, the moment map defines an infinitesimal holomorphic action of G on . We assume that this infinitesimal action can be exponentiated. These assumptions are the same as in [GS82] and are all the assumptions relevant to Theorem 1.1.  We need additional assumptions for Theorems 1.2 and 1.3. Let K = n√(T ∗ M)1,0 denote the canonical bundle of M. We assume that K admits a square root K → M (this is equivalent to the vanishing of the second Stiefel-Whitney class of M), and finally we assume that √ the G-action on K (which is induced from that on M) can be lifted to an action on K . 1.2. Prior results. To the best of our knowledge, the first work that systematically addresses the question of unitarity in the context of quantization and reduction of Kähler manifolds is the Ph.D. thesis of Flude [Flu98]. Flude gives a formal computation of the leading-order asymptotics of the density Ik as k tends to infinity, without the metaplectic correction. Flude’s main computation is essentially the same as ours (in the case without the metaplectic correction): an application of Laplace’s method. However, Flude’s result is not rigorous, because he considers the magnitude of an invariant section only in a small neighborhood of the zero-set. This should be compared to our Theorem 4.1, which gives an expression for the magnitude of an invariant section that is valid everywhere. (Even after Theorem 4.1 is obtained, some control is required over the blow-up of certain Jacobians near the unstable set.) Nevertheless, Flude does identify the volume of the G-orbits in −1 (0) as the obstruction to the asymptotic unitarity of the Guillemin–Sternberg map. Next, there is the work of L. Charles [Char06b], who considers the case in which the group G is a torus and without half-forms. (Charles does consider half-forms in a related context in [Char06a].) In the torus case, Proposition 4.17 of [Char06b], with f identically equal to 1 in a neighborhood of the zero-set, is essentially the same as our Theorem 4.2. Proposition 4.18 of [Char06b] is then essentially the same as the first part of our Theorem 5.1. We do, however, go slightly beyond the approach of Charles, in that we give an exact (not just asymptotic to all orders) expression for the norm of an invariant section upstairs as an integral over the downstairs manifold. This requires some control over the blow-up of certain Jacobians as one approaches the unstable set. Then there is the work of Paoletti [Pao05]. He takes a different approach to measuring unitarity, looking at the behavior of orthonormal bases. Nevertheless, his result agrees with ours (in the case without half-forms) in that he identifies the volume of the G-orbits in the zero-set of the moment map as an obstruction to unitarity of the Guillemin–Sternberg map. Finally, there is the work of Ma and Zhang in [MZ06]. In this paper, the authors compute an asymptotic expansion of the equivariant Bergman kernel, and our Theorem 1.1 is a special case of their Theorem 0.1. In contrast to our work and the works of Flude and Charles (all of which are based on Laplace’s method), those of Paoletti and Ma–Zhang are based on the microlocal analysis developed by L. Boutet de Monvel and Guillemin in [BdMG81]. To the best of our knowledge, ours is the first paper that constructs a Guillemin– Sternberg-type map in the presence of the metaplectic correction and therefore the first to show in a general setting that the metaplectic correction improves the situation in

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regard to unitarity. (Such an improvement had already been suggested by the special examples considered in [DH99] and [Hal02]. See the last section for a discussion of these examples.) We close this section by mentioning the work of Gotay [Got86], who considers the relationship between quantization and reduction in the context of a cotangent bundle equipped with the vertical polarization (a real polarization). Gotay considers the cotangent bundle of an n-manifold Q and assumes that a compact group G of dimension d acts freely on Q. The induced action of G on T ∗ Q is then free and Hamiltonian. Gotay includes half-forms in the quantization (as one must in the case of the vertical polarization) and obtains a unitary map between the quantize-then-reduce space and the reduce-then-quantize space. Gotay’s work does not overlap with ours, because we consider only complex polarizations. Nevertheless, it is worth noting that Gotay obtains exact unitarity, whereas we obtain unitarity only asymptotically as Planck’s constant tends to zero, even with halfforms. The reason that the cotangent bundle case is nicer than the Kähler case seems to be that there is less differential geometry involved. There is no additional structure on the manifold Q (metric or measure or complex structure) that enters. Gotay’s unitarity result comes down to the relationship between the integral over Q of a G-invariant n-form α and the integral over Q/G of an (n − d)-form αˆ obtained by contracting α with the generators of the G-action and then letting the result descend to Q/G. (Here α is the square of a half-form.) In the Kähler case, we have a mapping between (n, 0)-forms on M and (n − d, 0)-forms on M//G defined in a manner very similar to that in [Got86]. However, it does not make sense to integrate an (n, 0)-form over M, because M has real dimension 2n. So the norm of a half-form cannot be computed by simply squaring and integrating. Rather, one uses a more complicated procedure involving both the complex structure on M and the Liouville volume measure. The relationship between the norm upstairs and the norm downstairs is correspondingly more complicated and involves the geometric structures that we have on M but not on Q. Another way of thinking about the Kähler case is to observe that the way one lifts a section downstairs to a section upstairs is by lifting to a G C -invariant (not just G-invariant) section upstairs. However, the action of G C preserves only the complex structure on M and not the symplectic structure on M or the Hermitian structure on the relevant line bundles. Thus the relationship between the upstairs norm and the downstairs norm involves the way the volume measure and the magnitude of an invariant section vary over each G C -orbit. In the cotangent bundle case, by contrast, only a G-action is involved and this preserves all the relevant structure. 2. Preliminaries We begin this section by recalling the method of geometric quantization (at the moment, without half-forms) as it applies to Kähler manifolds. We then recall the notion of the Marsden–Weinstein or symplectic quotient and explain the special form this construction takes in the setting of Kähler manifolds. Finally, we describe the natural invertible map, due to Guillemin and Sternberg, between the “quantize then reduce” space and the “reduce then quantize” space. 2.1. Kähler quantization. Let (M, ω, J, B = ω(·, J ·)) be a Kähler manifold with symplectic form ω, complex structure J , and Riemannian metric B = ω(·, J ·). We assume

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that M is connected, compact, and integral (i.e., that [ω/2π ] is an integral cohomology class). We fix once and for all a Hermitian line bundle  with compatible connection ∇, chosen in such a way that the curvature of ∇ is equal to −iω. The connection on  induces a connection on the k th tensor power ⊗k , also denoted ∇. For any k, ⊗k may be given the structure of a holomorphic line bundle in such a way that the holomorphic sections are precisely those that are covariantly constant in the (0, 1) (or z¯ ) directions. We interpret the tensor power k as the reciprocal of Planck’s constant
. For each fixed value of k (or
), the quantum Hilbert space is the space of holomorphic sections of ⊗k , denoted H(M; ⊗k ). If εω denotes the Liouville volume form (1.1), then we use the following natural inner product on H(M; ⊗k ),
n/2 s1 , s2  := (k/2π ) (s1 , s2 ) εω , M

as in (1.2), where (s1 , s2 ) is the (pointwise) Hermitian structure on ⊗k . 2.2. Kähler reduction. We now assume that we are given a smooth action of a connected compact Lie group G of dimension d on M. We assume that the action preserves all of the structure (symplectic, complex, Riemannian) of M. Let g denote the Lie algebra of G and for each ξ ∈ g, let X ξ denote the  vector field describing the infinitesimal action d tξ of ξ on M. That is, X ξ (x) = dt e · x t=0 . We assume that for each ξ ∈ g there exists a smooth function φξ on M such that X ξ is the Hamiltonian vector field associated to φξ ; i.e., such that i(X ξ )ω = dφξ . (This is automatically the case if G is semisimple or if M is simply connected.) For each ξ, the function φξ is unique up to a constant. Since M is compact, it is always possible to choose the constants in such a way that the map ξ → φξ is linear and that {φξ , φη } = −φ[ξ,η] , and we fix one particular choice of constants with these two properties. (For any G, one way to choose the constants with these two properties is to require each φξ to have integral zero over M. If G is semisimple, then these two properties uniquely determine the choice of the constants.) We may put together the functions φξ into a “moment map”  : M → g∗ given by (x)(ξ ) = φξ (m) for each ξ ∈ g and m ∈ M. The condition {φξ , φη } = −φ[ξ,η] ensures that the moment map is equivariant with respect to the action of G on M and the coadjoint action of G on g∗ . We assume that 0 is in the image of the moment map and that it is a regular value, so that the zero-set −1 (0) is a submanifold of M. We assume, moreover, that G acts freely on −1 (0), so that the quotient is a manifold. The quotient M//G := −1 (0)/G is then called the symplectic or Marsden–Weinstein [MW74] quotient of M by G. The quotient M//G inherits a symplectic structure from M: there is a unique symplectic form  ω ∈ 2 (M//G) such that i ∗ ω = π ∗  ω, where i : −1 (0) → M denotes the inclusion map and π : −1 (0) → −1 (0)/G is the quotient map. Furthermore, [ ω/2π ] is an integral cohomology class on M//G.

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So far, we have described how the symplectic structure on M induces a symplectic structure on M//G. We now show how the complex structure on M induces a complex structure on M//G. The descent of the complex structure can be understood either “infinitesimally” (by describing how the distribution of (1, 0)-vectors descends) or “globally” (by realizing M//G as the quotient of the “stable set” in M by the complexification of G). We begin with the infinitesimal approach. Let T 1,0 M be the Kähler polarization on M; i.e., T 1,0 M is the n-dimensional (complex) distribution consisting of type-(1, 0) vector fields on M. For future use, we note that the projection π+ : T C M → T 1,0 M is given by π+ X =

1 (1 − i J )X. 2

In [GS82], Guillemin and Sternberg show that the G-orbits in the zero-set are totally real submanifolds; i.e., for all x0 ∈ −1 (0), M ∩ {X xξ0 : ξ ∈ g} = {0}. Tx1,0 0 Moreover, they show that T C (−1 (0)) ∩ T 1,0 M is a G-invariant complex distribution of complex rank n − d, and that π∗ (T C (−1 (0)) ∩ T 1,0 M) is a well-defined integrable complex distribution of rank n − d on M//G. In fact, the complex structure induced by this distribution defines a Kähler structure on M//G, and we henceforth make the identification T 1,0 (M//G) = π∗ (T C (−1 (0)) ∩ T 1,0 M). (2.1) We now turn to the global approach, which is more intuitive and much more useful for the computations we will carry out in this paper. We have assumed that the action of G is holomorphic. Using this, it can be shown [GS82] that the action of G can be analytically continued to a holomorphic action of the “complexified” group G C on M; the infinitesimal action of this continuation is defined by X iξ := J X ξ , ξ ∈ g.

(2.2)

Here G C is a connected complex Lie group containing G as a maximal compact subgroup, and the Lie algebra gC of G C is the complexification of g. The Cartan decomposition is a diffeomorphism G C exp(ig)G. Moreover, the set exp(ig) is diffeomorphic to the vector space g, the diffeomorphism being the exponential map, and so as smooth manifolds we have G C = g × G. See [Kna02, Sect. 6.3] for details. We let the stable set Ms denote the saturation of the zero-set −1 (0) by the action of G C : Ms := G C · −1 (0). (2.3) That is, Ms is the set of points in M that can be moved into −1 (0) by the action of G C . (See (2.5) below for another characterization of the stable set.) The following properties of Ms follow from results in [GS82]: (1) Ms is an open set of full measure in M; (2) G C acts freely on Ms ; and (3) each G C -orbit in Ms intersects −1 (0) in precisely one G-orbit. We will show shortly (Theorem 2.1) that Ms is in fact a principal G C -bundle over −1 (0)/G. This implies (Corollary 2.1) that the action of G C on Ms is proper. (Keep in mind that we are assuming that 0 is a regular value of the moment map and that G acts freely on −1 (0).) Because the action of G C is free, proper and holomorphic, the quotient Ms /G C has the structure of a complex manifold. On the other hand, since each

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G C -orbit in Ms intersects −1 (0) in precisely one G-orbit, we have a natural bijective identification −1 (0)/G = Ms /G C . (2.4) Once we know that Ms /G C has the structure of a complex manifold, it is not hard to see that this complex structure agrees (under the above identification) with the one obtained infinitesimally in (2.1). (If πC : Ms → Ms /G C is the quotient map and we identify Ms /G C with −1 (0)/G, then πC agrees with π on −1 (0). Thus (πC )∗ agrees with π∗ on vectors tangent to −1 (0). But it is not hard to show that every (1, 0)-vector at a point in −1 (0) is the sum of a (1, 0)-vector tangent to −1 (0) and a (1, 0)-vector tangent to the G C orbit.) Although we do not require this result, we explain briefly how the quotient Ms /G C can be identified with the quotient in geometric invariant theory (GIT). In GIT, there are several (in general, inequivalent) notions of stable points for the action of a complex reductive group (such as G C ) on a compact Kähler manifold. (In [MFK94], these are called semistable, stable, and properly stable points.) Under our assumptions—that 0 is a regular value of the moment map and that G acts freely on −1 (0)—these different notions of stability in GIT turn out to be equivalent to one another and to the notion of stability in (2.3). (As a consequence of a result of Kempf and Ness [KN79], the different notions of stability of y ∈ M in GIT are equivalent to: (1) the closure of G C · y intersects −1 (0), (2) G C · y itself intersects −1 (0), and (3) G C · y intersects −1 (0) and the stabilizer of y is finite. Condition 2 is our definition of the stable set. Since in our case Ms is open and acted on freely by G C , the other two conditions are equivalent to Condition 2.) The GIT quotient coincides, as a set and topologically, with Ms /G C . See for example Theorem 8.3 of [MFK94] or Sect. 3 of [HHL94]. The stable set can also be characterized in terms of the G-invariant holomorphic sections of ⊗k (see for example Donaldson’s work in [Don04], or [GS82, Theorem 5.5]): Ms = {x ∈ M : s(x) = 0 for some s ∈ H(M; ⊗k )G for some k}. (2.5) Suppose, then, that s is an element of s ∈ H(M; ⊗k )G that is not identically zero. (That such a section exists for some k follows from (2.5) and is established in [GS82, Appendix].) Then the unstable set M \ Ms is contained in the zero-set of s and is therefore a set of codimension at least one. This implies, for example, that the unstable set has measure zero. We now establish a structure result for Ms that implies, among other things, that the action of G C on Ms is proper. Let πC : Ms → M//G be the complex quotient map under the identification of Ms /G C with M//G. That is to say, πC (z) = π(x) if x belongs to the unique G-orbit in −1 (0) contained in the G C -orbit of z. Theorem 2.1. The map πC : Ms → M//G is smooth (in particular, continuous), and Ms has the structure of a smooth principal G C -bundle with respect to this map. Corollary 2.1. The action of G C on Ms is proper. Corollary 2.2. The map : g × −1 (0) → Ms given by (ξ, x) = eiξ · x is bijective and a diffeomorphism.

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Proof (Proof of Corollary 2.1). Suppose we have yk ∈ Ms and gk ∈ G C such that yk converges to y ∈ Ms and such that gk · yk converges to z ∈ Ms . We will show that gk must then be convergent, which implies that the action of G C on Ms is proper [DK00, pp. 53 ff. and Chap. 2]. Note that πC (gk yk ) = πC (yk ). Taking limits and using the continuity of πC gives that πC (z) = πC (y). Choose a neighborhood U of πC (z) such that πC−1 (U ) is homeomorphic to G C × U in such a way that the G C -action corresponds to left multiplication in G C . Under this homeomorphism, yk corresponds to (h k , u k ) and y corresponds to (h, u), with h k → h and u k → u. Then gk · yk corresponds to (gk h k , u k ) and z corresponds to ( f, u), with gk h k → f. Thus, gk converges to h −1 f.   Proof (Proof of Corollary 2.2). The bijectivity of follows from: (1) the freeness of the action of G C on Ms , (2) the fact that each G C -orbit intersects −1 (0) in a single G-orbit, and (3) the bijectivity of the Cartan decomposition of G C [Kna02, Thm 6.31, p. 362]. To see that is a diffeomorphism, decompose πC−1 (U ) as G C × U. Since the Cartan decomposition of G C is a diffeomorphism, we obtain πC−1 (U ) g × G × U. The quotient map π : −1 (0) → M//G is a principal G-bundle, so there is a slice N in −1 (0) which is transverse to the G-action and diffeomorphic to U . The free G-action on −1 (0) then yields a diffeomorphism from G × N to a G-invariant neighborhood M0 in the zero-set. Hence we obtain a local diffeomorphism g× M0 g×G × N πC−1 (U ) which corresponds to the map |g×M0 . This shows, in particular, that the Jacobian of is invertible at each point and hence that is a global diffeomorphism.   Proof (Proof of Theorem 2.1). Given a point u in −1 (0)/G, choose a neighborhood U of u, a point x ∈ −1 (0) with π(x) = u, and an embedded submanifold N through x which is transverse to the G-action, so that π maps N injectively and diffeomorphically onto U. Let π −1 : U → N be the smooth inverse to π. Now consider the map

: G C × U → Ms given by

(g, u) = g · π −1 (u). First, we establish that is injective (globally, not just locally). If u 1 and u 2 are distinct elements of N , then π −1 (u 1 ) and π −1 (u 2 ) are in distinct G-orbits. But distinct G-orbits in −1 (0) lie in distinct G C -orbits (see the comments following the proof of Theorem 4.1(a)). Thus, (g1 , u 1 ) = (g2 , u 2 ) if u 1 = u 2 . Then if g1 = g2 ,

(g1 , u) = (g2 , u), because G C acts freely on Ms . Next, we establish that the differential of is injective at each point. Since π maps N diffeomorphically onto U, it suffices to prove the same thing for the map  : G C × N → Ms given by (g, n) = g · n. So consider (g, n) ∈ G C × N . Since π maps N diffeomorphically onto U, Tn (−1 (0)) is the direct sum of Tn (N ) and Tn (G · n). Furthermore, for any ξ ∈ g, the vector field J X ξ is nonzero at n and orthogonal to the tangent space to −1 (0). It follows that Tn (M) = Tn (G C · n) ⊕ Tn (N ) and thus Tgn (M) = g∗ (Tn (G C · n)) ⊕ g∗ (Tn (N )) = Tgn (G C · n) ⊕ g∗ (Tn (N ))

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since the action of g takes G C · n into itself. Now suppose that α(t) = (g(t), n(t)) is a curve in G C × N passing through (g, n) at t = 0 and such that g  (0)g −1 = ξ ∈ gC and n  (t) = X. Then applying  and differentiating at t = 0 gives ξ ∗ (α  (0)) = X g·n + g∗ (X ),

which is nonzero provided that α  (0) is nonzero. Finally, we prove the theorem. Since is globally injective, it has an inverse map defined on its image, where the image of is simply πC−1 (U ). The inverse function theorem tells us, then, that πC−1 (U ) is an open set and that the inverse map to is smooth. Thus, πC−1 (U ) is diffeomorphic to G C × U in such a way that the action of G C corresponds to the left action of G C on itself. Under this diffeomorphism, πC corresponds to projection onto the second factor, which is smooth. Thus the diffeomorphism

shows that Ms → M//G has the necessary smooth local triviality property to be a smooth principal G C -bundle.   2.3. Quantum reduction and the Guillemin–Sternberg map. In the previous section, we considered reduction at the classical level, which amounts to passing from M to M//G. Alternatively, we may first quantize M, by looking at the space of holomorphic sections of l ⊗k over M, and then perform reduction at the quantum level. According to the philosophy of Dirac [Dir64, Lecture 2, pp. 34 ff.], reduction at the quantum level amounts to looking at holomorphic sections that are invariant under an appropriate action of the group G; that is, the quantum reduced space is the null-space of the quantized moment map ((2.6) below). We now describe how the action of G on the space of sections is constructed. Following the program of geometric quantization, we first define an action of the Lie algebra g on the space of smooth sections of  by Q ξ := ∇ X ξ − i φξ , ξ ∈ g. These operators satisfy [Q ξ , Q η ] = Q [ξ,η] . The prequantum bundle  is said to be G-invariant if this action of g can be exponentiated to an action of the group G. (If G is simply connected, this can always be done.) We henceforth assume that  is G-invariant. Since  is a holomorphic line bundle (i.e., the total space is a complex manifold), it is not hard to show that the G-action on  can be analytically continued to an action of G C (by following an argument similar to that of Guillemin and Sternberg that the action of G on M can be analytically continued to an action of G C on M [GS82, Theorem 4.4]). For each k, we then define an action of g on the space of smooth sections of ⊗k by Q ξ := ∇ X ξ − ikφξ , ξ ∈ g.

(2.6)

(We suppress the dependence on k.) These also satisfy [Q ξ , Q η ] = Q [ξ,η] and they preserve the quantum Hilbert space H(M; ⊗k ) of holomorphic sections of ⊗k . Since  is G-invariant, it follows that these operators can be exponentiated to an action of G that preserves H(M; ⊗k ). The space obtained by performing reduction at the quantum level is then the space of G-invariant holomorphic sections of ⊗k , denoted H(M; ⊗k )G . We see, then, that if we first quantize M and then reduce by G, we obtain the space H(M; ⊗k )G of G-invariant sections of ⊗k over M. On the other hand, we may first reduce M by G at the classical level and then perform quantization of the reduced manifold M//G. Assuming that the bundle  is G-invariant, it is not hard to see that it descends

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naturally to a Hermitian line bundle ˆ with connection over M//G, whose curvature is equal to −i ω. The bundle ˆ can be made into a holomorphic line bundle in the same way ˆ by decreeing that the holomorphic sections are those that are covariantly constant as , in the (0, 1)-directions. The space, then, obtained by first reducing and then quantizing is the space of all holomorphic sections of ˆ⊗k over M//G, denoted H(M//G; ˆ⊗k ). In the paper [GS82], Guillemin and Sternberg consider a geometrically natural linear map Ak between the “first quantize and then reduce” space H(M; ⊗k )G and the “first reduce and then quantize” space H(M//G; ˆ⊗k ). This map consists simply of taking a G-invariant holomorphic section over M, restricting it to −1 (0), and then letting it descend to M//G = −1 (0)/G. The remarkable result established in [GS82, Thm 5.2] is that this natural map is invertible. This result has been generalized in various ways, to symplectic manifolds that are not Kähler and to situations where the quotient is singular. See, for example, [Hue06, JK97, Sja95, Sja96, Mei98 and TZ98]. In view of the importance of the map Ak , we briefly sketch the proof of its invertibility. It is easy to see that a section of ˆ defines a G-invariant section of  over the zero-set by pullback along the canonical projection π : −1 (0) → M//G, and vice-versa. Moreover, a G-invariant section over the zero-set has a unique analytic continuation to the stable set, obtained from the G C -action on M. The difficulty is in showing that the resulting analytic continuation extends smoothly across the unstable set. To construct this extension, Guillemin and Sternberg show that the magnitude of a G-invariant holomorphic section over the stable set approaches zero as one approaches the unstable set. The Riemann Extension Theorem then shows that such a section extends holomorphically across the unstable set. The invertibility of the map Ak shows that, in a certain sense, quantization commutes with reduction. That is, Ak gives a natural identification of the vector space obtained by first quantizing and then reducing with the vector space obtained by first reducing and then quantizing. However, in quantum mechanics, the structure of the spaces as Hilbert spaces, not just vector spaces, is essential. For example, the expectation value of an operator D in the “state” s is given by s, Ds , where s is a unit vector in the relevant Hilbert space. It is natural, then, to consider the extent to which the map Ak is unitary with respect to the natural inner products (1.2) and (1.3) on the two spaces. Our Theorem 4.2 suggests that it is very rare for Ak (or any constant multiple of Ak ) to be unitary. If the G-orbits in −1 (0) have nonconstant volume, Theorem 5.3 shows that for all sufficiently large k, Ak is not a constant multiple of a unitary map. Indeed, in such cases, Ak is not even asymptotic to a multiple of a unitary map in the limit as Planck’s constant tends to zero (i.e., as k tends to infinity). We may say therefore that, in the setting of [GS82], quantization does not commute with reduction in the strongest desirable sense. 3. A Map of Guillemin–Sternberg Type in the Presence of the Metaplectic Correction In this section, we consider the “metaplectic correction,” which consists of tensoring the original line bundle  by the square root of the canonical bundle of M, and similarly for ˆ We introduce here an analog Bk of the Guillemin–Sternberg map in the presence of the . metaplectic correction and we show (Theorem 3.2 of this section) that Bk is invertible for all sufficiently large k. We will see eventually (Theorems 5.2 and 5.3) that the maps Bk , unlike the maps Ak , become approximately unitary as Planck’s constant tends to zero (i.e., as k tends to infinity).

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There are two conceptual differences between the original map Ak and the “metaplectically corrected” map Bk . First, G-invariant holomorphic sections of the square root of the canonical bundle behave badly as one approaches the unstable set. To compensate for this, we need rapid decay of holomorphic sections of G-invariant holomorphic sections of ⊗k . This means that we obtain invertibility of the maps Bk for all sufficiently large k, rather than for all k, as is the case for the maps Ak . Second, the pointwise magnitude of a section Bk s does not agree with the pointwise magnitude of the original section s on −1 (0), in contrast to the map Ak . Rather, the pointwise magnitudes differ by a factor involving the volume of the G-orbits in −1 (0) (Theorem 3.3). The volume factor in Theorem 3.3 ultimately cancels an unwanted volume factor in the asymptotics of the maps Ak (Sect. 5), allowing the maps Bk to be asymptotically unitary even though maps Ak are not. ∗   3.1. Half-form bundles on M and M//G. Let K = n T 1,0 M denote the canonical bundle of M, that is, the top exterior power of the bundle of (1, 0)-forms. A smooth section of K is called an (n, 0)-form, and the set of such forms is denoted n,0 (M). An (n, 0)-form is called holomorphic if in each holomorphic local coordinate system, the coefficient of dz 1 ∧ · · · ∧ dz n is a holomorphic function. Equivalently, we may define a “partial connection” (defined only for vector fields of type (0, 1)) on the space of (n, 0)-forms by setting ∇ X α = i X (dα) whenever X is of type (0, 1). It is easily verified that α is holomorphic if and only if ∇ X α = 0 for all vector fields √ of type (0, 1). (Compare [Woo91, Sect. 9.3].) A choice of square root K of the canonical bundle, if it exists, is called a half-form bundle. Since the first Chern class of the canonical bundle is −c1 (M), the canonical bundle will admit a square root if and only if −c1 (M)/2 is an integral class; that is, if and only if the second Stiefel–Whitney class w2 (M) (which is the reduction mod 2 of the first Chern class for a complex manifold) vanishes. We now assume that −c1 (M)/2 √ is integral and we fix a choice of K . (It is likely that results similar to the ones in this paper could be obtained assuming that [ω/2π ] − c1 (M)/2 is integral, rather than assuming, as we do, that [ω/2π ] and c1 (M)/2 are separately integral. See [Czy78] or [Woo91, Sect. 10.4].) √ One can define a partial connection acting on sections ν of K by requiring that 2 (∇ X ν) ν = ∇ X (ν 2 ). (3.1) √ (See again [Woo91, Sect. 9.3].) The bundle K can √then be made into a holomorphic line bundle by defining the holomorphic sections of K to be those for which ∇ X ν = 0 for all vector fields of type (0, 1). Because the action of G on M is holomorphic, the action of G on n-forms preserves the space of (n, 0)-forms and the space of holomorphic n-forms. The associated action of a Lie algebra element ξ ∈ g on (n, 0)-forms is √ by the Lie derivative L X ξ . There is an associated action of g on the space of sections of K , also denoted L X ξ , satisfying (by analogy with (3.1))   2 L X ξ ν ν = L X ξ (ν 2 ), (3.2) and this action preserves the space of holomorphic sections.

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√ There is a natural way to define a Hermitian structure on K , which is standard in geometric quantization. (It is a special case of the BKS pairing; see [Woo91, Sect. √  10.4].) If ν, µ ∈ ( K ) are half-forms, then ν 2 ∧ µ2 ∈ ( 2n TC∗ (M)). The volume  form εω = ω∧n /n! is a global trivializing section of the determinant bundle 2n T ∗ M, and so there is a function, denoted by (ν, µ), such that ν 2 ∧ µ2 = (ν, µ)2 εω .

(3.3) √ We can use this pairing to √ define a Hermitian form on the tensor product ⊗k ⊗ K : ⊗k for sections t1 , t2 ∈ ( ⊗ K ) which are locally represented by t j (x) = s j (x)µ(x), we define (t1 , t2 )(x) = (s1 ν, s2 µ)(x) = (s1 (x), s2 (x)) (ν, µ) (x). Denote the pairing of a half-form with itself by |ν|2 = (ν, ν).  denote the canonical bundle over the reduced manifold M//G. We now assume Let K √ that the action (3.2) of g on sections of K exponentiates to an action of the group G (this is automatic, for example, if G is simply connected). In the next subsection,√we  will show that this assumption allows us to construct in a natural way a square root √ K √   of K that is related in a nice way to the chosen square √ root K of K . Once K is  analogously to (3.3): constructed, we will define a Hermitian structure on K ν 2 ∧ µ2 = (ν, µ)2 ε ω,

(3.4)

ω∧(n−d) /(n − d)! is the volume form on M//G. where ε ω = 3.2. The modified Guillemin–Sternberg map. We continue to assume that the canonical bundle K of M√admits a square root and that we have chosen one such square root and denoted it by √ K . We also continue to assume that the action of the Lie algebra g on sections of K , given by (3.2), exponentiates to an action of the group G. In this subsection, we describe a natural map Bk from the space of G-invariant √ √ . holomorphic sections of ⊗k ⊗ K to the space of holomorphic sections of ˆ⊗k ⊗ K This map is the analog of the Guillemin–Sternberg map Ak between the G-invariant holomorphic sections of ⊗k and the space of holomorphic sections of ˆ⊗k . However, √ the construction of Bk is more involved than that √ of Ak . After all, sections of K are  are square roots of (n−d, 0)-forms. square roots of (n, 0)-forms, whereas sections of K Thus the map Bk must include a mechanism for changing the degree of a half-form. There is one further, vitally important difference between the map Bk and the map Ak . The bundle ˆ inherits its Hermitian structure from that of . As a result, the pointwise magnitude of a section a G-invariant section of ⊗k is the same on −1 (0) as the pointwise magnitude of the corresponding section of ˆ⊗k . That is, for each s ∈ H(M; ⊗k )G and each x0 ∈ −1 (0) we have √

|s|2 (x0 ) = |Ak s|2 ([x0 ]).

(3.5)

 has its own intrinsically defined Hermitian structure given by (3.3) By contrast, K It turns out, then, that Bk does not satisfy √ the analog of (3.5). Rather, we will show (Theorem 3.3) that for r ∈ H(M; ⊗k ⊗ K )G and each x0 ∈ −1 (0) we have |r |2 (x0 ) = 2d/2 vol(G · x0 )−1 |Bk r |2 ([x0 ]).

(3.6)

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We will see eventually that the volume factor in (3.6) cancels an unwanted volume factor in the asymptotic behavior of Ak . The result is that maps Bk (unlike the maps Ak ) are asymptotically unitary as k tends to infinity. Finally, we address the invertibility of the map Bk . Again, the situation is slightly different from that with Ak . Half-forms typically have bad behavior near the unstable set, and as a result, we are only able to prove that Bk is invertible for sufficiently large k. (This is in contrast to the map Ak , which is invertible for all k.) Before we get to half-forms, it will be useful to describe the descent of G C -invariant (n, 0)-forms on the stable set Ms to (n −d, 0)-forms on M//G. We cannot simply restrict an (n, 0)-form to the zero-set and then let it descend to M//G, as we did for the bundle ⊗k , because the result would not be an (n − d)-form on M//G. The process, which we describe in detail below, is to first contract with the infinitesimal G-directions and then use πC to push the result down to the quotient. It turns out that this process is invertible and preserves holomorphicity. Choose an Ad-invariant inner product on g which is normalized so that the volume of G with respect to the associated Haar measure is 1. Then fix a basis  = {ξ1 , ..., ξd } of the Lie algebra g which is orthonormal with respect to this inner product. Given a G C -invariant (n, 0)-form defined on the stable set Ms , define an (n − d, 0)-form β on Ms by  β = i( X ξ j )α. j

We claim that β has the properties that for each X ∈ T (G C · x), i(X )β = 0 and i(X )dβ = 0.

(3.7) (3.8)

ξ It is clear that (3.7) holds when √ X is of the form X with ξ ∈ g. But since β is an (n − d, 0)-form, i(J X ξ )β = −1i(X ξ )β, and so (3.7) holds for X = X ξ with ξ ∈ gC . To verify (3.8), we first note that in the presence of (3.7), (3.8) is equivalent to L X (β) = 0, i.e., to the condition that β be G C -invariant. So we  need to verify that β is invariant if α is. Since α is of type (n, 0), contracting α with j X ξ j is the same as contracting   it with j π+ X ξ j , and a simple computation shows that the polyvector j π+ X ξ j is G C -invariant. Contracting a G C -invariant form with a G C -invariant polyvector gives another G C -invariant form. Now, given any (n − d, 0)-form β on Ms satisfying (3.7) and (3.8), it is not hard to  on M//G = Ms /G C such that π ∗ (β  show that there is a unique (n − d, 0)-form β C ) = β, where πC∗ : Ms → Ms /G C is the quotient map. So we have given a procedure for turning  on M//G. a G C -invariant (n, 0)-form α on M into an (n − d, 0)-form β  is an (n − d, 0)-form on M//G. Then the pullIn the other direction, suppose β ) is an (n − d, 0)-form on Ms that (as is easily verified) satisfies (3.7) back β := πC∗ (β and (3.8). We can construct from β an (n, 0)-form α as follows: given a local frame {X ξ1 , . . . , X ξd , Y1 , . . . , Yn−d } for Tx Ms , set

(Y1 , . . . , Yn−d ) α(X ξ1 , . . . , X ξd , Y1 , . . . , Yn−d ) = πC∗ β

(3.9)

and define α on any other frame by G L(n, C)-equivariance and the requirement that α be an (n, 0)-form. (Note that the tangent space at a point in the stable set is a direct sum of the tangent space to the G C -orbit through that point and the transverse directions, so

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that every frame is equivalent to a linear combination of frames which are G L(n, C)equivalent to one of the√form (W1 , . . . , Wd , Y1 , . . . , Yn−d ), where W j = X ξ j or J X ξ j ; we define i(J X ξ )α = −1 i(X ξ )α.) It is easily verified that α is again G C -invariant. The  two processes we have defined—contracting a G C -invariant (n, 0)-form on Ms with j X ξ j and then letting it descend to the quotient, and pulling back an (n − d, 0)form on M//G and then “expanding” it by (3.9)—are clearly inverse to each other. They therefore define a bijective map α ∈ n,0 (Ms )G C → B(α) ∈ n−d,0 (M//G),

(3.10)

 where B(α) is the unique (n − d, 0)-form on M//G such that πC∗ B(α) = i( j X ξ j )α. This bijective map has the further property that α is  locally holomorphic if and only if B(α) is locally holomorphic since contracting with j X ξ j is the same as contracting  with j π+ X ξ j and the vector fields π+ X ξ are holomorphic.     Finally, note that the contraction i X ξ j α is proportional to the contraction i J X ξ j α. By the results of Sect. 2.2, pushing down a G C -invariant (n, 0)-form by the above process is equivalent to first restricting it to −1 (0) and then pushing it down by the analogous process using the quotient map π : −1 (0) → M//G −1 (0)/G. The vectors J X ξ span the normal bundle of −1 (0) in M, so the contraction step can then be understood as contracting with the directions normal to the zero-set; this is perhaps the obvious way to “restrict” a top-dimensional form to a submanifold. Since the Guillemin–Sternberg map Ak is defined as “restrict to −1 (0) and then descend to M//G”, we will sometimes interpret the map B as first contracting, then restricting the result to −1 (0), and finally pushing the result to the quotient. We now turn to the descent map for half-forms. Recall that we assume that the infin√ √ itesimal action of g on K defined by (3.2) exponentiates to an action of G on K which is compatible with the action of G on K . Following essentially the same argument that Guillemin and Sternberg make for the analytic continuation of the action of G on M, it can be shown that this action lifts to an action of G C which covers the G C -action on K . A moment’s thought shows that the map B actually provides an identification of K x [x] , x ∈ Ms , and this identification commutes with the action of G C , that is, for with K each g ∈ G C , g ∗ B(α) = B(g ∗ α). [x] . The contraction map B therefore identifies K |G C ·x with K √  denote a line bundle over M//G whose fiber is the equivalence class of Let K √  K under the G C -action. Since tensoring commutes with the G C -action, it folG C ·x √ lows that this bundle is a square root of the canonical bundle on M//G. √ We let (M, K ) √ √ ) denote the space of smooth sections of K and K  and (M//G, K √ , respectively, √ G and we let (M, K ) denote the space of G-invariant sections of K . By construction, then, the pullback of the half-form bundle on the quotient by the quotient map is isomorphic to the half-form bundle on the stable set; i.e.,  √   K  πC∗ K

Ms

.

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Moreover, this isomorphism defines a map  √ ) B : (M, K )G C → (M//G, K such that (Bνx , Bνx ) = B(ν 2 ). We have thus proved the following. √ √ ), unique Theorem 3.1. There exists a linear map B : (M, K )G → (M//G, K up to an overall sign, with the property that



  X ξ j (ν 2 )  . πC∗ (Bν)2 = i j

Ms

For any open set U in M//G, if ν is holomorphic in a neighborhood V of πC−1 (U ), then Bν is holomorphic on U. √ √ ), unique For each k, there is a linear map Bk : (⊗k ⊗ K )G → (ˆ⊗k ⊗ K up to an overall sign, with the property that Bk (s ⊗ ν) = Ak (s) ⊗ B(ν) √ √  for all s ∈ (⊗k ) and ν ∈ ( K ). This map takes holomorphic sections of ⊗k ⊗ K  V √   . to holomorphic sections of ˆ⊗k ⊗ K U

The last√claim in Theorem 3.1 follows from the definition (3.1) of the partial connection on K . We obtain a version of Guillemin and Sternberg’s “quantization commutes with reduction” using the modified map Bk . Our version is necessarily weaker, since the √ pointwise behavior of G-invariant elements of H(M; ⊗k ⊗ K ) is worse than the behavior G-invariant sections of H(M; ⊗k ) (Theorem 4.1). Theorem 3.2. For k sufficiently large, the map  √ ) Bk : H(M; ⊗k ⊗ K )G → H(M//G; ˆ⊗k ⊗ K is bijective. The proof of this result (below) is similar to the proof in [GS82] of the invertibility of the map Ak , with a few modifications to deal with the half-forms. The inclusion of half-forms modifies √ the Hermitian form on sections of the reduced  in a way that is not proportional to the modicorrected prequantum bundle ˆ⊗k ⊗ K √ fication of the Hermitian form on sections of ⊗k ⊗ K . We can actually compute the difference: √ Theorem 3.3. Suppose r ∈ H(M; ⊗k ⊗ K ). Then for x0 ∈ −1 (0), |Bk r |2 ([x0 ]) = 2−d/2 vol(G · x0 ) |r |2 (x0 ). The factor of 2−d/2 vol(G ·x0 ) appearing in Theorem 3.3 is the ultimate reason that the modified Guillemin–Sternberg map is asymptotically unitary; as we will see in Sect. 5, it precisely cancels the leading order asymptotic value of the uncorrected density Ik .

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Proof (Proof of Theorem 3.2). The natural map is injective because two holomorphic sections which agree on the stable set Ms , which is an open dense subset of M, must necessarily be√equal. We now establish surjectivity for large k. A section ) determines a G-invariant section π ∗ r over the zero-set  r ∈ H(M//G; ˆ⊗k ⊗ K −1 ∗  (0). Using the action of G C on the bundles, we can extend π r uniquely to a holo√ morphic, G C -invariant section of ⊗k ⊗ K defined over the stable set Ms . As shown in [GS82, Appendix], there exists some k for which H(M; ⊗k ) contains some nonzero G-invariant section s. By (2.5), the unstable set is contained in the zero-set of s. Thus, if the magnitude of r remains bounded as we approach the unstable set, the Riemann Extension Theorem (e.g., [GH78, pp. 9]) will imply that r extends holomorphically to all of M. Suppose ξ ∈ g has |ξ | = 1. Then we will show in the next section (Theorem 4.1) that the variation of the magnitude |r |2 along the curve eitξ · x0 is

 d L ξ εω itξ |r |2 (eitξ · x0 ) = |r |2 (x0 ) −2k φξ (eitξ · x0 ) − J X (e · x0 ) , dt 2εω where εω = ω∧n /n! is the Liouville volume form. Furthermore, we will show in (4.6) that φξ (eitξ · x0 ) increases with t for t ≥ 0. As a result, the function φξ (eiξ · x0 ), as ξ varies over unit vectors in g and x0 varies over −1 (0), is strictly positive and thus bounded below by compactness. Meanwhile, L J X ξ εω /εω is bounded over M uniformly in ξ with |ξ | = 1, again by compactness. Using the monotonicity of φξ (eitξ · x0 ) in t, it follows that for all sufficiently large k, 2kφξ (e

itξ

   L J X ξ εω itξ   · x0 ) ≥  (e · x0 ) 2εω

for all x0 ∈ −1 (0), all ξ ∈ g with |ξ | = 1, and all t ≥ 1. It follows that if k is large enough, every r obtained in the above way will extend holomorphically to all of M.   The proof of Theorem 3.3 boils down to the following two lemmas. Recall that Ξ = {ξ1 , . . . , ξd } is a basis of g for which the associated Haar measure on G is normalized to 1. Lemma 3.1. The function det Ξ Bx := det(B(X ξ j , X ξk )) is constant along each G-orbit. Moreover,  det Ξ Bx = vol(G · x). Proof. The fixed basis Ξ of g defines a left-invariant coframe ϑ on G · x. With respect to this coframe, the Riemannian volume associated to B is dvol =

 det  B ϑ 1 ∧ · · · ∧ ϑ k .

By our definition of Ξ, the pullback dg0 = ϕ ∗ ϑ 1 ∧· · ·∧ϑ k to G by the map ϕ : G → G·x is a left Haar measure on G for which the corresponding volume is vol0 (G) = 1. Denote 1 k by dvol0 (G) = ϑ ∧ · · · ∧ ϑ . Then since our choice √of inner product on g yields dvol (G) = 1, integrating the equation dvol = det  B dvol0 (G) over G · x 0 G·x yields the desired result.  

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Observe that if we choose a basis Ξ which is not orthonormal with respect to our fixed normalized inner  product on G, or if we choose an √inner product on g that is not normalized so that G dg = 1, then Lemma 3.1 yields det  Bx = Cvol(G · x). The final effect of our seemingly natural choices (which make C equal 1) is that the density Jk (which takes into account the metaplectic correction) approaches 1 as k → ∞ (in general it would be C −1 ). Our second lemma is technical; it is a straightforward though somewhat tedious calculation. Lemma 3.2. Let Z j = π+ X ξ j . Then for x0 ∈ −1 (0) we have i(



Z ) ◦ i( j

j

 k

  Z¯ j ) εω (x0 )

−1 (0)

=2

−d

(vol(G · x0 ))

2

  ωn−d (x0 ) . (n − d)! −1 (0)

Proof. First, since M is Kähler, the symplectic form ω is of (1, 1) -type, that is, ω contracted or two antiholomorphic vectors is zero. With our convention  on two holomorphic 1  σ (1) ⊗ · · · ⊗ w σ (d) we get i( j 1 d that j w j := d! w σ ∈Sd j w )α = α(w , . . . , w , d 1 ·, . . . , ·) = i(w ) ◦ · · · ◦ i(w )α. Using the fact that the interior product is an antiderivation, we compute i( Z¯ d ) ◦ · · · ◦ i( Z¯ 1 )ωn = (−1)d(d−1)/2+d

 n! ω(·, Z¯ j ) ∧ ωn−d . j (n − d)!

Continuing on, we contract the above with the holomorphic polyvector: i(

 j

Z j )◦i(

 k

Z¯ k ) ωn = n!  det(ω(Z j , Z¯ k ))ωn−d (−1)d(d+1)/2 (n − d)!    + (−1)d ω(·, Z¯ j ) ∧ (n − d)! ω(Z k , ·) ∧ ωn−2d . j

k

(3.11)

A short computation shows that ω(Z j , Z¯ k ) = 2i B(X ξ j , X ξk ). When restricted to −1 (0), the moment map is constant. Hence  ω(·, π± X ξ j )−1 (0) = and so



¯j j ω(·, Z ) ∧

i(

 j

  ∓ 2i ω(·, J X ξ j )−1 (0) = ∓ 2i ω(·, J X ξ j )−1 (0) ,

 k , ·) ω(Z  k



Z ) ◦ i( j

1 2 dφξ j

 k

−1 (0)

= 0. Equation (3.11) is now

  n ωn−d  k ω  −d ¯ = 2 (det  B) . Z ) n! −1 (0) (n − d)! −1 (0)

Combining this with Lemma 3.1 yields the desired result.   Proof (Proof of Theorem 3.3). Near x0 we can write r = sν where s is a local G-invariant √ holomorphic section of ⊗k and ν is a local G-invariant holomorphic section of K .

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Let α = ν 2 . Then since i(π+ X ξ ) α¯ = i(π− X ξ ) α = 0, we have  ωn−d  (n − d)! −1 (0)     = i( Z j ) α ∧ i( Z¯ k ) α¯ 

  2 π |Bν|2 εωˆ = π ∗ Bν, π ∗ Bν ∗

j



k



−1 (0)  ωn 

) n! −1 (0)  ωn−d  2 −d = (ν, ν) 2 det  B . (n − d)!  −1 = i(

j

Zj∧

k

Z¯ k )((ν, ν)2



(0)

The result now follows upon dividing by ωn−d /(n −d)!, taking the square root and using Lemma 3.1 and the fact that |Ak s|2 ([x0 ]) = |s|2 (x0 ).   4. Norm Decompositions In this section, we show how the norm-squared of a G-invariant holomorphic section over the upstairs manifold M can be computed as an integral over the downstairs manifold M//G. To do this, we first show that the pointwise magnitude of a G-invariant holomorphic section (with or without the metaplectic correction) varies in a predictable way as we move off the zero-set of the moment map along the orbits of G C . This means that the behavior of an invariant section on −1 (0) determines its behavior on the whole stable set Ms . We then integrate the resulting expressions over the stable set using the decomposition in Theorem 2.2. By (2.5), the unstable set is contained in a set of complex codimension at least 1 and hence the stable set is a set of full measure in M. A similar computation shows that a Toeplitz operator on the upstairs manifold with a G-invariant symbol, when restricted to the space of G-invariant sections, is equivalent to a certain Toeplitz on the downstairs manifold. Recall from Sect. 2.2 that the stable set Ms consists of those points in M that can be moved into the zero-set of the moment map by means of the action of G C . Recall also (Theorem 2.2) that every point in Ms can be expressed uniquely in the form eiξ · x0 , for some ξ ∈ g and x0 ∈ −1 (0). The first main result of this section shows how the (pointwise) magnitude of a G-invariant holomorphic section, evaluated at eiξ · x0 , varies with respect to ξ. ⊗k Theorem 4.1. Let s be a G-invariant √ holomorphic section of−1 and let r be a G-invariant ⊗k holomorphic section of  ⊗ K . Let x0 be a point in  (0) and ξ an element of g. Then the magnitudes of the sections at eiξ · x0 are related to the magnitudes at x0 as follows:


(a) |s|2 (ei ξ · x0 ) = |s|2 (x0 ) exp − 0

 (b) |r | (e · x0 ) = |r | (x0 ) exp 2

iξ

2

1


−

0

 2kφξ (eitξ · x0 ) dt ,

1

2kφξ (e

itξ

(4.1)

  L J X ξ εω itξ · x0 ) + (e · x0 ) dt . 2εω (4.2)

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B. C. Hall, W. D. Kirwin

Furthermore, if we consider


1

ρ(ξ, x0 ) := 2 0

φξ (eitξ · x0 ) dt,

then for each fixed x0 , ρ(ξ, x0 ) achieves its unique minimum at ξ = 0 and the Hessian of ρ(ξ, x0 ) at ξ = 0 is given by  Dξ1 Dξ2 ρ(ξ, x0 )ξ =0 = 2Bx0 (J X ξ1 , J X ξ2 ), ξ1 , ξ2 ∈ g. Note that the extra factor in (4.2), as compared to (4.1), is independent of k and is equal to 1 when ξ is equal to 0. This extra factor, therefore, does not affect the leading order asymptotics. On the other hand, this extra factor can become unbounded near the unstable set which means that pointwise behavior of r tends to be worse than that of s. Let : g × −1 (0) → Ms denote the diffeomorphism (ξ, x0 ) = eiξ · x0 of Theorem 2.2. Recall also that we have chosen an orthonormal basis Ξ = {ξ j }dj=1 of g to which there corresponds a Lebesgue measure d d ξ on g. The Liouville volume εω = ω∧n /n! on M (which is the same as the Riemannian volume) decomposes as ∗ (εω )(ξ,x0 ) = τ (ξ, x0 ) d d ξ ∧ dvol(−1 (0))x0 for some G-invariant smooth Jacobian function τ ∈ C ∞ (g × −1 (0)). We are now ready to state the remaining main results of this section. Let γξ : [0, 1] → Ms be the  1 path γξ (t) = eitξ · x0 ; then γξ φξ = 0 φξ (eitξ · x0 ) dt. Theorem 4.2. Suppose s is a G-invariant holomorphic section of ⊗k . Then the norm of s can be computed by

2 n/2 2 (n−d)/2 s := (k/2π ) |s| εω = (k/2π ) |Ak s|2 Ik ε (4.3) ω, M//G

M

where, with [x0 ] = G · x0 , 


Ik ([x0 ]) = vol(G · x0 ) (k/2π )d/2

g

τ (ξ, x0 ) exp −2k




γξ

φξ

d d ξ.

(4.4)

In the case of a Hamiltonian torus action (i.e., the case when G is commutative), a similar formula was obtained by Charles in [Char06b, Sect. 4.5]. Nevertheless, our result is slightly stronger than that of Charles, in that Charles inserts into the first integral in (4.3) a function f that is assumed to be supported away from the unstable set. In general, this allows him to obtain asymptotics about Toeplitz operators (as we do also in Theorem 5.4). To obtain results about the norm of a section, Charles takes f to be identically equal to one in a neighborhood of the zero-set, but still supported away from the unstable set. The insertion of such a cutoff function leaves unchanged the asymptotics to all orders of the first integral in (4.3). Still, our formula actually gives an exact (not just asymptotic) expression for the norm of an invariant section upstairs as an integral over the downstairs manifold. To get this, we require some estimates for the blow-up of certain Jacobians as we approach the unstable set.

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√ Theorem 4.3. Suppose r is a G-invariant holomorphic section of ⊗k ⊗ K . Then the norm of r can be computed by

r 2 := (k/2π )n/2 |r |2 εω = (k/2π )(n−d)/2 |Bk r |2 Jk ε ω, M

where
Jk ([x0 ]) = (k/2π )

d/2 d/2

2

M//G


  L J X ξ εω 2k φξ + τ (ξ, x0 ) exp − d d ξ. (4.5) 2εω g γξ

Note the volume factor that is present in the expression for Ik but not in the expression for Jk . In computing Ik , we decompose Ms as g × −1 (0) and then integrate out the ξ -dependent part of (4.1), leaving an integration over −1 (0). Since the resulting integrand on −1 (0) will be G-invariant, the integration over −1 (0) can be turned into an integration over −1 (0)/G. The volume factor in (4.4) arises because the projection map π : −1 (0) → −1 (0)/G is a Riemannian submersion, whence the Riemannian volume measure on −1 (0) maps to the Riemannian volume measure on −1 (0)/G, multiplied by a density given by the volume factor. This crucial fact is the basic reason that the Guillemin–Sternberg map (without half-forms) is not asymptotically unitary. Meanwhile, the volume factor fails to arise in (4.5) because it is canceled by the volume factor in Theorem 3.3, which relates the pointwise magnitude of r at x0 to the pointwise magnitude of Bk r at [x0 ]. In Sect. 5, we will calculate the asymptotic behavior of the expressions for densities Ik and Jk as k tends to infinity. The calculation is done by means of Laplace’s method and uses the Hessian computed in Theorem 4.1. Because the extra factor in the expression for |r |2 (eiξ · x0 ) is independent of k and equal to 1 at ξ = 0, this factor does not affect the leading-order asymptotics of the norm. The different asymptotic behavior of the maps Ak and Bk is due to the volume factor in the expression for Ik that is not present in the expression for Jk . If it should happen that Ik is constant, then Ak (which is in any case invertible for all k) will be a constant multiple of a unitary map. Similarly, if Jk is constant for some large k (large enough that Bk is invertible), then Bk will be a constant multiple of a unitary map. There does not, however, seem to be any reason that either Ik or Jk should typically be constant. Nevertheless, we will see in Sect. 5 that Jk asymptotically approaches 1 for large k, which implies that Bk is asymptotically unitary. On the other hand, Ik is not asymptotically constant unless all the G-orbits in −1 (0) happen to have the same volume. In the case where the G-orbits do not all have the same volume, it is not hard to show (Theorem 5.3) that Ak is not asymptotic to any constant multiple of a unitary map. A similar analysis shows that if we consider a Toeplitz operator with a G-invariant symbol f upstairs, then the matrix entries for such an operator can be expressed as an integral over the downstairs manifold involving a certain density Ik ( f ), which reduces to Ik when f ≡ 1. Theorem 5.4 shows that in the case with half forms we obtain an asymptotic equivalence of a very simple form between Toeplitz operators upstairs and downstairs. We now turn to the proofs of Theorems 4.1 and 4.2. First, we will decompose the integral over Ms into integrals over g and −1 (0) and the integral over −1 (0) into integrals over M//G and G · x0 . Both of these integral decompositions follow from the coarea formula [Chav06, pp. 159–160]:

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Lemma 4.1. (Coarea formula) Let Q and N be smooth Riemannian manifolds with dim Q ≥ dim N , and let p : Q → N . Then for any f ∈ L 1 (M) one has








J p f dvol(Q) = Q

dvol(N )(y) N

 where the Jacobian is J p :=

p −1 (y)

( f | p−1 (y) ) dvol( p −1 (y)),

adj

det p∗ ◦ p∗ .

Applying the coarea formula to the map pr 2 ◦ −1 : eiξ · x0 → x0 from Ms to M0 , and identifying exp(ig) · x0 with g, we have that for every f ∈ L 1 (Ms ),








f dvol(Ms ) = Ms

M0

( f |exp(i g)·x0 ) τ d d ξ dvol(M0 ),

g

where τ is the Jacobian J of the map : g × −1 (0) → Ms . Since the volume form on the quotient M//G is d vol(M//G) =  ω n−d /(n − d)! and −1 since  (0) → M//G is a Riemannian submersion, we have [GLP99, Sect. 2.2] dvol(−1 (0)) = dvol(G · x0 ) ∧

∗  πhor ω n−d , (n − d)!

∗  where πhor ω n−d denotes the horizontal lift (i.e., pullback composed with projection to the horizontal subspace Wx⊥0 B ) of the Liouville form on the quotient. The 2-form π ∗ ω = i ∗ ω is already horizontal, though, since

i(X ξ ) π ∗  ω = i(X ξ ) i ∗ ω = dφξ |−1 (0) = 0. This proves the following. Lemma 4.2. Let π : −1 (0) → M//G := −1 (0)/G denote the canonical projection. Then Jπ = 1; in particular, for every G-invariant function f ∈ L 1 (−1 (0))G we have


−1

−1 (0)

f dvol(


(0)) =

M//G

vol(G · x0 ) f ([x0 ]) dvol(M//G) .

To prove Theorem 4.1, we first make the following computations: Lemma 4.3. The first two derivatives of the norm of a G-invariant holomorphic section s ∈ H(M; ⊗k )G along γξ are: J X ξ |s|2 = −2k φξ |s|2 , and J X ξ1 J X ξ2 |s|2 = −2k B(X ξ1 , X ξ2 ) |s|2 + 4k 2 φξ1 φξ2 |s|2 . The last statement of Theorem 4.1 follows immediately upon combining the two above equations and restricting to the zero-set.

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425

Proof. Since the connection on ⊗k is Hermitian, we have J X ξ |s|2 = (∇ J X ξ s, s) + (s, ∇ J X ξ s). By assumption, s is G-invariant. Since the action of G on H(M; ⊗k ) is given by Eq. (2.6), we have   0 = Q ξ s = ∇ X ξ − ik φξ s so that ∇ X ξ s = −ik φξ s. The projection of a vector field onto its (0, 1)-part is X → 1 2 (1−i J )X. We assume that s is holomorphic, i.e., ∇(1−i J )X ξ s = 0. Therefore ∇ J X ξ s = i∇ X ξ s. Putting this all together, we get     J X ξ (s, s) = −i ik φξ s, s + i s, ik φξ s = −2k φξ (s, s). Next, the Leibniz rule yields the second derivative: J X ξ j J X ξl |s|2 = −2k J X ξ j (φξl ) + 4k 2 φξl φξ j |s|2 . Since dφξl = i(X ξl ) ω we have J X ξ j φξl = ω(X ξl , J X ξ j ) = B(X ξ j , X ξl ). Therefore,

J X ξ j J X ξl |s|2 = −2k B(X ξ j , X ξl ) |s|2 + 4k 2 φξ j φξl |s|2

as desired.   Proof (Proof of Theorem 4.1(a)). Applying Lemma 4.3, the derivative of |s|2 along γξ is d ∗ 2 γ |s| (t) = J X ξ |s|2 (eitξ · x0 ) = −2k φξ (eitξ · x0 ) |s|2 (eitξ · x0 ), dt ξ so that d log γξ∗ |s|2 (t) = −2k φ(eitξ · x0 ). dt Integrating this equation from t = 0 to t = 1 yields the desired result.   As Guillemin and Sternberg pointed out, an important consequence of Theorem 4.1 is that a G-invariant holomorphic s ∈ H(M; ⊗k ) attains its (unique) maximum value in the orbit G C · x0 on the zero-set of the moment map (G C · x0 ) ∩ −1 (0) = G · x0 . The reason is that J X ξ is the gradient vector of the function φξ , since B(J X ξ , ·) = ω(J X ξ , J ·) = ω(X ξ , ·) = dφξ .

(4.6)

Hence, φξ is increasing along γξ and so |s|2 is decreasing away from −1 (0). A useful and direct consequence of this fact is that each G C -orbit intersects the zeroset in exactly one G-orbit; since if not, then there exist points x, x  ∈ −1 (0) which lie in distinct G-orbits such that x  = eiξ · x for some ξ ∈ g. But this is impossible since the moment map is strictly increasing along the path t → eitξ · x. we express the norm of a G-invariant half-form corrected section r ∈ H(M; ⊗k⊗ √ Next G K ) along γξ in terms of its value at x0 .

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√ Proof (Proof of Theorem 4.1(b)). Locally, a section r ∈ H(M; ⊗k ⊗ K )G can be written as r = sν for some G-invariant holomorphic half-form ν . Since ν is a G-invariant holomorphic half-form, the combination ν 2 ∧ ν 2 is G C -invariant, whence   0 = L J X ξ (ν 2 ∧ ν 2 ) = L J X ξ ((ν, ν)2 εω ) = 2 (ν, ν) J X ξ (ν, ν) εω + (ν, ν)2 L J X ξ εω , so that d L ξ εω itξ J X ξ (ν, ν) itξ log γξ∗ (ν, ν) (t) = (e · x0 ) = − J X (e · x0 ). dt 2εω (ν, ν) Integrating this along γξ and combining it with Theorem 4.1(a) yields the desired result.   By decomposing the Liouville measure on Ms in terms of the global decomposition : g × −1 (0) → Ms (Theorem 2.2) and the fibration G → −1 (0) → M//G, we will obtain our desired integral. Recall that we have chosen a basis Ξ = {ξ j }dj=1 of g to which there corresponds a Lebesgue measure d d ξ on g. The Liouville volume εω = ω∧n /n!, which is the same as the Riemannian volume, decomposes as ∗ (εω )(ξ,x0 ) = τ (ξ, x0 ) d d ξ ∧ dvol(−1 (0))x0 (4.7) for some G-invariant smooth Jacobian function τ ∈ C ∞ (g × −1 (0)) (this is just the coarea formula (4.1) applied to the map Ms → M0 given by eiξ · x0 → x0 ). Proof (Proof of Theorem 4.2). Combining (4.7) and Theorem 4.1 yields
|s|2 εω (k/2π )n/2 M 


(n−d)/2

= (k/2π )

(k/2π ) M0

∗

d/2

g

|π Ak s| (x0 ) exp −2k 2

γξ

 φξ

× τ (ξ, x0 ) d d ξ dvol(M0 ), (4.8) since s(x0 ) = π ∗ Ak s(x0 ). The terms |π ∗ Ak s|2 and τ are obviously G -invariant. To see that the exponential factor in the integrand is G -invariant, define

1  ρ(eiξ · x0 ) = 2 (eitξ · x0 ), ξ dt. φξ = 2 γξ

0

To evaluate g ∗ ρ, we rewrite geiξ · x0 = ei Ad(g)ξ g · x0 . Then
1  (ei Ad(g)ξ t g · x0 ), Ad(g)ξ dt g ∗ ρ(eiξ · x0 ) = ρ(ei Ad(g)ξ g · x0 ) = 2
=2

1



0

(eitξ · x0 ), ξ dt = ρ(eiξ · x0 ),

0

where we have used the G-equivariance of the moment map (g · p) = Ad∗ (g −1 )( p). Hence, the integrand in (4.8) is G-invariant, so by the coarea formula (Lemma 4.2) the integral over M0 reduces to an integral over M//G with an extra factor of vol(G · x0 ), and we obtain the desired result.  

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We now compute the norm of a G-invariant holomorphic section as an integral over the zero-set of the moment map taking into account the metaplectic correction. Proof (Proof of Theorem 4.3). The proof is similar to the above proof of Theorem 4.2; we first write the integral over M as a multiple integral:

|r |2 εω = (k/2π )(n−d)/2 (k/2π )n/2 (k/2π )d/2 M M0 

×

g

|r |2 (x0 ) exp −

γξ

(2k φξ + L J X ξ εω /2εω )

×τ (ξ, x0 ) d d ξ dvol(M0 ). The main difference from the proof of Theorem 4.2 is that by Theorem 3.3 we have |r |2 (x0 ) = 2d/2 vol(G · x0 )−1 |Bk r |2 ([x0 ]). Inserting this into the above integral, and noting that the remaining integrand is G-invariant (following the same argument as in the proof of Theorem 4.2) we obtain (again using the coarea formula Lemma 4.1) that

|r |2 εω = (k/2π )(n−d)/2 |r |2 ([x0 ]) (k/2π )n/2 M M//G 

L J X ξ εω d/2 d/2 exp − (2k φξ + ) × (k/2π ) 2 2εω g γξ ×τ (ξ, x0 ) dvol(exp(ig) · x0 ) d d ξ ε ω whence r 2 = (k/2π )(n−d)/2


M//G

|Bk r |2 ([x0 ])Jk ([x0 ]) ε ω

as desired.   5. Asymptotics In this section we compute the leading order asymptotics of the densities  

d/2 Ik ( f )([x0 ]) = vol(G · x0 )(k/2π ) τ (ξ, x0 ) f (x0 , ξ ) exp −2k φξ d d ξ, g

and

γξ


Jk ( f )([x0 ]) = (k/2π )d/2 2d/2 τ (ξ, x0 ) f (x0 , ξ ) g 
  L J X ξ εω 2k φξ + × exp − d d ξ, 2εω γξ

where f is a smooth, G-invariant function on M. In the case that f ≡ 1, these asymptotics will give asymptotics of the Guillemin–Sternberg-type maps. For arbitrary smooth f ∈ C ∞ (M)G , the asymptotics will give information about Toeplitz operators. The main result is the following.

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Theorem 5.1. For f ∈ C ∞ (M)G , the densities Ik ( f ) and Jk ( f ) satisfy lim Ik ( f )([x0 ]) = 2−d/2 f (x0 )vol(G · x0 ), and

k→∞

lim Jk ( f )([x0 ]) = f (x0 )

k→∞

for each x0 ∈ −1 (0), and the limits are uniform. Our main tool will be Laplace’s approximation [BlH75, Chap. 10], also frequently referred to as the stationary phase approximation or the method of steepest descent. Let D ⊂ Rd be a bounded domain, and consider ρ ∈ C 2 (D) and σ ∈ C(D). Suppose ρ attains a unique minimum at x0 ∈ D \ ∂ D (i.e., the interior of D). Laplace’s approximation gives the leading order asymptotic limit  d/2
  2π det Hρ (x0 )−1/2 σ (x0 ), k → ∞, I (k) = σ (x) e−k ρ(x) d d x ∼ ek ρ(x0 ) k D (5.1) where Hρ denotes the Hessian of ρ. The formula for the large-k limits of Ik and Jk come from applying Laplace’s method to the integral over g Rd in Theorems 4.2 and 4.3, using the computation of the relevant Hessian, which we have already performed in Theorem 4.1. Conceptually, then, it is easy to understand where the limiting formulas come from. There are, however, some technicalities to attend to, and it is these technicalities that will occupy the bulk of this section. First, we need to consider the part of the integrals near infinity as well as the part near the origin. The density τ (x0 , ξ ) can (apparently) blow up as ξ tends to infinity for certain values of x0 . As a result, we need some estimates (Lemma 5.2) to ensure that the ξ -integrals in (4.4) or (4.5) are finite for all (as opposed to almost all) x0 , at least for large k. Second, we wish to be careful in verifying that the limits are uniform over M//G, which is needed to obtain the asymptotic unitarity of the maps Bk . Before coming to these technicalities we state the consequences of the above asymptotic formulas for the unitarity of the maps Ak and Bk and indicate some applications to the asymptotics of Toeplitz operators. Theorem 5.2. The maps Bk are asymptotically unitary, in the sense that     lim  Bk∗ Bk − I  = lim  Bk Bk∗ − I  = 0, k→∞

k→∞

where · refers to the operator norm. √ √ ). Then for all t ∈ H(M; ⊗k ⊗ K ) we have Proof. Let  r ∈ H(M//G; ˆ⊗k ⊗ K  ∗  Bk  r , t M =  r , Bk t M//G . On the other hand, Theorem 4.3 implies √  ∗    Bk  r , t M = Jk Bk Bk∗ r , Bk t M//G , ∀t ∈ H(M; ⊗k ⊗ K ). √ ) is of the Since Bk is bijective (for k sufficiently large), every  t ∈ H(M//G; ˆ⊗k ⊗ √ K √ ) we have form Bk t for some t ∈ H(M; ⊗k ⊗ K ); so for all  t ∈ H(M//G; ˆ⊗k ⊗ K     r , t M//G .  r , t M//G = Jk Bk Bk∗

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Applying Theorem 5.1 then yields the claim that Bk is asymptotically unitary:            lim  Bk∗ Bk − I  ≤ lim mult. by (Jk−1 − 1) ≤ lim max  Jk−1 − 1 = 0.  k→∞

k→∞ M//G

k→∞

A very similar argument establishes the other inequality in the theorem. Theorem 5.3. If vol(G  · x0 ) is not constant on M//G, then there is no sequence ck of constants for which  Ak A∗k − ck I  tends to zero as k tends to infinity. In the torus case, a similar result was proved by Charles in [Char06b, Remark 4.29]. Proof. This theorem follows easily from the existence of localized sections (see [MM04, pp. 36–37]). For each [x] ∈ M//G, there exists a sequence of holomorphic sections (k) {S[x] ∈ H(M//G; ˆ⊗k )} which are asymptotically concentrated near [x] in the sense that for every f ∈ C(M//G),
   (k) 2 (n−d)/2 f S[x]  ε lim (k/2π ) ω = f ([x]). k→∞

M//G

If vol(G · x0 ) is not constant on M//G, there exist points [xmax ] and [xmin ], where vol(G · x0 ) achieves its maximum and minimum values. Consider a sequence of peak sections localized at [xmax ]. Let {ck } be a sequence of constants. Then  2       −1 (k) 2 ∗ (k)  Ak A∗ − ck I 2 ≥  A S − c S → ([x ]) − c   , k → ∞. A I k k max k k k [xmax ] [xmax ] k Similarly, we obtain  2       −1 (k) 2 ∗ (k)  Ak A∗ − ck I 2 ≥  A S − c S → ([x ]) − c  A I k k [xmin ] k [xmin ] min k  , k → ∞. k k Since Ik ([xmax ]) → vol(G · xmax ) = vol(G · xmin ) ← Ik ([xmin ]), k → ∞, we cannot have both 2    −1 Ik ([xmax ]) − ck  → 0 2  Hence  Ak A∗k − ck I  → 0.

and

 2  −1  Ik ([xmin ]) − ck  → 0.

 

The asymptotic estimates of Theorem 5.1 can be used to derive results about Toeplitz operators with G-invariant symbols. We mention here the nicest result, concerning the asymptotics of Toeplitz operators in the case where half-forms are included. Let f ∈ C ∞ (M),√and recall that the Toeplitz operator with symbol f is the map T f from H(M, ⊗k ⊗ K ) to itself defined by T f s = proj( f s), where “proj” denotes the orthogonal projection from the space of all square-integrable sections onto the holomorphic subspace. If f is G-invariant, then precisely the same

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B. C. Hall, W. D. Kirwin

argument as in the proof of Theorem 4.3 shows that for r1 , r2 ∈ H(M, ⊗k ⊗ have
  r1 , T f r2 = (k/2π )n/2 f · (r1 , r2 ) εω M
= (k/2π )(n−d)/2 (Bk r1 , Bk r2 ) Jk ( f ) ε ω.

√

K ) we

M//G

If we denote by Tˆφ the Toeplitz operator on H(M//G, ˆ⊗k ⊗ φ ∈ C ∞ (M//G), then the above formula becomes     r1 , T f r2 = Bk r1 , TˆJk ( f ) Bk r2 .

√

) with symbol K

The asymptotics of Jk ( f ) then immediately imply the following. Theorem 5.4. Let f ∈ C ∞ (M)G and let fˆ denote the restriction of f to −1 (0), regarded as a function on −1 (0)/G. Let T fG denote the restriction of the Toeplitz operator T f to the space of G-invariant sections. Then T fG is asymptotically equivalent to Tˆ ˆ in the sense that f

  ˆ  T fˆ − Bk ◦ T fG ◦ Bk−1  → 0 as k → ∞. Since we also prove that the operators Bk are asymptotically unitary, we may say that T fG and Tˆ fˆ are “asymptotically unitarily equivalent.” Such a result certainly does not hold without half-forms, indicating again the utility of the metaplectic correction. √ ), we have Proof. For all  r , t ∈ H(M//G, ˆ⊗k ⊗ K     ˆ − Jk ( f ))  ˆ − Bk ◦ T fG ◦ B −1 ) t =  r , ( f t .  r , (T k f       G ◦ B −1  =  ˆ by ( f − J Hence, T − B ◦ T ( f )) mult.   . It then follows from k k f k fˆ Theorem 5.1 that        −1  G ˆ = lim mult. by ( f − J lim T − B ◦ T ◦ B ( f ))    k k f k fˆ k→∞ k→∞     ≤ lim max  fˆ − Jk ( f ) = 0.   k→∞ M//G

We now turn to the proof of Theorem 5.1. Our first task will be to control the part of the integral near infinity. From the definition of τ, it is apparent that the integral of τ (ξ, x0 ) over g is finite for almost all x0 , but it is not obvious that this integral is finite for all x0 . We will, however, give uniform exponential bounds on the behavior of τ (ξ, x0 )  as ξ tends to infinity. This is sufficient to show that τ (ξ, x0 ) exp −2k γξ φξ is finite for all x0 , provided k is sufficiently large, and that the part of the integral outside a neighborhood of the origin is uniformly negligible as k tends to infinity.

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Our second task will be to show that Laplace’s approximation can be applied so as to give uniform limits, which we need to prove asymptotic unitarity of Bk . Theorems 4.2 and 4.3 express the densities Ik and Jk in the form of (5.1), where in the case of the expression for Jk , the k-independent term in the exponent should be factored out and grouped with τ (ξ, x0 ). We develop a uniform version of Laplace’s approximation that allows for the desired uniformity. (It is probably well known that this is possible, but we were not able to find a written proof.) Our argument will be based on the Morse–Bott Lemma (a parameterized version of the usual Morse lemma). Once these two tasks are accomplished, it is a straightforward matter to plug in the Hessian computation in Theorem 4.1 to obtain Theorem 5.1.

5.1. Growth estimates. In this section, we show that the contribution to Ik coming from the complement of a tubular neighborhood of the zero-set is negligible as k tends to infinity: Theorem 5.5. There exist constants b, D > 0 such that for all x0 ∈ −1 (0) and for all R and k sufficiently large, 

g\B R (0)

f (ξ, x0 )τ (ξ, x0 ) exp −

γξ

d d ξ ≤ be−R Dk ,

2k φξ

where B R (0) is a ball of radius R centered at 0 ∈ g. Our first step is to show that the integrand appearing in Ik decays exponentially in the radial directions. Lemma 5.1. There exists C > 0 such that for all sufficiently large t, 
 exp −

γt ξˆ

2k φt ξˆ

≤ e−2ktC

    uniformly on −1 (0), where ξˆ ∈ g with ξˆ  = 1. Proof. By definition, we have



γt ξˆ

φt ξˆ =

1

ˆ (eiτ t ξ · x0 ), t ξˆ  dτ = t

0




1

ˆ (eiτ t ξ · x0 ), ξˆ  dτ.

0

Hence, it is enough to show that there exists some C > 0 such that for t sufficiently large and for all x0 ∈ −1 (0),


1

ˆ

(eiτ t ξ · x0 ), ξˆ  dτ ≥ C.

0

For t ≥ 0, define

     Mt = eitξ · x0 : ξˆ  = 1, x0 ∈ −1 (0) .

(5.2)

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B. C. Hall, W. D. Kirwin

Then Mt , t > 0, is a sphere bundle over −1 (0) and is hence compact. Define ρt : Mt → R by
1 ˆ ˆ φξˆ (eiτ t ξ · x0 ) dτ. ρt (eit ξ · x0 ) = 2 0

Fix t > 0. Then since ρt is smooth with compact domain there is some m ∈ Mt , where ˆ ρt achieves its minimum. Recall that eitτ ξ · x0 is the gradient line of φt ξˆ . Moreover, ξˆ

(grad φξˆ )eiξ ·x0 = J X eiξ x

0

and

   ˆ   ξˆ   ξ   J X eiξ ·x  =  X eiξ ·x  = 0, 0

0

ˆ

since G acts freely on Ms . Therefore, φt ξˆ is strictly increasing along eitτ ξ · x0 . In particular, 2C := ρt (m) > 0 so that, by the definition of m, we have ρt (m  ) ≥ 2C for 1 ˆ all m  ∈ Mt . Hence we obtain 0 (eiτ t ξ · x0 ), t ξˆ  dτ ≥ C for all t  > t and for all  x0 ∈ −1 (0).  We now turn to the task of estimating τ, which we will do by embedding M into projective space. We recall some basic projective geometry. The function τ is the Jacobian of the map : g × −1 (0) → M. Since M is a compact Kähler manifold and  is an equivariant ample line bundle, we can equivariantly embed M into CP N for some N (this is Kodaira’s embedding theorem enhanced by the presence of a G-invariant Kähler structure and line bundle; the embedding is realized by considering holomorphic sections of high tensor powers of ). We identify M with its embedded image. Under this embedding, we identify G with a compact subgroup of PU (N + 1). The metric on the embedded image of M induced by the Kähler metric on M differs from the Fubini–Study metric on CP N by smooth factors. In particular, the difference between computing a determinant with the Fubini–Study metric and the induced Kähler metric is the determinant of the linear map which takes a Fubini–Study orthogonal basis of the tangent space to a basis which is orthogonal with respect to the induced metric. This map is invertible and smooth on M and hence is bounded and bounded away from zero, and so it suffices for our purposes to compute the Jacobian τ using the Fubini–Study metric. We use homogeneous coordinates on CP N ; i.e., CP N = {[z = (z 0 , z 1 , . . . , z N )] : j z ∈ C}. The tangent space T[z] CP N can be identified with the orthogonal complement of the line in C N +1 determined by z. The Fubini–Study metric on CP N is [Zhe00, Sect. 7.4] ! 2 "  |z| d z¯ ⊗ dz− Nj,k=0 z j d z¯ j ⊗ z¯ k dz k gFS = Re . |z|4 Now we show that τ (ξ, x0 ) grows at most exponentially in the radial directions away from the zero-set. Lemma 5.2. There exist constants a, b > 0 such that for all t sufficiently large τ (t ξˆ , x0 ) ≤ bt −d eat     uniformly on −1 (0) and for all ξˆ ∈ g with ξˆ  = 1.

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Proof. Recall that Ξ = {ξ j }dj=1 is an orthonormal basis of g < pu(N + 1) with respect 2n to an Ad-invariant inner product on g. Let {X j }2n−d j=1 and {Y j } j=1 be orthonormal bases of Tx0 −1 (0) and Teit ξˆ ·x M, respectively. Write a vector in terms of its components with 0

respect to the appropriate basis using the usual convention; for example, v = v j Y j ∈ Teiξ ·x0 M. We want to estimate τ (t ξˆ , x0 ) = (det ∗ )(t ξˆ , x0 ). We can compute the components of the images of the basis Ξ ∪ {X j } by ∗ (ξ j )l = gFS (Yl , ∗ (ξ j )),

and

∗ (X j )l = gFS (Yl , ∗ (X j )).

(5.3)

ˆ

There is a unique R ∈ su(N +1) such that ei ξ [z] = [ei R z]. Since R is skew-Hermitian, it has pure imaginary eigenvalues and these √ eigenvalues vary √continuously with respect to the choice of ξ . Let the eigenvalues be − −1λ0 , . . . , − −1λ N with λ0 ≤ · · · ≤ λ N . Then for generic z ∈ C N +1 , we have (eit R z) j  = O(eλ N t ). In order to compute the components in Eq. (5.3), we need to estimate the growth of both Yl and the pushforwards ˆ ∗ (ξ j ) and ∗ (X j ). We begin with Yl . Let [wt ] = eit ξ · x0 . Since Y j = (Y j0 , . . . , Y jN ) is a unit vector, we have ⎛ ⎞ N   % 1 2 Y j  =1= Re ⎝|wt |2 Y j · Y¯ j − wtl w¯ tm Y¯ lj Y jm ⎠ . FS eitξ x |wt |4 0 l,m=0       In particular, since |wt |4 = O(e4λ N t ) and wtl  = O(eλ N t ), we must have Y lj  = O(eλ N t ). To estimate the growth rate of the pushforward ∗ (X j ), we observe that if X j = [x j ], then ∗ (X j ) = [eit R x j ] from which we obtain

⎛ ⎞ % % 1 Re ⎝|wt |2 wtl w¯ tm Y¯ll (eit R x j )m ⎠ . Y¯lm (eit R x j )m − ∗ (X j )l = |wt |4 m l,m

(5.4)

  By the same argument as above, we see that (eit R x j )m  = O(eλ N t ). We must be a little careful with our growth estimates; if λ0 > 0, then at worst |wt |−1 = O(e−λ0 t ) = O(1), but if λ0 < 0, then there may be certain points x0 , where |wt |−1 = O(e−λ0 t ), which is exponential growth. With this in mind, we can use Eq. (5.4) to make our first growth estimates:      ∗ (X j )l it ξˆ  = O(e4(λ N −λ0 )t ). e

·x0

Of course, it is clear from Eq. (5.4) that at generic points in generic directions, the growth will actually be much less than this estimate indicates since the numerator and denominator will both grow at the same rate. Finally we consider the components ∗ (ξ j )l . Since we have embedded M in CP N , the exponential eit R is a matrix exponential, and there is an explicit formula for its derivative. Write x0 = [z] and let S j ∈ su(N + 1) be

434

B. C. Hall, W. D. Kirwin ˆ

the matrix such that ei(t ξ +sξ j ) · x0 = [ei(t R+s S j ) z]. Using the formula for the derivative of the exponential, we compute ( ) ! " −it ad(R) d i(t ξˆ +sξ j ) it R 1 − e ∗ (ξ j ) = |s=0 e · x0 = e Sjz . ds t ad(R) As a linear operator on su(N + 1), the matrix ad(R) is skew and hence has purely imaginary √ eigenvalues√(which vary continuously with respect to choice of ξ ). Denote them by −1µ0 , . . . , −1µd with µ0 ≤ · · · ≤ µd . Then ! ! "m  ! " "   −it ad(R) e(λ N +µd )t   it R 1 − e Sjz  = O .  e   t ad(R) t Finally, we can use this to estimate the growth of the component ∗ (ξ j )l by computing with the Fubini–Study metric as above: "m " 1 − e−it ad(R) |wt | e Sjz t ad(R) m "m ⎞ ! ! " −it ad(R) % 1 − e − wtl Y¯lm w¯ tm eit R Sjz ⎠ t ad(R) l,m  (4λ0 +3λ N +µd )t  e . =O t

1 ∗ (ξ j ) = |wt |4 l

!

2

%

!

Y¯lm

!

it R

Putting this all together, we obtain an estimate of the growth of the Jacobian τ . Define a(ξˆ ) = d(4λ0 + 3λ N + µd ) + 4(2n − d)(λ N − λ0 ). Then for all x0 ∈ −1 (0),    d  2n−d    ˆ −1 (4λ0 +3λ N +µd )t 4(λ N −λ0 )t = O t (t ξ , x e ) e τ 0    ˆ = O t −d e(d(4λ0 +3λ N +µd )+4(2n−d)(λ N −λ0 ))t = O t −d ea(ξ )t . Since the eigenvalues λ j and µ j depend continuously on the choice of ξ , and the ˆ ˆ unit sphere {ξˆ ∈ g : ξˆ = 1} is compact,  there issome point ξmax ∈ g where a(ξ ) attains   −d at its maximum; let a = a(ξˆmax ). Then τ (t ξˆ , x0 ) = O(t e ) uniformly in x0 and ξˆ as desired.   Next, we combine the previous two lemmas and integrate over the complement of a neighborhood of 0 ∈ g to prove Theorem 5.5. Proof (Proof of Theorem 5.5). Introduce the polar coordinate ξ = (t, ξˆ ), t ∈ R+ on g. Then d d ξ = t d−1 J (ξˆ )dt ∧ d ξˆ , where d ξˆ is the solid angle element and J (ξˆ ) is the

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435

appropriate Jacobian. By the previous two lemmas and the fact that f ∈ C ∞ (M) implies f is bounded on M, there is some b > 0 such that for R and k sufficiently large, 
  

g\B R (0)


   f τ exp − 2k φξ d d ξ  γξ



 
  ∞    = τ (t ξˆ , x0 ) exp − 2k φt ξˆ t d−1 J (ξˆ )d ξˆ dt   R S d−1  γt ξˆ
∞
≤ b t −d eat e−2kCt t d−1 J (ξˆ ) d ξˆ dt. R

S d−1

For k sufficiently large there is some b > 0 such that


∞
R

S d−1

e(a−2kC)t J (ξˆ ) dt ∧ d ξˆ = vol(S d−1 ) t




∞ R

e(a−2kC)t dt ≤ b e−R Dk . t

 Letting b = b b , we obtain our desired result. 

5.2. Laplace’s approximation. We now turn to the issue of uniformity in Laplace’s approximation. We need to show that when applied to the expressions for Ik ( f )([x0 ]) and Jk ( f )([x0 ]), we obtain limits that are uniform in x0 . To do this, follow a standard argument for Laplace’s approximation using the Morse Lemma, but we use a parameterized version (the Morse–Bott Lemma). Then if we simply are careful to bound our errors at each stage, we will obtain the desired uniform limits. We now state the Morse–Bott Lemma. (A careful proof can be found in [BaH04].) A point p ∈ M is a critical point of a function ρ : M → R if the differential of ρ at p is zero. A function ρ is said to be a Morse–Bott function if Crit (ρ), the set of critical points of ρ, is a disjoint union of connected smooth submanifolds and for each critical point, the Hessian H (ρ)( p), in the directions normal to C, is non-degenerate. Lemma 5.3 (Morse–Bott Lemma). Let ρ : M → R be a Morse–Bott function, C a connected component of Crit ( f ) of dimension 2n − d, and p ∈ C. Then there exists an open neighborhood U of p and a smooth chart ϕ : U → Rd × R2n−d such that 1. ϕ( p) = 0, 2. ϕ(U ∩ C) = {(x, y) ∈ Rd × R2n−d : x = 0}, and 2 + · · · + z2 3. (ρ ◦ ϕ −1 )(z, y) = ρ(C) − 21 (z 12 + z 22 + · · · + z k2 ) + 21 (z k+1 2n−d ), where k ≤ 2n − d is the index of H (ρ)( p) and f (C) is the common value of ρ on C. Moreover, if ϕ  is another smooth chart satisfying (1) and (2) near p, the dz Jacobian of the coordinate change in the z-directions, evaluated at p, is det dz  = √ det H (ρ)( p). To apply Laplace’s approximation in our case, we need to know the value of the density τ (ξ, x0 ) when ξ = 0. Lemma 5.4. The function τ equals vol(G · x0 ) on the zero-set of the moment map.

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Proof. Let Wx = {X x : ξ ∈ g}. Then we claim that the complexified tangent space to M, at a point x0 ∈ −1 (0), decomposes as a B-orthogonal direct sum M ∩ Tx0 −1 (0)). TxC0 M = WxC0 ⊕ J WxC0 ⊕ (Tx1,0 0

(5.5)

To see this, first note that by the definition of the G C -action, the tangent space to the G C -orbit G C · x is Wx ⊕ J Wx . Since G · x0 is a totally real submanifold, Wx0 ∩ Tx1,0 M = {0}, 0 and so a dimension count shows that the complexified tangent space decomposes into the claimed direct sum. We need to show that the direct sums are in fact B-orthogonal. First, if v ∈ Tx0 −1 (0), then v(φξ ) = 0 since the moment map is constant (and in fact identically zero) on −1 (0). Hence B(v, J X ξ ) = ω(v, X ξ ) = −dφξ (v) = −v(φξ ) = 0 and so J Wx0 is −1 (0)). Then J v = i v B-orthogonal to Tx0 −1 (0). Next, suppose v ∈ (Tx1,0 0 M ∩ Tx 0  so that 



B(X ξ + J X ξ , v) = i ω(X ξ , v) + iω(J X ξ , v) = iv(φξ ) + v(φξ ) = 0, since v is a linear combination of vectors in Tx0 −1 (0). This shows WxC0 ⊕ J WxC0 −1 (0)). Finally, B(X ξ , J X ξ  ) = ω(X ξ , X ξ  ) = is B-orthogonal to (Tx1,0 0 M ∩ Tx 0  {φξ , φξ  } = φ[ξ,ξ  ] = 0 on the zero-set, so Wx0 is B-orthogonal to J X x0 . Having established (5.5), we now turn to the computation of τ. On the zero-set of the moment map, the tangent space to the orbit exp(ig) · x0 is J Wx0 , which, by (5.5), is B-orthogonal to the tangent space of the zero-set. So if we choose coordinates {x j } on M near x0 ∈ −1 (0) such that exp(ig) · x0 = {x d+1 = · · · = x 2n = 0}, and x j (eiξ · x0 ) = ξ j , ξ  (i.e., x j is the image in M of the linear coordinate in the direction of ξ j on g), then B is block diagonal and so for x ∈ −1 (0), det Bx = (det Bx )|exp(i g)·x0 (det Bx )|−1 (0) . By the standard formula for the volume form in local coordinates, we have  (εω )x = dvol(M)x = det Bx d x 1 ∧ · · · ∧ d x 2n   = det Bx |exp(i g)·x0 d x 1 ∧ · · · ∧ d x d ∧ det Bx |−1 (0) d x d+1 ∧ · · · ∧ d x 2n  = det Bx |exp(i g)·x0 d d ξ ∧ dvol(−1 (0))x0 .   We can rewrite det Bx |exp(i g)·x0 = det B ∂ ∂x j , ∂ ∂x k in the desired form by noting that ∂ ∂x j

=

d itξ j x dt |t=0 e

ξ

= J X x j so that          ∂ ∂ = det B J X ξ j , J X ξk = det B X ξ j , X ξk = det  B. det B , j k ∂x ∂x

The lemma now follows from Lemma 3.1.   We are now ready to perform a uniform Laplace’s approximation. We follow a standard argument (see, for example, [BlH75, Sect. 8.3]), although we keep careful track of the error terms to prove uniformity of the limit. Denote the ball of radius R > 0 centered at 0 ∈ g by B R (0).

Unitarity in “Quantization Commutes with Reduction”

Lemma 5.5. Let f ∈ C ∞ (M) and define Ik,R ( f )(x0 ) = (k/2π )d/2 vol(G · x0 ) where ρ(ξ, x0 ) = 2

437


B R (0)

f (ξ, x0 )e−kρ(ξ,x0 ) τ (ξ, x0 ) d d ξ,

1

φξ (eitξ · x0 ) dt. Then there exists some R > 0 such that     lim  Ik,R ( f )(x0 ) − 2−d/2 vol(G · x0 ) f (x0 ) = 0 0

k→∞

uniformly on −1 (0), where H (ρ) is the Hessian of ρ. Proof. Laplace’s method is based on an application of the Morse–Bott lemma; the zeroset {0} × −1 (0) is a critical submanifold for the function ρ—in fact, ρ(0, −1 (0)) = 0 is a minimum since eitξ x0 is the gradient flow line of φξ —and so there exist local coordinates (z, y) in some neighborhood of x0 such that (ρ ◦ ϕ −1 )(z, y) =

1 z · z. 2

(5.6)

For each x0 the coordinates (z, y) are defined in some ball of positive radius; since the zero-set {0} × −1 (0) is compact we can define R > 0 to be the minimum of these radii. If we regard the coordinates z as new coordinates on g then there is a Jacobian J (z) = det ∂ξ ∂z of the coordinate change in the g -directions, whence
k 2 d/2 Ik,R ( f )(x0 ) = (k/2π ) vol(G · x0 ) e− 2 z f (z, x0 )τ (z, x0 )J (z) d d z. B R (0)

The exact first-order Taylor series of the function T (z, x0 ) = f (z, x0 )τ (z, x0 )J (z) 1 d T (tz, x0 )dt to yield T (z, x0 ) = is given by expanding the identity T (z, x0 ) = 0 dt 1 ∂ T (0, x0 ) + z · S(z, x0 ), where S j (z, x0 ) = 0 ∂z j T (tz, x0 )dt. We now have

k 2 d/2 Ik,R (x 0 ) − k2 z2 d = T (0, x0 ) e d z+ e− 2 z (z · S)d d z. (2π/k) vol(G · x0 ) B R (0) B R (0)  Let ∇ = ∂z∂ 1 , . . . , ∂z∂ d be the vector calculus gradient. Then k 2 k 2 k k 2 − e− 2 z (z · S) = ∇ · (Se− 2 z ) − (∇ · S)e− 2 z . 2

Substituting this into the last term above and applying the divergence theorem yields


k 2 k 2 − k2 z2 d−1 ˆ ˆ d d−1 , ∇ · (Se− 2 z )d d z = (n·S)e d  ≤ e− 2 r (n·S) B R (0)

∂ B R (0)

∂ B R (0)

where nˆ is an outward pointing unit normal vector to ∂ B R (0) and d d−1  is the solid angle element. The last integral above is a continuous function of x0 ∈ −1 (0), so it is bounded by a constant Q 1 > 0. Similarly, there is some Q 2 > 0 such that Q 2 = max |∇ · S| . z∈B R (0) x0 ∈−1 (0)

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We can now estimate that for any x0 ∈ −1 (0),  
  − k2 z2 d  (2π/k)d/2 Ik,R (x0 ) − T (0, x0 ) e d z  vol(G · x0 ) B R (0)
k 2 k 2 Q2 Q 1 ≤ e− 2 z d d z+ e− 2 r . k B R (0) k It is a standard result [BlH75, Lemma 8.3.1] that for any bounded domain D and for every m > 0, there exists a constant Cm > 0 such that  d/2
k 2 2π e− 2 z d d z ≤ + Cm λ−m . k D Putting this all together, we have for every x0 ∈ −1 (0) and for every m > 0,     (2π/k)d/2 Ik,R (x0 ) − T (0, x0 )   vol(G · x0 )  
  d/2 − k2 z2 d   ≤ (2π/k) Ik,R (x0 ) − T (0, x0 ) e d z B R (0)  
  k 2 −2z d d/2   + |T (0, x0 )|  e d z − (2π/k)  B R (0)

Q2 Q1 − k r 2 ≤ (2π/k)d/2 + e 2 + (Q 2 + Q 3 )Cm k −m , k k where Q 3 = max x0 ∈−1 (0) |T (0, x0 )| . All that remains is to note that by the Morse–Bott Lemma and Lemma 5.4, f (x0 )vol(G · x0 ) T (0, x0 ) = √ . det H (ρ)(0, x0 ) By Theorem 4.1 and Lemma 3.1, the denominator is 2d/2 vol(G · x0 ). Putting this into the above inequality and taking the limit k → ∞ we obtain our desired result.   If we include the metaplectic then a similar proof applies if we replace   Lcorrection, ξ εω . Since the argument of the exponent is zero τ (ξ, x0 ) by τ (ξ, x0 ) exp − γξ J2εX ω on the zero-set, we obtain a similar result: Lemma 5.6. Let f ∈ C ∞ (M) and define 
  d/2
L J X ξ εω k −kρ(ξ,x0 ) Jk,R ( f )(x0 ) = e f (ξ, x0 ) exp − τ (ξ, x0 )d d ξ. 2π 2εω B R (0) γξ Then

  lim  Jk,R (x0 ) − f (x0 ) = 0

k→∞

uniformly on

−1 (0).

We are now ready to prove our main result.

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Proof (Proof of Theorem 5.1). We write Ik as the sum of an integral over B R (0) and an integral over the complement of B R (0). Combining Theorem 4.1 and Lemmas 5.4 and 5.5, the first term approaches 2−d/2 f (x0 )vol(G · x0 ) uniformly as k → ∞. By Lemma 5.5 the second term approaches 0 uniformly as k → ∞. If we include the metaplectic correction, the proof is similar, except the first term approaches f (x0 ) uniformly.   6. Discussion and Examples We begin by noting that compactness is not essential to the issues considered in this paper. Given a noncompact Kähler manifold M acted on by a compact group G, one can define a natural map between the G-invariant holomorphic sections of the relevant line bundle over M and the space of all holomorphic sections of the “quotient” bundle over M//G. (Simply restrict the section over M to −1 (0) and then let it descend to M//G = −1 (0)/G.) Although it is unlikely that this map is invertible for arbitrary noncompact M, it is likely to be invertible in many interesting examples, and similarly in the presence of the metaplectic correction. It therefore makes sense to investigate the issue of unitarity at least in the more favorable noncompact examples. Indeed, it even makes sense, to some extent, to consider quantization and reduction for certain infinite-dimensional Kähler manifolds. This sort of problem arises naturally in quantum field theories, where the holomorphic approach to quantization is often the most natural one and where one usually has to reduce by an (infinite-dimensional) group of gauge symmetries. Of course, it inevitably requires some creativity to give a sensible meaning to the quantization and to the reduction in infinite-dimensional settings. Nevertheless, there are some interesting examples (discussed below) where this can be done. In the rest of this subsection, we discuss some noncompact (and, in some cases, infinite dimensional) examples in which the issue of unitarity in quantization versus reduction is of interest. In some of these cases, a Guillemin–Sternberg-type map (with the metaplectic correction) actually turns out to be exactly unitary. Our first example is the quantization of Chern–Simons theory, as considered in the paper [ADPW91] of Axelrod, della Pietra, and Witten. The authors perform a Kähler quantization of the moduli space M of flat connections modulo gauge transformations over a Riemann surface . Much of the analysis is done by regarding this moduli space as the symplectic quotient of the space A of all connections by the group G of gauge transformations, using the result of Atiyah and Bott that the moment map for the action of G is simply the curvature. The main result of [ADPW91] is the construction of a natural projectively flat connection that serves to identify the Hilbert spaces obtained by using different complex structures on . The existence of the projectively flat connection shows that the quantization procedure is independent of the choice of complex structure on , since it allows one to identify (projectively) all the different Hilbert spaces with one another. There is, however, one important issue that is not fully resolved in [ADPW91], namely the issue of the unitarity of the connection. The projectively flat connection “upstairs” on A is unitary, at least formally. The authors suggest, then, that one should simply define the norm of a section downstairs to be the norm of the corresponding section upstairs, in which case, the connection would (formally) be unitary. However, because A is infinitedimensional, it is not entirely clear that this prescription makes sense. Now, if it were true that the Guillemin–Sternberg map was unitary in this context, that would mean that

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computing the norm upstairs is the same as computing the norm downstairs. In that case, one would expect to have unitarity using the natural downstairs norm on the space of sections. Since (as we show in this paper) one cannot expect the Guillimen–Sternberg map to be unitary in general, it is not clear what norm one should use in order to have the projectively flat connection be unitary. Thus, the failure of unitarity (in general) has consequences in this case. A second example is the canonical quantization of (1 + 1)-dimensional Yang–Mills theory. In this case, the upstairs space is an infinite-dimensional affine space, namely the cotangent bundle of the space of connections over the spatial circle. Because the upstairs space is an affine space, there is a well-defined (Kähler) quantization, namely a Segal–Bargmann space over an infinite-dimensional vector space (as in [BSZ92]). Unfortunately, as is often the case in field theories, there are no nonzero vectors in this space that are invariant under the action of the gauge group [DH00]. Thus, if one wants to do quantization first and then reduction, some regularization procedure must be used when performing the reduction. Two different regularization procedures have been considered, that of Wren [LW97, Wre98] (using Landsman’s generalized Rieffel induction [Lan95]) and that of Driver–Hall [DH99] (using a Gaussian measure of large variance to approximate the nonexistent Lebesgue measure). The two procedures give the same result, that the “first quantize and then reduce” space can be identified with a certain L 2 -space of holomorphic functions over the complexified structure group G C . Meanwhile, the results of [Hal02] indicate that the same L 2 -space of holomorphic functions can be obtained by geometric quantization of G C , provided that the metaplectic correction is used. This means that in this case, a Guillemin–Sternberg-type map, with metaplectic correction, does turn out to be unitary. See also [Hal01]. A third example, related to the second one, is reduction from G C (which is naturally identified with the cotangent bundle T ∗ (G)) to G C /HC (identified with T ∗ (G/H )). Here H is the fixed-point subgroup of an involution, which means that G/H is a compact symmetric space. Results of Hall [Hal02] and Stenzel [Ste99] show that the “first quantize and then reduce” space can be identified with an L 2 -space of holomorphic functions on G C /HC with respect to a certain heat kernel measure. (This result can also be obtained from the computation of the relevant orbital integral by Flensted-Jensen [FJ78, Eq. 6.20].) If G/H is again isometric to a compact Lie group (i.e., if G = U × U and H is the diagonal copy of U ), then the above-mentioned results show that the “first reduce and then quantize” can be identified in a natural unitary fashion with the “first quantize and then reduce” space, provided that the metaplectic correction is included in both constructions. So this situation provides another example in which a Guillemin– Sternberg-type map (with metaplectic correction) is unitary. On the other hand, if G/H is not isometric to a compact Lie group, then the norms on the two spaces are not the same. Rather, the measure obtained in the “first reduce and then quantize” space is the leading term in the asymptotic expansion of the heat kernel measure in the “first quantize and then reduce” space; see the discussion on pp. 244–245 of [Hal02]. (In the case that G/H is isometric to a compact Lie group, the relevant heat kernel is actually equal to the leading term in the expansion, up to an overall constant.) In this case, we see that, even with the metaplectic correction, the measures used to compute the two norms are not equal; they are, however, equal to leading order in
. A final example, considered in [FMMN03], is reduction from T ∗ (G) to T ∗ (G/AdG). We may identify G/AdG with T /W, where T is a maximal torus in G and W is the Weyl group. Although in this case the action of G is generically nonfree, one should be able to construct a natural map (with the metaplectic correction) by a construction

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441

similar to the one we give in this paper. We expect that this map will turn out to be the one given by F → σ F|T in the notation of [FMMN03]. If this is the case, then Theorems 2.2 and 2.3 of [FMMN03] will show that the identification of the “first quantize and then reduce space” with the “first reduce and then quantize” space is unitary, up to a constant. The results of [FMMN03] are related to the genus-one case of the quantization of Chern–Simons theory considered in [ADPW91]. Acknowledgements. The authors are grateful to Dennis Snow for suggesting to use an embedding into projective space to prove Lemma 5.2. They would also like to thank Alejandro Uribe for bringing to their attention the papers [Char06b] and [Char06a], and Liviu Nicolaescu for pointing us toward references for the coarea formula.

References [ADPW91] Axelrod, S., Della Pietra, S., Witten, E.: Geometric quantization of Chern-Simons gauge theory. J. Diff. Geom. 33, 787–902 (1991) [BSZ92] Baez, J.C., Segal, I.E., Zhou, Z.-F.: Introduction to algebraic and constructive quantum field theory. Princeton Series in Physics. Princeton, NJ: Princeton University Press, 1992 [BaH04] Banyaga, A., Hurtubise, D.E.: A proof of the Morse–Bott lemma. Exp. Math. 22(4), 365– 373 (2004) [BlH75] Bleistein, N., Handelsman, R.A.: Asymptotic Expansions of Integrals. New York: Dover Publications, Inc., 1975 [BPU95] Borthwick, D., Paul, T., Uribe, A.: Legendrian distributions with applications to relative Poincaré series. Invent. Math. 122(2), 359–402 (1995) [BU96] Borthwick, D., Uribe, A.: Almost complex strucutres and geometric quantization. Math. Res. Lett. 3(6), 845–861 (1996) [BU00] Borthwick, D., Uribe, A.: Nearly Kählerian embeddings of symplectic manifolds. Asian J. Math. 4(3), 599–620 (2000) [BdMG81] Boutet de Monvel, L., Guillemin, V.: The Spectral Theory of Toeplitz Operators, Vol. 99 Ann. of Math. Studies, Princeton, NJ: Princeton University Press, 1981 [Char06a] Charles, L.: Semi-classical properties of geometric quantization with metaplectic correction. Comm. Math. Phys. 270(2), 445–480 (2007) [Char06b] Charles, L.: Toeplitz operators and Hamiltonian torus actions. J. Func. Anal. 236(1), 299–350 (2006) [Chav06] Chavel, I.: Riemannian Geometry: A Modern Introduction, Second Edition, Vol. 98 Cambridge studies in advanced mathematics. New York: Cambridge University Press, 2006 [Czy78] Czyz, J.: On some approach to geometric quantization. Diff. Geom. Methods in Math. Phys. 676, 315–328 (1978) [Dir64] Dirac, P.A.M.: Lectures on Quantum Mechanics. New York: Yeshiva University, 1964 [Don04] Donaldson, S.K.: Remarks on gauge theory, complex geometry and 4-manifold topology. In: S.M. Atiyah, D. Iagolnitzer, editors, Fields Medallists’ Lectures, 2nd Edition, no. 9 in World Scientific Series in 20th Century Mathematics, 2004 [DH99] Driver, B.K., Hall, B.C.: Yang–Mills theory and the Segal–Bargmann transform. Commun. Math. Phys. 201, 249–290 (1999) [DH00] Driver, B.K., Hall, B.C.: The energy representation has no non-zero fixed vectors. In: Stochastic processes, physics and geometry: new interplays, II, number 29 in Conference Proceedings, Providence, RI: Amer. Math. Soc. 2000 [DK00] Duistermaat, J., Kolk, J.: Lie Groups. Berlin: Springer-Verlag, 2000 [FJ78] Flensted-Jensen, M.: Spherical functions of a real semisimple Lie group. A method of reduction to the complex case. J. Func. Anal. 30(1), 106–146 (1978) [FMMN03] Florentino, C., Matias, P., Mourão, J., Nunes, J.P.: Coherent state transforms and vector bundles on elliptic curves. J. Func. Anal. 204(2), 355–398 (2003) [Flu98] Flude, J.P.M.: Geometric asymptotics of spin. Thesis, U. Nottingham, UK, 1998 [GLP99] Gilkey, P.B., Leahy, J.V., Park, J.: Spectral geometry, Riemannian submersions, and the Gromov– Lawson conjecture. Studies in Advanced Mathematics. Boca Raton, FL: Chapman & Hall/CRC, 1999 [Got86] Gotay, M.J.: Constraints, reduction, and quantization. J. Math. Phys. 27(8), 2051–2066 (1986) [GH78] Griffiths, P., Harris, J.: Principles of Algebraic Geometry. New York: John Wiley & Sons, 1978

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B. C. Hall, W. D. Kirwin

Guillemin, V., Sternberg, S.: Geometric quantization and multiplicities of group representations. Invent. Math. 67, 515–538 (1982) Hall, B.C.: Coherent states and the quantization of (1 + 1)-dimensional Yang–Mills theory. Rev. Math. Phys. 13(10), 1281–1305 (2001) Hall, B.C.: Geometric quantization and the generalized Segal–Bargmann transform for Lie groups of compact type. Commun. Math. Phys. 226, 233–268 (2002) Heinzner, P., Huckleberry, A., Loose, F.: Kählerian extensions of the symplectic reduction. J. Reine Angew. Math. 455, 123–140 (1994) Huebschmann, J.: Kähler quantization and reduction. J. Reine Angew. Math. 591, 75–109 (2006) Jeffrey, L.C., Kirwan, F.C.: Localization and the quantization conjecture. Topology 36(3), 647– 693 (1997) Kempf, G., Ness, L.: The length of vectors in representation spaces. In: Algebraic Geometry (Proceedings of the Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), no. 732 Lecture Notes in Math., Berlin: Springer, 1979, pp 233–243 Knapp, A.: Lie Groups: Beyond an Introduction, 2nd Edition, Vol. 140 Progress in Mathematics. Basel-Boston: Birkhäuser, 2002 Landsman, N.: Rieffel induction as generalized quantum Marsden–Weinstein reduction. J. Geom. Phys. 15(4), 285–319 (1995) Landsman, N., Wren, K.: Constrained quantization and θ -angles. Nuc. Phys. B 502(3), 537– 560 (1997) Ma, X., Marinescu, G.: Generalized Bergman kernels on symplectic manifolds. C. R. Acad. Sci. Paris, Ser. I 339 (7), 493–498 (2004). Full Version: http://arxiv.org/list/math.DG/0411559, 2004 Ma, X., Zhang, W.: Bergman kernels and symplectic reduction. C. R. Acad. Sci. Paris, Ser. I 341, 297–302 (2005) Ma, X., Zhang, W.: Bergman kernels and symplectic reduction. http://arxiv.org/list/math.DG/0607605, 2006 Marsden, J., Weinstein, A.: Reduction of symplectic manifolds with symmetry. Rep. Math. Phys. 5(1), 121–130 (1974) Meinrenken, E.: Symplectic surgery and the Spinc -Dirac operator. Adv. Math. 134(2), 240–277 (1998) Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory, Third Edition, Volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)]. Berlin: Springer-Verlag, 1994 Paoletti, R.: The Szëgo Kernel of a symplectic quotient. Adv. Math. 197(2), 523–553 (2005) Sjamaar, R.: Holomorphic slices, symplectic reduction and multiplicities of representations. Ann. Math. (2) 141(1), 87–129 (1995) Sjamaar, R.: Symplectic reduction and Riemann–Roch formulas for multiplicities. Bull. Amer. Math. Soc. (N.S.) 33(3), 327–388 (1996) Stenzel, M.: The Segal–Bargmann transform on a symmetric space of compact type. J. Func. Anal. 165, 44–58 (1999) Tian, Y., Zhang, W.: An analytic proof of the geometric quantization conjecture of Guillemin– Sternberg. Invent. Math. 132, 229–259 (1998) Woodhouse, N.M.J.: Geometric Quantization, 2nd Edition. New York: Oxford University Press, Inc., 1991 Wren, K.: Constrained quantization and θ -angles. II. Nuc. Phys. B 521(3), 471–502 (1998) Zheng, F.: Complex Differential Geometry, Volume 18 of Studies in Advanced Mathematics. Providence, RI: Amer. Math. Soc./ Int. Press, 2000

Communicated by J.Z. Imbrie

Commun. Math. Phys. 275, 443–477 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0305-4

Communications in

Mathematical Physics

Asymptotic Completeness for N -Body Quantum Systems with Long-Range Interactions in a Time-Periodic Electric Field Tadayoshi Adachi
Department of Mathematics, Graduate School of Science, Kobe University, 1-1, Rokkodai-cho, Nada-ku, Kobe-shi, Hyogo 657-8501, Japan. E-mail: [email protected] Received: 4 October 2006 / Accepted: 11 January 2007 Published online: 10 August 2007 – © Springer-Verlag 2007

Abstract: We show the asymptotic completeness for N -body quantum systems with long-range interactions in a time-periodic electric field whose mean in time is non-zero, where N ≥ 2. One of the main ingredients of this paper is to give some propagation estimates for physical propagators generated by time-periodic Hamiltonians which govern the systems under consideration. 1. Introduction In this paper, we study the scattering theory for N -body quantum systems with longrange pair interactions in a time-periodic electric field whose mean in time is non-zero, where N ≥ 2. In our previous work [A3], under the assumption that pair interactions between particles whose specific charges are different are short-range, the asymptotic completeness of (modified) wave operators was obtained. We here note that [A3] does allow that pair potentials between particles whose specific charge are the same belong to a certain class of long-range potentials. In this paper, we will improve the above result by permitting that pair potentials between particles whose specific charges are different also belong to some classes of long-range potentials. First we will give the notation in the N -body scattering theory for describing our results. We consider a system of N particles moving in a given time-periodic electric field E (t) ∈ Rd , E (t) ≡ 0. We suppose that E (t) ∈ C 0 (R; Rd ) has a period T > 0, that is, E (t + T ) = E (t) for any t ∈ R, and its mean E in time is non-zero, i.e.
1 T E = E (t) dt = 0. T 0 Let m j , e j and r j ∈ Rd , 1 ≤ j ≤ N , denote the mass, charge and position vector of the j th particle, respectively. We suppose that the particles under consideration interact
Research partially supported by the Grant-in-Aid for Young Scientists of MEXT #17740078.

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with one another through the pair potentials V jk (r j − rk ), 1 ≤ j < k ≤ N . We assume that these pair potentials are independent of time t. Then the total Hamiltonian for the system is given by     1 − ∆r j − e j E (t), r j  + V jk (r j − rk ), H˜ (t) = 2m j 1≤ j≤N

1≤ j
  where ξ, η = dj=1 ξ j η j for ξ , η ∈ Rd . 1≤ j
π : Rd×N → X denotes the orthogonal projection onto X . We put x = πr for r ∈ Rd×N , and  
e1 eN 1 T E(t) = π E (t), . . . , E (t) , E = E(t) dt. m1 mN T 0 Throughout this paper, we assume that there exists at least one pair ( j, k) whose specific charges are different, that is, e j /m j = ek /m k . By virtue of this assumption, one sees that E(t) = 0 whenever E (t) = 0, and that E = 0. By separating the part associated with the center of mass motion from H˜ (t), we obtain the Hamiltonian 1 H (t) = − ∆ − E(t) · x + V 2 on L 2 (X ), where ∆ is the Laplace-Beltrami operator on X . We will study the scattering theory for this Hamiltonian H (t). A non-empty subset of the set {1, . . . , N } is called a cluster. Let C j , 1 ≤ j ≤ m, be clusters. If ∪1≤ j≤m C j = {1, . . . , N } and C j ∩ Ck = ∅ for 1 ≤ j < k ≤ m, a = {C1 , . . . , Cm } is called a cluster decomposition. #(a) denotes the number of clusters in a. Let A be the set of all cluster decompositions. Suppose a, b ∈ A . If b is obtained as a refinement of a, that is, if each cluster in b is a subset of a cluster in a, we say b ⊂ a, and its negation is denoted by b ⊂ a. Any a is regarded as a refinement of itself. The one and N -cluster decompositions are denoted by amax and amin , respectively. The pair ( j, k) is identified with the (N − 1)-cluster decomposition {( j, k), (1), . . . , ( j), . . . , ( k), . . . , (N )}. Next we introduce two subspaces X a and X a of X for a ∈ A :     a m j r j = 0 for each cluster C in a , X a = X  X a . X = r ∈ X  j∈C

In particular, X ( j,k) is identified with the configuration space for the relative position of ( j,k) ) = V (r − r ). It is well known j th and k th particles. k  Hence one can put V( j,k) (x  jk j that X a = r ∈ X  r j = rk for each pair ( j, k) ⊂ a , and that L 2 (X ) is decomposed

Asymptotic Completeness for N -Body Quantum Systems

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into L 2 (X a ) ⊗ L 2 (X a ). π a : X → X a and πa : X → X a denote the orthogonal projections onto X a and X a , respectively. We put x a = π a x and xa = πa x for x ∈ X . We now define the cluster Hamiltonian  1 V( j,k) (x ( j,k) ), Ha (t) = − ∆ − E(t) · x + V a , V a = 2 ( j,k)⊂a

which governs the motion of the system broken into non-interacting clusters of particles. The intercluster potential Ia is given by  Ia (x) = V (x) − V a (x) = V( j,k) (x ( j,k) ). ( j,k)⊂a

Put E a (t) = π a E(t) and E a (t) on L 2 (X ) is decomposed into

= πa E(t). Then the cluster Hamiltonian Ha (t) acting

Ha (t) = H a (t) ⊗ Id + Id ⊗ Ta (t) on L 2 (X a ) ⊗ L 2 (X a ), where Id are the identity operators, 1 1 H a (t) = − ∆a − E a (t) · x a + V a , Ta (t) = − ∆a − E a (t) · xa , 2 2 and ∆a (resp. ∆a ) is the Laplace-Beltrami operator on X a (resp. X a ). Now we will state 

the assumptions  on the pair potentials. Let c stand for a maximal element of the set a ∈ A  E a = 0 with respect to the relation ⊂, where E a = π a E. Such a cluster decomposition uniquely exists, and it follows that ( j, k) ⊂ c is equivalent to e j /m j = ek /m k . If, in particular, e j /m j = ek /m k for any ( j, k) ∈ A , then c = amin . Since E = 0 as mentioned above, we see that c = amax . We will impose different assumptions on V jk according as ( j, k) ⊂ c or ( j, k) ⊂ c: Let ρ > 0. (V )c,L V jk (r ) ∈ C ∞ (Rd ), ( j, k) ⊂ c, is a real-valued function and satisfies 

|∂ β V jk (r )| ≤ Cβ r −(ρ +|β|)

√ with 3 − 1 < ρ  ≤ 1. (V )c,G V jk (r ) ∈ C ∞ (Rd ), ( j, k) ⊂ c, is a real-valued function and satisfies ¯ |∂ β V jk (r )| ≤ Cβ r −(ρG +|β|) , |β| ≤ 1, |∂ β V jk (r )| ≤ Cβ , |β| ≥ 2, with 0 < ρG ≤ 1/2. (V )c,D,ρ V jk (r ) ∈ C ∞ (Rd ), ( j, k) ⊂ c, is a real-valued function and satisfies ¯ |∂ β V jk (r )| ≤ Cβ r −(ρ+|β|/2) . Under these assumptions, all the Hamiltonians defined above are essentially selfadjoint on C0∞ . Their closures are denoted by the same notations. If V jk , ( j, k) ⊂ c, satisfies (V )c,L , then V jk is called a long-range potential. We note that if V jk , ( j, k) ⊂ c, satisfies (V )c,G with ρ ≤ 1/2, then V jk should be called a “Stark long-range” ¯ or (V )c,D,ρ ¯ potential. To formulate the obtained results precisely, we will define modified wave operators: Let U (t, s), Ua (t, s) and U¯ a (t, s), a ⊂ c, be unitary propagators generated by timedependent Hamiltonians H (t), Ha (t) and Ta (t), respectively. Here the unitary propagator U (t, s) generated by the time-dependent Hamiltonian H (t) means the family of unitary operators {U (t, s)}t,s∈ R on L 2 (X ) with the following properties:

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(1) (t, s) → U (t, s) is strongly continuous. (2) U (t, s) = U (t, r )U (r, s) holds for any r , s, t ∈ R. (3) For ψ ∈ D, d d U (t, s)ψ = −i H (t)U (t, s)ψ, U (t, s)ψ = iU (t, s)H (s)ψ dt ds hold, where D ⊂ t∈ R D(H (t)) is a certain dense subspace of L 2 (X ) such that U (t, s)D ⊂ D for any s, t ∈ R. We note that U (s, s) = Id and U (t, s)∗ = U (s, t) for any s, t ∈ R. If H (t) is periodic in t with a period T > 0, then U (t + T, s + T ) = U (t, s) holds for any s, t ∈ R. The existence and uniqueness of U (t, s) are guaranteed by virtue of results of Yajima [Ya2] and the Avron-Herbst formula [CFKS] as follows: We introduce a strongly continuous family of unitary operators on L 2 (X ) by ˜ ˜ ˜ p T˜ (t) = e−i a(t) ei b(t)·x e−i c(t)· ,

where ˜ = b(t)


0

t




t

E(τ ) dτ, c(t) ˜ = 0

1 ˜ ) dτ, a(t) b(τ ˜ = 2

(1.1)


t

˜ )2 dτ. b(τ

(1.2)

0

We also introduce the time-dependent Hamiltonian H Sc (t) on L 2 (X ) by 1 ˜ H Sc (t) = − ∆ + V (x + c(t)). 2 Since the propagator generated by H Sc (t) exists uniquely by virtue of results of [Ya2], we write it as U Sc (t, s). Then one sees that the propagator U (t, s) generated by H (t) also exists uniquely by virtue of the Avron-Herbst formula U (t, s) = T˜ (t)U Sc (t, s)T˜ (s)∗ .

(1.3)

We here emphasize that U (t, s) enjoys the domain invariance property, U (t, s)D(( p 2 + x 2 )n ) ⊂ D(( p 2 + x 2 )n ), n ∈ N,

(1.4)

and that U (t, s) is strongly continuous in D(( p 2 + x 2 )n ) with respect to (t, s) under the assumptions (V )c,L , and (V )c,G or (V )c,D,ρ . Equation (1.4) is proved as follows: By ¯ ¯ virtue of results of [Ya2], one sees that U Sc (t, s)D( p 2 ) ⊂ D( p 2 ) and that U Sc (t, s) is strongly continuous in D( p 2 ) with respect to (t, s). Since V is smooth with bounded derivatives by assumption, one sees that U Sc (t, s)D(( p 2 + x 2 )n ) ⊂ D(( p 2 + x 2 )n ), n ∈ N holds and that U Sc (t, s) is strongly continuous in D(( p 2 + x 2 )n ) with respect to (t, s), by following the argument in Hunziker [Hu], Radin-Simon [RaS] and Graf [Gr1, Gr2]. Noting that T˜ (t)D(( p 2 + x 2 )n ) ⊂ D(( p 2 + x 2 )n ) and that T˜ (t) is strongly continuous in D(( p 2 + x 2 )n ) with respect to t, we obtain (1.4) by virtue of (1.3). We now note that for a ⊂ c, H a (t) is independent of time t because of E a (t) ≡ 0. Thus we write it as H a . Then Ua (t, s) is written as Ua (t, s) = e−i(t−s)H ⊗ U¯ a (t, s). a

(1.5)

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447

We here introduce Ua,D (t, 0) = Ua (t, 0)e−i

t

c 0 Ia ( pa τ ) dτ

(1.6)

for a ⊂ c. Here Iac = Ia − Ic and pa = −i∇a is the velocity operator on L 2 (X a ). Under D,± the assumptions (V )c,L and (V )c,G ¯ , we define the modified wave operators Wa,G , a ⊂ c, by D,± Wa,G = s-lim U (t, 0)∗ Ua,D (t, 0)e−i

t

˜ )) dτ 0 Ic (c(τ

t→±∞

(P a ⊗ Id),

(1.7)

where P a : L 2 (X a ) → L 2 (X a ) is the eigenprojection associated with H a . We call

t ˜ )) dτ the Graf (or Zorbas)-type modifier (see [A1, AT1, Gr3, HMS2 and Zo]). e−i 0 Ic (c(τ One of the main results of this paper is the following theorem: are fulfilled. Then the modified wave Theorem 1.1. Assume that (V )c,L and (V )c,G ¯ D,± operators Wa,G , a ⊂ c, exist, and are asymptotically complete L 2 (X ) =

 D,± ⊕ Ran Wa,G . a⊂c

with 0 < ρ ≤ 1/2 instead of (V )c,G is satisfied. Next we suppose that (V )c,D,ρ ¯ ¯ First we consider the case where c = amin , that is, #(c) = N . Since 2 ≤ #(c) < N by assumption, N ≥ 3 is assumed here. Under the assumptions (V )c,L and (V )c,D,ρ with ¯ √ D,± ( 3 − 1)/2 < ρ ≤ 1/2, we define the modified wave operators Wa,D , a ⊂ c, by D,± Wa,D = s-lim U (t, 0)∗ Ua,D (t, 0)e−i

t

˜ )) dτ 0 Ic ( pc τ +c(τ

t→±∞

(P a ⊗ Id).

(1.8)

Then we have the following theorem: √ Theorem 1.2. Assume that c = amin and that (V )c,L and (V )c,D,ρ with ( 3 − 1)/2 < ¯ D,± ρ ≤ 1/2 are fulfilled. Then the modified wave operators Wa,D , a ⊂ c, exist, and are asymptotically complete  D,± ⊕ Ran Wa,D . L 2 (X ) = a⊂c

Finally, we consider the case where c = amin . For example, when N = 2, c = amin is satisfied by assumption. We here note that if c = amin , 1 Hc (t) = − ∆ − E(t) · x ≡ H0 (t), 2 Ic (x) = V (x), xc = x and pc = p, where p = −i∇ is the velocity operator on L 2 (X ). U0 (t, s) denotes the unitary propagator generated by H0 (t). Under the assumption (V )c,D,ρ with 0 < ρ ≤ 1/2, an approximate solution of the Hamilton-Jacobi ¯ equation (∂t K )(t, ξ ) =

1 2 ˜ (ξ + b(t)) + V ((∇ξ K )(t, ξ )) 2

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can be constructed (see §6). If V ≡ 0 and K (0, ξ ) ≡ 0, K (t, ξ ) is written as K (t, ξ ) = K 0 (t, ξ ) ≡

t 2 ˜ · ξ + a(t), ˜ ξ + c(t) 2

(1.9)

where a(t) ˜ and c(t) ˜ are as in (1.2). We here note that (∇ξ K 0 )(t, ξ ) is written as (∇ξ K 0 )(t, ξ ) = ξ t + c(t). ˜

(1.10)

Under the assumptions c = amin and (V )c,D,ρ with 0 < ρ ≤ 1/2, we define the modified ¯ ± wave operators W0,D by ± W0,D = s-lim U (t, 0)∗ U0 (t, 0)e−i

t T

V (∇ξ K (τ, p)) dτ

t→±∞

.

(1.11)

t

As seen in §6, in particular if 1/4 < ρ ≤ 1/2, e−i T V (∇ξ K (τ, p)) dτ in (1.11) can be

t

t ˜ )) dτ , which is called the Dollardreplaced by e−i 0 V ((∇ξ K 0 )(τ, p)) dτ = e−i 0 V ( pτ +c(τ type modifier (see [A1, AT2, JO, JY, S and W]). Then we have the following theorem: Theorem 1.3. Assume that c = amin and (V )c,D,ρ with 0 < ρ ≤ 1/2 are fulfilled. Then ¯ ± the modified wave operators W0,D exist and are unitary on L 2 (X ). Remark 1.4. In our analysis, we need a certain regularity of V jk like being at least in Cb8 (Rd ) in order to obtain some propagation estimates which are useful for proving the asymptotic completeness of wave operators (see §3, in particular Lemma 3.7). The initial time 0 can be replaced by any s ∈ R. The problem of the asymptotic completeness for N -body quantum systems has been studied by many mathematicians and they have achieved a great success. As for results for N -body Schrödinger operators, see e.g. the book by Derezi´nski-Gérard [DG1]. As for results for N -body Schrödinger operators in a constant magnetic field, see e.g. the book by Gérard-Łaba [GŁ]. As for results for N -body Schrödinger operators in a constant electric field, whose effect is called the Stark effect, see e.g. Adachi-Tamura [AT1, AT2] and Herbst-Møller-Skibsted [HMS1, HMS2]). These results are concerned with time-independent Hamiltonians. On the other hand, for time-dependent Hamiltonians, the lack of energy conservation is a barrier in studying this problem. For instance, the time-boundedness of the kinetic energy was the key fact for studying the charge transfer model (see e.g. [Gr1]). Howland [Ho1] proposed the stationary scattering theory for time-dependent Hamiltonians, whose formulation was the quantum analogue to the procedure in the classical mechanics in order to ‘recover’ the conservation of energy. Yajima [Ya1] applied this Howland method to the two-body quantum systems with time-periodic short-range potentials and studied the problem of the asymptotic completeness for the systems (see also [Ho2] and [Yo1]). His result was extended to the three-body case by Nakamura [N] later (as for the spectral theory for general N -body systems, see Møller-Skibsted [MøS]). Under the same assumption on E (t) as in this paper, Møller [Mø] studied the scattering theory for two-body quantum systems with short-range interactions, and Adachi [A3] also studied the scattering theory for N -body quantum systems with short-range interactions between particles whose specific charges are different as mentioned before, by using the so-called Howland-Yajima method.

Asymptotic Completeness for N -Body Quantum Systems

449

The Howland-Yajima method reduces the problem under consideration to the problem of the asymptotic completeness of the usual wave operators associated with the Floquet Hamiltonian given by K = −i∂t + H (t) on L 2 (T ; L 2 (X )) formally. Thus this method matches the quantum scattering theory for time-periodic short-range interactions, but seems not sufficient for the time-periodic long-range ones. For instance, Kitada-Yajima [KY] dealt with the so-called AC Stark effect, in which the mean of E (t) in t is zero, for two-body quantum systems with long-range interactions, by using the so-called Enss method. As implied by this, in studying the scattering theory for time-periodic long-range interactions, one needs to know some propagation properties of the physical propagator U (t, s). One of purposes of this paper is to give some propagation estimates for U (t, s) (see §3), that was not done in [Mø] and [A3]. In the case where E (t) = E + o(1), which is not time-periodic, this was done by Yokoyama [Yo2] for two-body systems with shortrange interactions. In the argument below, we will consider the case where t → ∞ only. The case where t → −∞ can be dealt with quite similarly. For an X -valued operator L, (L 2 )1/2 is denoted by |L| for brevity’s sake. 2. Asymptotic Clustering In this section, we prove the so-called asymptotic clustering for the system under consideration, which is the key to showing Theorems 1.1, 1.2 and 1.3. Throughout this and the next sections, we suppose that (V )c,L and (V )c,D,ρ V jk (r ) ∈ C ∞ (Rd ), ( j, k) ⊂ c, is a real-valued function and satisfies ¯ |∂ β V jk (r )| ≤ Cβ r −(ρ+|β|/2) , |β| ≤ 1, |∂ β V jk (r )| ≤ Cβ , |β| ≥ 2, with ρ = ρG or (V )c,D,ρ , with 0 < ρ ≤ 1/2 are fulfilled. We note that under (V )c,G ¯ ¯ (V )c,D,ρ is fulfilled. ¯ In this paper, we often use the following convention for smooth cut-off functions F with 0 ≤ F ≤ 1: For sufficiently small δ > 0, we define F(s ≤ d) = 1 for s ≤ d − δ, = 0 for s ≥ d, F(s ≥ d) = 1 for s ≥ d + δ, = 0 for s ≤ d, and F(d1 ≤ s ≤ d2 ) = F(s ≥ d1 ) F(s ≤ d2 ). To clarify the dependence on δ > 0 in the definition of F, we often write Fδ for F. We now introduce the time-dependent intercluster potential Ic (t, x) as ˜ ≤ 2ε1 ) Ic (t, x) = Ic (x)Fε1 (t −2 |x − c(t)| ˜ is defined by (1.2). Since with some sufficiently small ε1 > 0, where c(t)
t E ˜ − Es) ds = O(t) c(t) ˜ − t2 = (b(s) 2 0

(2.1)

(2.2)

˜ − Et by the definition of E, we see that Ic (t, x) by virtue of the periodicity of b(t) enjoys the estimate |∂xβ Ic (t, x)| ≤ Cβ (t + x1/2 )−(2ρ+|β|) , |β| ≤ 1,

(2.3)

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T. Adachi

for t > 0, if 0 < ε1 < minα⊂c |E α |/4. Then we define the time-dependent Hamiltonian H˜ c (t) by H˜ c (t) = Hc (t) + Ic (t, x),

(2.4) ˜ ˜ ˜ and denote by Uc (t), t ≥ T , the unitary propagator generated by Hc (t) such that Uc (T ) = Id. We here note that the domain invariance property of U˜ c (t), U˜ c (t)D( p 2 + x 2 ) ⊂ D( p 2 + x 2 ) holds and that U˜ c (t) is strongly continuous in D( p 2 + x 2 ) with respect to t, which can be proved by the argument in the proof of (1.4). In order to prove Theorems 1.1, 1.2 and 1.3, we will claim that the following asymptotic clustering holds: Theorem 2.1 (Asymptotic Clustering). Assume that (V )c,L and (V )c,D,ρ with ¯ 0 < ρ ≤ 1/2 are fulfilled. Then the strong limit Ω˜ c = s-lim U (t, 0)∗ U˜ c (t) t→∞

(2.5)

exists and is unitary on L 2 (X ). This property played an important role to prove the asymptotic completeness of N -body quantum systems in a (time-independent or time-periodic) homogeneous electric field in the works of Adachi and Tamura [AT1, AT2], and Adachi [A3] (see also [A1] and [HMS2]). In order to prove Theorem 2.1, we need the following propagation estimates for both U˜ c (t) and U (t, 0). From now on the norm and scalar product in a Hilbert space H1 are denoted by  · H1 and (·, ·)H1 , respectively. The norm of bounded operators on H1 is also denoted by  · B (H1 ) : Proposition 2.2. The following estimates hold for φ ∈ D( p 2 + x 2 ) as t → ∞:   | p − b(t)| ˜ U˜ c (t)φ  2 = O(1), L (X )

  |x − c(t)| ˜ U˜ c (t)φ  L 2 (X ) = O(t).

(2.6) (2.7)

Proof. In this paper, the Heisenberg derivative of Φ(t) associated with H (t) is denoted by ∂Φ (t) + i[H (t), Φ(t)]. D H (t) (Φ(t)) = ∂t We first note that (U˜ c (t)φ, H c U˜ c (t)φ) L 2 (X )
= (φ, H c φ) L 2 (X ) + pc U˜ c (t)φ = U˜ c (t) pc φ + x U˜ c (t)φ = U˜ c (t)xφ +




t T




t T

t T

(U˜ c (τ )φ, D H˜ c (τ ) (H c )U˜ c (τ )φ) L 2 (X ) dτ,

U˜ c (t)U˜ c (τ )∗ D H˜ c (τ ) ( pc )U˜ c (τ )φ dτ,

U˜ c (t)U˜ c (τ )∗ D H˜ c (τ ) (x)U˜ c (τ )φ dτ

Asymptotic Completeness for N -Body Quantum Systems

451

hold, which can be proved by regularizing pc and x (see e.g. [Gr1, Gr2]). Taking M > 0 so large that H c + M = −∆c /2 + V c + M ≥ 1, by the first equality and (2.3), we have (H c + M)1/2 U˜ c (t)φ2L 2 (X )




≤ (H c + M)1/2 φ2L 2 (X ) + C

t T

τ −(2ρ+1) (H c + M)1/2 U˜ c (τ )φ2L 2 (X ) dτ.

We here used that D H˜ c (τ ) (H c ) = −{(∇ c Ic )(τ, x) · p c + p c · (∇ c Ic )(τ, x)}/2 and that p c (H c + M)−1/2 is bounded on L 2 (X ). By virtue of Gronwall’s lemma, (H c + M)1/2 U˜ c (t)φ2L 2 (X ) = O(1) is obtained because (∇ c Ic )(τ, x) = O(τ −(2ρ+1) ) is integrable on [T, ∞) by virtue of ρ > 0. This implies | p c |U˜ c (t)φ L 2 (X ) = O(1) because | p c |(H c + M)−1/2 is bounded on L 2 (X ). On the other hand, since ˜ ˜ ))φ ( pc − b(t)) U˜ c (t)φ = U˜ c (t)( pc − b(T
t ˜ ))U˜ c (τ )φ dτ + U˜ c (t)U˜ c (τ )∗ D H˜ c (τ ) ( pc − b(τ T

˜ )) = −(∇c Ic )(τ, x) = O(τ −(2ρ+1) ), we have and D H˜ c (τ ) ( pc − b(τ ˜ U˜ c (t)φ L 2 (X ) = O(1). | pc − b(t)| These two estimates imply (2.6). Moreover, since ˜ ))φ (x − c(t)) ˜ U˜ c (t)φ = U˜ c (t)(x − c(T
t + U˜ c (t)U˜ c (τ )∗ D T

H˜ c (τ ) (x

˜ ), (2.6) implies (2.7). and D H˜ c (τ ) (x − c(τ ˜ )) = p − b(τ

− c(τ ˜ ))U˜ c (τ )φ dτ

 

By virtue of (2.7), the following corollary can be obtained immediately. Corollary 2.3. Let ε > 0. Then the following estimate holds for φ ∈ D( p 2 + x 2 ) as t → ∞:    Fε (t −2 |x − c(t)| ˜ ≥ ε)U˜ c (t)φ  L 2 (X ) = O(t −1 ). (2.8) Theorem 2.4. Let 0 < ε < minα⊂c |E α |/4. Then the following estimates hold for φ ∈ D(( p 2 + x 2 )2 ) as t → ∞:    Fε (t −2 |x − c(t)| ˜ ≥ ε)U (t, 0)φ  L 2 (X ) = O(t −1/2 ), (2.9)   −2 1/2 | p − b(t)|F ˜ ˜ ≤ 2ε)U (t, 0)φ  L 2 (X ) = O(t ), (2.10) ε (t |x − c(t)|   −2 |x − c(t)|F ˜ ˜ ≤ 2ε)U (t, 0)φ  L 2 (X ) = O(t 3/2 ). (2.11) ε (t |x − c(t)|

452

T. Adachi

Theorem 2.4 is one of the main results of this paper. The next section will be devoted to showing this theorem. We will now prove Theorem 2.1 under the assumption that Theorem 2.4 holds. Proof of Theorem 2.1. We have only to prove the existence of the limits lim U (t, 0)∗ U˜ c (t)φ,

t→∞

lim U˜ c (t)∗ U (t, 0)φ

t→∞

for φ ∈ D(( p 2 + x 2 )2 ), because D(( p 2 + x 2 )2 ) is dense in L 2 (X ). We here put η(t) = Fε1 /2 (t −2 |x − c(t)| ˜ ≤ ε1 ). By virtue of Corollary 2.3 and Theorem 2.4, we see that lim U (t, 0)∗ (1 − η(t))U˜ c (t)φ = 0,

t→∞

lim U˜ c (t)∗ (1 − η(t))U (t, 0)φ = 0.

t→∞

Thus we have only to show the existence of the limits lim U (t, 0)∗ η(t)U˜ c (t)φ,

t→∞

lim U˜ c (t)∗ η(t)U (t, 0)φ.

t→∞

(2.12)

We here note that Ic (x)η(t) = Ic (t, x)η(t) for t > 0, which is the key in the proof. Since d (U (t, 0)∗ η(t)U˜ c (t)φ) dt ˜ ˜ + t −2 ( p − b(t))} + O(t −4 )]U˜ c (t)φ, = U (t, 0)∗ [η1 (t) · {−2t −3 (x − c(t)) d ˜ (Uc (t)∗ η(t)U (t, 0)φ) dt ˜ = U˜ c (t)∗ [{−2t −3 (x − c(t)) ˜ + t −2 ( p − b(t))} · η1 (t) + O(t −4 )]U (t, 0)φ with η1 (t) = Fε1 /2 (t −2 |x − c(t)| ˜ ≤ ε1 )(x − c(t))/|x ˜ − c(t)|, ˜ we obtain from Proposition 2.2 and Theorem 2.4,   d   (U (t, 0)∗ η(t)U˜ c (t)φ) = O(t −2 ),  dt  2 L (X )   d   (U˜ c (t)∗ η(t)U (t, 0)φ) = O(t −3/2 ),  dt  2 L (X )

which implies the existence of (2.12) by virtue of the Cook-Kuroda method. Thus the proof is completed.   Remark 2.5. If ρ > 1/2, that is, if all V jk ’s with ( j, k) ⊂ c are Stark short-range, s-lim U˜ c (t)∗ Uc (t, 0) t→∞

exists and is unitary on L 2 (X ), by virtue of (2.3) with −2ρ < −1. Therefore it follows from this and Theorem 2.1 that Ωc = s-lim U (t, 0)∗ Uc (t, 0) t→∞

(2.13)

exists and is unitary on L 2 (X ). This gives an alternative proof of the asymptotic completeness obtained in Møller [Mø] and Adachi [A3].

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3. Propagation Estimates for U(t, 0) We first move the oscillation arising from E(t) − E into the potential V , and reduce the present problem to the one for a so-called N -body Stark Hamiltonian with a certain time-periodic potential, by using a version of the Avron-Herbst formula initiated by Møller [Mø]: We define T -periodic functions on R:
t

1 T t b(t) = (E(s) − E) ds − b0 , b0 = (E(s) − E) dsdt, T 0 0 0 
t
t
T 1 1 E 2 c(t) = − |b(t)| + b(s) ds − c0 , c0 = E · b(s) ds dt , (3.1) 2 T 2 |E| 0 0 0 
t 1 |b(s)|2 − E · c(s) ds, a(t) = 2 0 where b(t), c(t) ∈ X and a(t) ∈ R, and a strongly continuous periodic family of unitary operators on L 2 (X ) by T (t) = e−ia(t) eib(t)·x e−ic(t)· p .

(3.2)

We here note that the constants b0 and c0 in (3.1) are chosen in order to make c(t) and a(t) T -periodic. Moreover we define the time-dependent Hamiltonian H S (t) on L 2 (X ) by 1 H0S = − ∆ − E · x. 2

H S (t) = H0S + V (x + c(t)),

(3.3)

H0S is called the free Stark Hamiltonian. We note that the time-periodic potential V (x + c(t)) is written as V (x + c(t)) = V c (x) + Ic (x + c(t)),

(3.4)

because c(t) ∈ X c by definition and V c (x) = V c (x c ) is independent of xc ∈ X c also by definition. Put
t
t E b S (t) = E dτ = Et, c S (t) = b S (τ ) dτ = t 2 , (3.5) 2 0 0 and define T S (t) as T (t) = e S

−ia S (t) ib S (t)·x −ic S (t)· p

e

e

1 , a (t) = 2




t

S

b S (τ )2 dτ.

(3.6)

0

It is well known that the original Avron-Herbst formula [AH] holds: e−it H0 = T S (t)e−it H0 , S

Sc

1 H0Sc = − ∆. 2

(3.7)

Let U S (t, s) be the unitary propagator generated by the Hamiltonian H S (t), whose existence and uniqueness can be guaranteed by the Avron-Herbst formula U (t, s) = T (t)U S (t, s)T (s)∗ , or U S (t, s) = T S (t)U Sc (t, s)T S (s)∗ . (3.8)

454

T. Adachi

We here note that the domain invariance property of U S (t, 0), U S (t, 0)D(( p 2 + x 2 )n ) ⊂ D(( p 2 + x 2 )n ), n ∈ N, holds and that U S (t, 0) is strongly continuous in D(( p 2 + x 2 )n ) with respect to t, by virtue of the property of U (t, s) mentioned in §1. Noting that ˜ ˜ + b(t) = p − b S (t) − b0 , T (t)∗ ( p − b(t))T (t) = p − b(t) T (t)∗ (x − c(t))T ˜ (t) = x − c(t) ˜ + c(t) = x − c S (t) − (b0 t + c0 ) by virtue of (3.1), we see that Theorem 2.4 is equivalent to the following: Theorem 3.1. Let 0 < ε < minα⊂c |E α |/4. Then the following estimates hold for φ ∈ D(( p 2 + x 2 )2 ) as t → ∞:    Fε (t −2 |x − c S (t)| ≥ ε)U S (t, 0)φ  2 = O(t −1/2 ), (3.9) L (X )   S −2 S S 1/2 | p − b (t)|Fε (t |x − c (t)| ≤ 2ε)U (t, 0)φ  2 = O(t ), (3.10) L (X )   |x − c S (t)|Fε (t −2 |x − c S (t)| ≤ 2ε)U S (t, 0)φ  2 = O(t 3/2 ). (3.11) L (X ) Therefore this section is devoted to showing Theorem 3.1. To this end, we first introduce the Floquet Hamiltonian associated with H S (t), which is key in the HowlandYajima method (see Howland [Ho1, Ho2] and Yajima [Ya1]). We let T = R/(T Z) be the torus and introduce H = L 2 (T ; L 2 (X )) ∼ = L 2 (T ) ⊗ L 2 (X ). We define a family of operators {Uˆ (σ )}σ ∈ R on H by (Uˆ (σ ) f )(t) = U S (t, t − σ ) f (t − σ )

(3.12)

for f ∈ H . Since {Uˆ (σ )}σ ∈ R forms a strongly continuous unitary group on H , Uˆ (σ ) is written as Uˆ (σ ) = e−iσ K ,

(3.13)

where K = Dt + H S (t) is a self-adjoint operator on H , where Dt = −i∂t is a selfadjoint operator on L 2 (T ) with its domain AC 2 (T ), which is the space of absolutely continuous functions on T with their derivatives being square integrable (following the notation in [RS]). K is called the Floquet Hamiltonian associated with H S (t). The following two theorems show some spectral properties of K , which can be proved in the same way as in [A3] (see also Herbst-Møller-Skibsted [HMS1]) by using |V jk (r )| + |∇V jk (r )| = o(1) as |r | → ∞, which is fulfilled under (V )c,L and (V )c,D,ρ with 0 < ρ ≤ 1/2. So we ¯ omit the proof. Theorem 3.2 (Absence of Bound States). The pure point spectrum σ pp (K ) of the Floquet Hamiltonian K is empty.

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Theorem 3.3 (Mourre Estimate). Let A = E · p/|E| and 0 < ν < |E| < ν  . Then one can take δ > 0 so small uniformly in λ ∈ R that ηδ (K − λ)i[K , A]ηδ (K − λ) ≥ νηδ (K − λ)2 , 

ηδ (K − λ)i[K , −A]ηδ (K − λ) ≥ −ν ηδ (K − λ)

(3.14) 2

(3.15)

hold, where ηδ ∈ C0∞ (R) satisfies 0 ≤ ηδ ≤ 1, ηδ (t) = 1 for |t| ≤ δ and ηδ (t) = 0 for |t| ≥ 2δ. In particular, the spectrum of K is purely absolutely continuous. Now we prepare the maximal and minimal acceleration bounds for e−iσ K , by following the abstract theory of Skibsted [Sk]: Definition 3.4. For given β, α ≥ 0 and ε > 0, take a function χα,ε (y) = Fε (y ≤ −ε) such that d d χα,ε (y) ≤ 0, αχα,ε (y) + y χα,ε (y) = κ 2 (y) dy dy for some κ ∈ C ∞ (R) with κ(y) ≥ 0. Put gβ,α,ε (y, σ ) = −σ −β (−y)α χα,ε (σ −1 y) for (n) y ∈ R and σ > 0. Write gβ,α,ε (y, σ ) = ∂ yn gβ,α,ε (y, σ ) for n ∈ N ∪ {0}. We recall the almost analytic extension method by Helffer-Sjöstrand [HeSj], which is useful in analyzing operators given by functions of self-adjoint operators. For two operators B1 and B2 , we define ad0B1 (B2 ) = B2 , adnB1 (B2 ) = [adn−1 B1 (B2 ), B1 ], n ≥ 1. For m ∈ R, let S m be the set of functions f ∈ C ∞ (R) such that | f (k) (s)| ≤ Ck sm−k , k ≥ 0. ∞ ˜ ˜ If f ∈ S m with  then there exists f ∈ C (C) such that f (s) = f (s) for s ∈ R,

m ∈ R,  ˜ supp f (ζ ) ⊂ ζ ∈ C |Im ζ | ≤ d(1 + |Re ζ |) for some d > 0 and

|∂ ζ f˜(ζ )| ≤ C M ζ m−1−M |Im ζ | M ,

M ≥ 0.

Such a function f˜(ζ ) is called an almost analytic extension of f . Let B be a self-adjoint operator. If f ∈ S −m with m > 0, then f (B) is represented by
1 ∂ ζ f˜(ζ )(ζ − B)−1 dζ ∧ dζ . f (B) = 2πi C For f ∈ S m with m ∈ R, we have the following formulas of the asymptotic expansion of the commutator: [B1 , f (B)] =

M−1  n=1

=

M−1  n=1

(−1)n−1 n ad B (B1 ) f (n) (B) + R M n! 1 (n) f (B)ad Bn (B1 ) + R M , n!

456

T. Adachi

where


(−1) M+1 ∂ ζ f˜(ζ )(ζ − B)−1 ad BM (B1 )(ζ − B)−M dζ ∧ dζ , 2πi C
1 = ∂ ζ f˜(ζ )(ζ − B)−M ad BM (B1 )(ζ − B)−1 dζ ∧ dζ , 2πi C

RM = R M

R M is bounded if there exists k such that m + k < M and ad BM (B1 )(B + i)−k is bounded. Similarly, R M is bounded if there exists k such that m + k < M and (B + i)−k ad BM (B1 ) is bounded. For the proof, see e.g. Gérard [G]. We use the above formulas frequently. Proposition 3.5 (Maximal Acceleration Bound). Let f ∈ C0∞ (R), s0 ≥ s1 ≥ 0, and ε > 0. Then there exists M > 0 such that the following estimate holds as σ → ∞:   −1 −s0 (σ  p)s1 Fε (σ −1  p ≥ M)e−iσ K f (K ) p−s0  ). (3.16) B (H ) = O(σ Proof. This proposition is proved in [A3] under the assumption that V jk , ( j, k) ⊂ c, is Stark short-range, that is, ρ > 1/2. In order to show that one has only to assume that ρ > 0 for the proof, we sketch the proof by following the argument in the proof of Theorem 2.4 of [Sk]. We take any n 0 ∈ N such that n 0 ≥ 2 and 0 < β0 < 1, and set α0 ∈ N such that α0 ≤ n 0 − 1, A(σ ) = vσ −  p and B(σ ) = σ −1 A(σ ). We here note that adnA(σ ) (K ) is bounded on H for any n such that 1 ≤ n ≤ n 0 , by pseudodifferential calculus. This fact makes the proof slightly simpler than the one given by the abstract theory of [Sk] directly (see [Yo2]). Let (β, α) = (0, α0 − 1) or (β, α) = (β0 , α0 ). For φ ∈ H , we put ψ(σ ) = e−iσ K f (K ) p−α/2 φ and use the convention P(σ )σ = (ψ(σ ), P(σ )ψ(σ ))H . Putting bα,ε (y) = (−y)α χα,ε (y), gβ,α,ε (y, σ ) is written as gβ,α,ε (y, σ ) = −σ α−β bα,ε (σ −1 y). Hence we have gβ,α,ε (A(σ ), σ ) = −σ α−β bα,ε (B(σ )). We abbreviate gβ,α,ε and bα,ε as g and b, respectively. Since b ∈ S α , an almost analytic ˜ )| ≤ C M ζ α−1−M |Im ζ | M , M ≥ 0. Since extension b˜ of b satisfies |∂ ζ b(ζ
σ D K (g(A(τ ), τ ))τ dτ, −g(A(σ ), σ )σ = −g(A(1), 1)1 − 1

we compute D K (g(A(τ ), τ )) = D K (−τ α−β b(B(τ ))) as D K (g(A(τ ), τ )) = −τ α−β D K (b(B(τ ))) − (α − β)τ α−β−1 b(B(τ )). By using


1 ˜ )(ζ − B(τ ))−1 D K (B(τ ))(ζ − B(τ ))−1 dζ ∧ dζ D K (b(B(τ )) = ∂ ζ b(ζ 2πi C

and D K (B(τ )) = D K (τ −1 A(τ )) = τ −1 D K (A(τ )) − τ −1 B(τ ), we have
τ −1 ˜ )(ζ − B(τ ))−1 D K (A(τ ))(ζ − B(τ ))−1 dζ ∧ dζ D K (b(B(τ )) = ∂ ζ b(ζ 2πi C − τ −1 B(τ )b(1) (B(τ )).

Asymptotic Completeness for N -Body Quantum Systems

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Since b(1) (y) = −α(−y)α−1 χ (y) + (−y)α χ (1) (y) = −(−y)α−1 κ 2 (y), we have E 1 (τ ) = τ α−β−1 B(τ )b(1) (B(τ )) − (α − β)τ α−β−1 b(B(τ )) = − τ α−β−1 (−B(τ ))α+1 χ (1) (B(τ )) + βτ α−β−1 b(B(τ )) ≥ 0

(3.17)

because of χ (1) ≤ 0 and β ≥ 0. We here note that −


τ α−β−1 ˜ )(ζ − B(τ ))−1 D K (A(τ ))(ζ − B(τ ))−1 dζ ∧ dζ ∂ ζ b(ζ 2πi C α0  τ α−β−1 (m) b (B(τ ))adm−1 − = B(τ ) (D K (A(τ ))) m! m=1
τ α−β−1 ˜ )(ζ − B(τ ))−(α0 +1) − ∂ ζ b(ζ 2πi C

=

α0 

−1 0 × adαB(τ dζ ∧ dζ ) (D K (A(τ )))(ζ − B(τ ))

(3.18)

τ α−β−m (m) b (B(τ ))adm−1 A(τ ) (D K (A(τ ))) m! m=1
τ α−α0 −β−1 ˜ )(ζ − B(τ ))−(α0 +1) − ∂ ζ b(ζ 2πi C −

−1 0 × adαA(τ dζ ∧ dζ . ) (D K (A(τ )))(ζ − B(τ ))

We now write the last term of the right hand side of (3.18) as E 7 (τ ). The first term −τ α−β−1 b(1) (B(τ ))D K (A(τ )) is written as the sum of the following E 2 (τ ), E 3 (τ ) and E 4 (τ ): E 2 (τ ) = τ α−β−1 γ (B(τ ))D K (A(τ ))γ (B(τ )), E 3 (τ ) = τ α−β−1 γ (B(τ ))

α1  (−1)m τ −m m ad A(τ ) (D K (A(τ )))γ (m) (B(τ )), m!

m=1

E 4 (τ ) = τ

α−β−1

γ (B(τ ))R1 (τ )

with α1 = (α − 1)/2 if α is odd, and α1 = α/2 if α is even, R1 (τ ) =


(−1)α1 +1 τ −(α1 +1) 1 +1 ∂ ζ γ˜ (ζ )(ζ − B(τ ))−1 adαA(τ ) (D K (A(τ ))) 2πi C × (ζ − B(τ ))−(α1 +1) dζ ∧ dζ ,

where we put γ (y) = (−b(1) (y))1/2 = (−y)(α−1)/2 κ(y) ∈ S (α−1)/2 . Noting that adnA(τ ) (D K (A(τ ))) = O(1) for 0 ≤ n ≤ n 0 − 1, we have R1 (τ ) = O(τ −(α1 +1) )

458

T. Adachi

if (α − 1)/2 < α1 + 1, which is satisfied obviously. The rest are written as the sum of E 5 (τ ) and E 6 (τ ), E 5 (τ ) =

α0 

−

m=2

×

τ α−β−m gm (B(τ ))h m (B(τ )) m!

α−m  m 1 =0

E 6 (τ ) =

α0  m=2

−

(−1)m 1 τ −m 1 m 1 +m−1 (m 1 ) (B(τ )), ad A(τ ) (D K (A(τ )))gm m1!

τ α−β−m gm (B(τ ))h m (B(τ ))Rm (τ ) m!

with Rm (τ ) =


(−1)α−m+1 τ −(α−m+1) ∂ ζ g˜ m (ζ )(ζ − B(τ ))−1 adαA(τ ) (D K (A(τ ))) 2πi C × (ζ − B(τ ))−(α−m+1) dζ ∧ dζ .

Here gm (y) = b(α−m)+ /2,ε/2 (y) ∈ S (α−m)+ /2 and h m (y) = (−y)−(α−m)+ b(m) (y) ∈ S 0 with (s)+ = max{0, s} for s ∈ R. We have Rm (τ ) = O(τ −(α−m+1) ) if (α − m)/2 < α − m + 1, that is, α > m − 2, for 2 ≤ m ≤ α0 , which is satisfied because α = α0 − 1 or α = α0 . In the same way, we have E 7 (τ ) = O(τ α−α0 −β−1 )

(3.19)

because of α < α0 + 1. Since D K (A(τ )) = v − i[K ,  p] ≥ 0 for sufficiently large v > 0, E 2 (τ ) ≥ 0

(3.20)

holds. By (3.17) and (3.20), we have
−g(A(σ ), σ )σ ≤ −g(A(1), 1)1 −

σ 1

7 

E j (τ )τ dτ.

j=3

Now we would like to show −g(A(σ ), σ )σ = O(1)φ2H

(3.21)

for (β, α) = (0, 1), . . . , (0, α0 − 1), (β0 , α0 ) by induction in α0 . When α0 = 1, since E 3 (τ ) = E 5 (τ ) = E 6 (τ ) = 0 and E 4 (τ ) = E 7 (τ ) = O(τ −β0 −1 ) with β0 > 0, we have (3.21) for (β, α) = (β0 , α0 ). Suppose that (3.21) in the case where α0 = n ∈ N with n ≤ n 0 − 2 is true. We now consider the case where α0 = n + 1. We first let (β, α) = (0, α0 − 1) = (0, n). By the assumption of induction, for any l ∈ R such that 0 ≤ l ≤ n, (−g0,n−l,ε (A(τ ), τ ))1/2 e−iτ K f (K ) p−n/2 = O(τ (β0 −l)/2 )

Asymptotic Completeness for N -Body Quantum Systems

459

holds because of −gβ0 ,n,ε (y, τ ) ≥ −τ −β0 (ετ )l g0,n−l,ε (y, τ ). By this, we have b(n−l)/2,ε (B(τ ))e−iτ K f (K ) p−n/2 = O(τ (β0 −n)/2 ).

(3.22)

It is important that the power of the order in the right hand side of (3.22) is independent of l. It follows from (3.22) that γ (m) (B(τ ))ψ(τ )H = O(τ (β0 −α)/2 )φH for 0 ≤ m ≤ α1 , (m 1 ) gm (B(τ ))ψ(τ )H = O(τ (β0 −α)/2 )φH for m ≥ 2, 0 ≤ m 1 ≤ α − m, h m (B(τ )) = O(1) for 2 ≤ m ≤ α0 .

Then we have E 3 (τ ) = 0 if n = 1, E 3 (τ )τ = O(τ β0 −2 )φ2H if n ≥ 2,

E 4 (τ )τ = O(τ β0 /2−3/2 )φ2H if n is odd,

E 4 (τ )τ = O(τ β0 /2−2 )φ2H if n is even,

E 5 (τ )τ = O(τ β0 −2 )φ2H , E 6 (τ )τ = O(τ (β0 −α)/2−1 )φ2H , E 7 (τ )τ = O(τ −2 )φ2H ,

which are all integrable on [1, ∞) by virtue of β0 < 1. Thus we obtain (3.21) for (β, α) = (0, n). We next let (β, α) = (β0 , α0 ) = (β0 , n + 1). In the same way as above, it follows from (3.21) for (β, α) = (0, n), which has been shown above, that γ (m) (B(τ ))ψ(τ )H = O(τ −(α0 −1)/2 ) for 0 ≤ m ≤ α1 , (m 1 ) gm (B(τ ))ψ(τ )H = O(τ −(α0 −1)/2 ) for m ≥ 2, 0 ≤ m 1 ≤ α − m.

Then we have E 3 (τ )τ = E 5 (τ )τ = E 7 (τ )τ = O(τ −β0 −1 )φ2H ,

E 4 (τ )τ = O(τ −β0 −3/2 )φ2H if n + 1 is odd, E 4 (τ )τ = O(τ −β0 −2 )φ2H if n + 1 is even, E 6 (τ )τ = O(τ −β0 −1−(α0 −1)/2 )φ2H ,

which are all integrable on [1, ∞) by virtue of β0 > 0. Thus we obtain (3.21) for (β, α) = (β0 , n + 1). These estimates improve the ones obtained in [Sk] slightly by virtue of the almost analytic extension method, and relax conditions on the relation among α0 , β0 and n 0 supposed in [Sk].   Proposition 3.6 (Minimal Acceleration Bound). Let f ∈ C0∞ (R), s0 ≥ s1 ≥ 0 and ε > 0. Let A, ν and ν  be as in Theorem 3.3. Then the following estimates hold as σ → ∞:   −s0 (ν − σ −1 A)s1 Fε (σ −1 A ≤ ν − ε)e−iσ K f (K )A−s0  ), (3.23) B (H ) = O(σ  −1  −s0 (σ A − ν  )s1 Fε (σ −1 A ≥ ν  + ε)e−iσ K f (K )A−s0  ). (3.24) B (H ) = O(σ

460

T. Adachi

Proof. This proposition can be proved as in the proof of Proposition 3.5. We sketch the proof of (3.23) only: Set A(σ ) = A − νσ . Then we have D K (A(τ )) = i[K , A] − ν. Without loss of generality, one may assume that f (s) = ηδ/2 (s − λ) ∈ C0∞ (R) with some λ ∈ R. Put f 1 (s) = ηδ (s − λ) ∈ C0∞ (R), which satisfies f 1 f = f . Then one should replace E 2 (τ ) in the proof of Proposition 3.5 by E 2 (τ ) = τ α−β−1 f 1 (K )γ (B(τ ))D K (A(τ ))γ (B(τ )) f 1 (K ). E 2 (τ ) can be written as E 2,0 (τ ) + E 2,1 (τ ) + E 2,2 (τ ) + E 2,3 (τ ) + E 2,4 (τ ), where E 2,0 (τ ) = τ α−β−1 γ (B(τ )) f 1 (K )D K (A(τ )) f 1 (K )γ (B(τ )), E 2,1 (τ ) = τ α−β−1 f 1 (K )γ (B(τ ))D K (A(τ )) α1  (−1)m τ −m m ad A(τ ) ( f 1 (K ))γ (m) (B(τ )), × m! m=1

E 2,2 (τ ) = τ α−β−1 f 1 (K )γ (B(τ ))D K (A(τ ))R1,1 (τ ), E 2,3 (τ ) = τ α−β−1

α1  τ −m (m) γ (B(τ ))admA(τ ) ( f 1 (K )) m!

m=1

× D K (A(τ )) f 1 (K )γ (B(τ )), E 2,4 (τ ) = τ

α−β−1

R1,2 (τ )D K (A(τ )) f 1 (K )γ (B(τ )),

with α1 = (α − 1)/2 if α is odd, and α1 = α/2 if α is even,
(−1)α1 +1 τ −(α1 +1) 1 +1 R1,1 (τ ) = ∂ ζ γ˜ (ζ )(ζ − B(τ ))−1 adαA(τ ) ( f 1 (K )) 2πi C × (ζ − B(τ ))−(α1 +1) dζ ∧ dζ ,

R1,2 (τ ) =


τ −(α1 +1) 1 +1 ∂ ζ γ˜ (ζ )(ζ − B(τ ))−(α1 +1) adαA(τ ) ( f 1 (K )) 2πi C × (ζ − B(τ ))−1 dζ ∧ dζ .

Noting that admA(τ ) ( f 1 (K )) = O(1) for 1 ≤ m ≤ α1 + 1 and E 2,0 (τ ) ≥ 0 by virtue of Theorem 3.3, the proof of (3.23) is quite similar to the one of Proposition 3.5.   In order to translate these propagation estimates for e−iσ K into the ones for U S (t, 0), we need the following lemma. Lemma 3.7. Let f ∈ C0∞ (R), s0 ≥ s1 ≥ 0, and ε > 0. Let A, ν and ν  be as in Theorem 3.3. Let M be as in Proposition 3.5. Let Jσ,s1 be one of the following three operators on H: (σ −1  p)s1 Fε (σ −1  p ≥ M), (ν − σ −1 A)s1 Fε (σ −1 A ≤ ν − ε), (σ −1 A − ν  )s1 Fε (σ −1 A ≥ ν  + ε). Then the following estimate holds as σ → ∞: Dt Jσ,s1 e−iσ K f (K ) p−s0 Dt −1 B (H ) = O(σ 1−s0 ).

(3.25)

Asymptotic Completeness for N -Body Quantum Systems

461

Proof. Since −iad Dt (K ) = ∇ Ic (x + c(t)) · b(t), (−i)2 ad2Dt (K ) = ∇ Ic (x + c(t)) · (E(t) − E) + b(t)∗ ∇ 2 Ic (x + c(t))b(t), are bounded on H , it can be shown easily that Dt 2 e−iσ K f (K )Dt −2 = O(σ 2 ), which implies Dt 2 Jσ,0 e−iσ K f (K )Dt −2 = O(σ 2 ) because p does commute with Dt . Noting that p does commute with Dt again, by complex interpolation between this and Jσ,2s1 e−iσ K f (K ) p−2s0 = O(σ −2s0 ) by virtue of Hadamard’s three line theorem, we obtain (3.25).

 

Remark 3.8. In the above proof, it is needed that E(t) ∈ L ∞ (R; Rd ), which is slightly stronger than the condition that E(t) ∈ L 1loc (R; Rd ) supposed in [Mø]. Now we will translate the obtained propagation estimates for e−iσ K into the ones for Take s0 = 2. Let φ ∈ D(( p 2 + x 2 )2 ) ⊂ L 2 (X ) and put φ(t) = U S (t, 0)φ. Then we see that φ(t) ∈ D(Dt ) and that Dt φ(t) ∈ D( p 2 + x 2 ) by virtue of the domain invariance property of U S (t, 0) mentioned before. Let U be the unitary operator on H defined by

U S (t, 0).

(U ψ)(t) = U S (t, 0)ψ(t), t ∈ T , ψ(t) ∈ H . It is known that e−i T K = U (Id ⊗ U S (T, 0))U ∗

(3.26)

holds on H ∼ = L 2 (T ) ⊗ L 2 (X ) (see Yajima-Kitada [YK]). Then we have ( f (K )φ)(t) = U S (t, 0)g(U S (T, 0))φ, t ∈ T , where f ∈ C0∞ (R) supported in (λ0 − π/T, λ0 + π/T ) for some λ0 ∈ R, and g is the function on the unit-circle defined by g(e−i T λ ) = f (λ) (see Møller-Skibsted [MøS]). We here note the following: Let J = J (t) be an operator on H , and ψ = ψ(t) ∈ H be such that e−iσ K ψ ∈ D(J ). Then
T −iσ K 2 ψH = J (t + σ )U S (t + σ, t)ψ(t)2L 2 (X ) dt J e 0

holds. Noting that Jσ,s1 in Lemma 3.7 is independent of t, we see that
T Jσ,s1 U S (t + σ, 0)g(U S (T, 0))φ2L 2 (X ) dt = O(σ −2 ),


0

T 0

∂t {Jσ,s1 U S (t + σ, 0)g(U S (T, 0))φ}2L 2 (X ) dt = O(σ −2 )

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hold with 0 ≤ s1 ≤ 2, by virtue of the above formula and Lemma 3.7. From these, we obtain Jσ,s1 U S (t + σ, 0)g(U S (T, 0))φ2L 2 (X ) ∈ W 1,1 (0, T ),    Jσ,s U S (t + σ, 0)g(U S (T, 0))φ2 2  1,1 = O(σ −2 ) 1 L (X ) W (0,T ) 

 by the Schwarz inequality. Here W 1,1 (0, T ) = u ∈ L 1 (0, T )  u  ∈ L 1 (0, T ) is a Sobolev space on the interval (0, T ). By using the Sobolev imbedding theorem (see e.g. [B]), we obtain    Jσ,s U S (t + σ, 0)g(U S (T, 0))φ2 2  ∞ = O(σ −2 ), 1 L (X ) L (0,T ) which implies Jσ,s1 U S (σ, 0)g(U S (T, 0))φ L 2 (X ) = O(σ −1 ). Therefore the following propagation estimates can be obtained by using a partition of unity on the unit-circle. Proposition 3.9. Let 0 ≤ s1 ≤ 2 and ε > 0. Let A, ν and ν  be as in Theorem 3.3. Let M be as in Proposition 3.5. Then the following estimates hold for φ ∈ D(( p 2 + x 2 )2 ) as t → ∞:   −1 (t  p)s1 Fε (t −1  p ≥ M)U S (t, 0)φ  2 = O(t −1 ), (3.27) L (X )   −1 s −1 S −1 (ν − t A) 1 Fε (t A ≤ ν − ε)U (t, 0)φ  2 = O(t ), (3.28) L (X )   −1 (t A − ν  )s1 Fε (t −1 A ≥ ν  + ε)U S (t, 0)φ  2 = O(t −1 ). (3.29) L (X ) Based on these estimates, we will derive some useful propagation estimates for U S (t, 0). Proposition 3.10 (Maximal Acceleration Bound). Let 0 ≤ s1 ≤ 1/2 and ε > 0. Then there exists M  > 0 such that following estimate holds for φ ∈ D(( p 2 +x 2 )2 ) as t → ∞:   −2 (t x)s1 Fε (t −2 x ≥ M  )U S (t, 0)φ  2 = O(t −1 ). (3.30) L (X ) Proof. The proof will proceed in the way similar to the one of Proposition 3.5. We will here use the notations in the proof of Proposition 3.5. Let n 0 = 3 and 0 < β0 < 1. We set M(t) = −(t −2 x − C)1/2 with C > 0 which will be determined below, B(t) = −b1,ε/2 (M(t)), and A(t) = t B(t). We here note that gβ,α,ε (A(t), t) = −t α−β (−M(t))α χα,ε (M(t)) by definition (see [Sk]). Let φ ∈ D(( p 2 + x 2 )2 ) and put φ(t) = U S (t, 0)φ. We use the convention P(t)t = (φ(t), P(t)φ(t)) L 2 (X ) . As in the proof of Proposition 3.5, we have
t 7 −g(A(t), t)t = −g(A(1), 1)1 − E j (τ )τ dτ, 1 j=1

where D K (A(τ )) in the definitions of E j (τ )’s is replaced by D H S (τ ) (A(τ )). We here note that D H S (τ ) (A(τ )) = −τ D H S (τ ) (b1,ε/2 (M(τ ))) − b1,ε/2 (M(τ )). Since ∂τ b1,ε/2 ,

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463

(1)

(M(τ )) = −b1,ε/2 (M(τ ))(τ −1 M(τ ) + Cτ −1 M(τ )−1 ) and ∇b1,ε/2 (M(τ )) = τ −2 (1)

b1,ε/2 (M(τ ))M(τ )−1 x/(2x), we have (1)

(1)

D H S (τ ) (A(τ )) = b1,ε/2 (M(τ ))M(τ ) − b1,ε/2 (M(τ )) + Cb1,ε/2 (M(τ ))M(τ )−1 (1) − (b1,ε/2 (M(τ ))M(τ )−1 )1/2 (x/x · (τ −1 p) + (τ −1 p) · x/x) (1)

× (b1,ε/2 (M(τ ))M(τ )−1 )1/2 /4. It follows from this that ad1A(τ ) (D H S (τ ) (A(τ ))) = O(τ −2 ), ad2A(τ ) (D H S (τ ) (A(τ ))) = 0. We here note that if α ≤ 2, (1)

(b1,ε/2 (M(τ ))M(τ )−1 )1/2 γ (B(τ )) = O(1),

(3.31)

because γ ∈ S (α−1)/2 . When (β, α) = (β0 , 1), we have E 1 (τ ) ≥ 0, E 3 (τ ) = E 5 (τ ) = E 6 (τ ) = 0 and E 4 (τ ) = E 7 (τ ) = O(τ −β0 −3 ) with β0 > 0. Now we consider E 2 (τ ). We first note that (1) (1) (M(τ ))M(τ ) − b1,ε/2 (M(τ )) = −(−M(τ ))2 χ1,ε/2 (M(τ )) ≥ 0 b1,ε/2 (1)

because of χ1,ε/2 ≤ 0. By using (3.31), Fδ (τ −1  p ≥ M) + Fδ (τ −1  p ≤ M + δ) = Id, (τ −1  p)2 Fδ (τ −1  p ≤ M + δ)2 ≤ (M + δ)2 , (1)

[(τ −1  p)Fδ (τ −1  p ≥ M), (b1,ε/2 (M(τ ))M(τ )−1 )1/2 γ (B(τ ))] = O(τ −3 ), and Proposition 3.9, we have −E 2 (τ )τ ≤ −E 2 (τ )τ + O(τ −1−β0 ), where (1)

E 2 (τ ) = (C − (1 + (M + δ)2 )/4)τ −β0 {(b1,ε/2 (M(τ ))M(τ )−1 )1/2 γ (B(τ ))}2 . If one makes C > 0 so large that C ≥ (1 + (M + δ)2 )/4, −g(A(t), t)t = O(1) is obtained because of β0 > 0. Next we consider the case where (β, α) = (0, 1). We here note that E 1 (τ ) ≥ 0, E 3 (τ ) = E 7 (τ ) = 0 and E 4 (τ ) = E 5 (τ ) = E 6 (τ ) = O(τ −3 ). As for E 2 (τ ), we have −E 2 (τ )τ ≤ O(τ (β0 −1)/2−1 ) by using the result obtained in the case where (β, α) = (β0 , 1). −g(A(t), t)t = O(1) is obtained because β0 < 1. Finally we consider the case where (β, α) = (0, 2). We here note that E 1 (τ ) ≥ 0 and E 4 (τ ) = E 6 (τ ) = E 7 (τ ) = 0. By using the result obtained in the case where (β, α) = (0, 1), we have E 3 (τ )τ = E 5 (τ )τ = O(τ −3 ). As for E 2 (τ ), we have −E 2 (τ )τ ≤ O(τ −3/2 ) by using the result obtained in the case where (β, α) = (0, 1) again. Thus we obtain  −g(A(t), t)t = O(1). This implies (3.30). 

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Proposition 3.11 (Minimal Acceleration Bound). Let 0 ≤ s1 ≤ 1/2 and ε > 0. Let ν and ν  be as in Theorem 3.3. Then the following estimates hold for φ ∈ D(( p 2 + x 2 )2 ) as t → ∞:   (ν/2 − t −2 z)s1 Fε (t −2 z ≤ ν/2 − ε)U S (t, 0)φ  2 = O(t −1 ), (3.32) L (X )   −2  s −2  S −1 (t z − ν /2) 1 Fε (t z ≥ ν /2 + ε)U (t, 0)φ  2 = O(t ), (3.33) L (X ) where z = E · x/|E|. Proof. The proof will proceed in the same way as the one of Proposition 3.10 (see also [A2]). We sketch the proof of (3.32) only by using the notations in the proof of Proposition 3.10. We set M(t) = −(ν/2 − t −2 z)1/2 , B(t) = −b1,ε/2 (M(t)), and A(t) = t B(t). (1) Since ∂τ b1,ε/2 (M(τ )) = −b1,ε/2 (M(τ ))(τ −1 M(τ ) − ντ −1 M(τ )−1 /2) and ∇b1,ε/2 (1)

(M(τ )) = −τ −2 b1,ε/2 (M(τ ))M(τ )−1 E/(2|E|), we have (1)

(1)

D H S (τ ) (A(τ )) = b1,ε/2 (M(τ ))M(τ ) − b1,ε/2 (M(τ )) − νb1,ε/2 (M(τ ))M(τ )−1 /2 (1)

(1)

+ (b1,ε/2 (M(τ ))M(τ )−1 )1/2 τ −1 A(b1,ε/2 (M(τ ))M(τ )−1 )1/2 /2, where A = E · p/|E|. Then it follows from this that ad1A(τ ) (D H S (τ ) (A(τ ))) = O(τ −2 ), ad2A(τ ) (D H S (τ ) (A(τ ))) = 0. In dealing with E 2 (τ )τ , one should notice that τ −1 A − ν = τ −1 A − ν  + (ν  − ν) ≥ (τ −1 A − ν  )Fδ (τ −1 A ≤ ν  − δ), (1)

[(ν  − τ −1 A)Fδ (τ −1 A ≤ ν  − δ), (b1,ε/2 (M(τ ))M(τ )−1 )1/2 γ (B(τ ))] = O(τ −3 ), if α ≤ 2. Here ν  is such that 0 < ν < ν  < |E|, and 2δ = ν  − ν > 0. Then by the same argument as in the proof of Proposition 3.10, (3.32) is obtained.   By the maximal and minimal acceleration bounds, the following key estimate can be obtained. Theorem 3.12. Let ε > 0. Then the following estimates hold for φ ∈ D(( p 2 + x 2 )2 ) as t → ∞:    Fε (t −1 | p − b S (t)| ≥ ε)U S (t, 0)φ  2 = O(t −1/2 ), (3.34) L (X )   S −1 S S 1/2 | p − b (t)|Fε (t | p − b (t)| ≥ ε)U (t, 0)φ  2 = O(t ). (3.35) L (X ) Proof. The proof is quite similar to the one of Theorem 1.1 in [A2]: We write ψ(t) = Fε (t −1 | p − b S (t)| ≥ ε)U S (t, 0)φ. Noting (3.5), we compute   | p − b S (t)|ψ(t)2 2 = (ψ(t), ( p − Et)2 ψ(t)) L 2 (X ) L (X ) = 2(ψ(t), {H0S + |E|(z − At + |E|t 2 /2)}ψ(t)) L 2 (X ) = 2(ψ(t), [H0S + |E|{(z − |E|t 2 /2) − t (A − |E|t)}]ψ(t)) L 2 (X ) .

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465

We first note that H S (t)U S (t, 0)φ L 2 (X ) = O(t) as t → ∞, because it can be proved that H S (t)U S (t, 0)φ




t

= U (t, 0)H (0)φ + S

S

U S (t, τ )b(τ ) · ∇V (x + c(τ ))U S (τ, 0)φ dτ

0

by using the Yosida approximation of H S (τ ), D H S (τ ) H S (τ ) = b(τ ) · ∇V (x + c(τ )) = O(1), and the dominated convergence theorem (see e.g. [Gr1, Gr2]). In particular, it follows from this that (H0S + i)U S (t, 0)φ L 2 (X ) = O(t),

(3.36)

because V (x + c(t)) = O(1). By (3.36) and H0S Fε (t −1 | p − b S (t)| ≥ ε)(H0S + i)−1 = O(1), we have H0S ψ(t) L 2 (X ) = O(t). By virtue of Propositions 3.9 and 3.11,  −1  |t A − |E||1/2 Fε (|t −1 A − |E|| ≥ ε )U S (t, 0)φ 

L 2 (X )

  −2 |t z − |E|/2|1/2 Fε (|t −2 z − |E|/2| ≥ ε )U S (t, 0)φ 

= O(t −1 ),

L 2 (X )

= O(t −1 )

hold. Since [|t −2 z − |E|/2|1/2 Fε (|t −2 z − |E|/2| ≥ ε ), Fε (t −1 | p − b S (t)| ≥ ε)] = O(t −3 ), we have   −1 |t A − |E||1/2 Fε (|t −1 A − |E|| ≥ ε )ψ(t) 2 = O(t −1 ), L (X )   −2 |t z − |E|/2|1/2 Fε (|t −2 z − |E|/2| ≥ ε )ψ(t) 2 = O(t −1 ). L (X ) By using these estimates, we obtain ε2 t 2 ψ(t)2L 2 (X ) ≤ O(t) + 8|E|ε t 2 ψ(t)2L 2 (X ) . If one makes ε > 0 so small that 8|E|ε ≤ ε2 /2, we have ε2 t 2 ψ(t)2L 2 (X ) /2 ≤ O(t), which implies (3.34). By the above argument, one has | p − b S (t)|ψ(t)2L 2 (X ) ≤ O(t) + 8|E|ε t 2 ψ(t)2L 2 (X ) = O(t) by virtue of (3.34). This implies (3.35).

 

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T. Adachi

Proof of (3.9). The proof is quite similar to the one of Proposition 3.1 in [A2]. We sketch the proof. By virtue of Proposition 3.10, there exists M > 0 such that the following estimate holds: Fε (t −2 |x − c S (t)| ≥ M)U S (t, 0)φ L 2 (X ) = O(t −1 ).

(3.37)

Here we set Jx (t) = Fε (ε ≤ t −2 |x − c S (t)| ≤ M + 2ε), J p (t) = Fε (t −1 | p − b S (t)| ≤ ε ), and put ψ(t) = Jx (t)J p (t)U S (t, 0)φ, where Fε (ε ≤ s ≤ M + 2ε) ≥ 0 for ε ≤ s ≤ 2ε. We will here estimate (ψ(t), (x − c S (t))2 ψ(t)) L 2 (X ) . The Heisenberg derivatives of x − c S (t), Jx (t) and J p (t) are computed as follows:   D H S (t) x − c S (t) = p − b S (t), D H S (t) (Jx (t)) = Fε (ε ≤ t −2 |x − c S (t)| ≤ M + 2ε)(x − c S (t))/|x − c S (t)| · {−2t −3 (x − c S (t)) + t −2 ( p − b S (t))} + O(t −4 ), D H S (t) (J p (t)) = O(t −1 )Fε /2 (t −1 | p − b S (t)| ≥ ε /2) + O(t −∞ ). Noting that Fε (ε ≤ s ≤ M + 2ε) ≥ 0 for ε ≤ s ≤ 2ε, and using (3.37), Theorem 3.12 and the above formulas, d(ψ(t), (x −c S (t))2 ψ(t)) L 2 (X ) /dt can be estimated as follows: d (ψ(t), (x − c S (t))2 ψ(t)) L 2 (X ) ≤ O(t 2 ) + Cε t 3 ψ(t)2L 2 (X ) . dt Hence we have ε2 t 4 ψ(t)2L 2 (X ) ≤ (ψ(t), (x − c S (t))2 ψ(t)) L 2 (X )
t ≤ O(t 3 ) + Cε τ 3 ψ(τ )2L 2 (X ) dτ. 1

Taking

ε

> 0 so small that

Cε

≤

ε2 ,

by virtue of Gronwall’s lemma, we obtain

ε2 t 4 ψ(t)2L 2 (X ) ≤ O(t 3 ). Combining this with Theorem 3.12 and (3.37), the theorem is obtained.

 

Proof of (3.10) and (3.11). The proof is quite similar to the one of Theorem 1.2 in [A2]: We put ψ(t) = Jx (t)U S (t, 0)φ with Jx (t) = Fε (t −2 |x − c S (t)| ≤ 2ε). We first claim |(ψ(t), (z − At + |E|t 2 /2)ψ(t)) L 2 (X ) | = O(t).

(3.38)

We here note that D H S (t) (z − At + |E|t 2 /2) = t E · (∇V )(x + c(t))/|E|. Hence, by assumption, we see that D H S (t) (z − At + |E|t 2 /2) = O(t −2ρ ) on the support of Jx (t). We write D H S (t) (Jx (t)) as I1 (t) + I2 (t), where I1 (t) = −2t −1 Fε (t −2 |x − c S (t)| ≤ 2ε)(t −2 |x − c S (t)|) − i∆(Jx (t))/2,

I2 (t) = t −2 Fε (t −2 |x − c S (t)| ≤ 2ε)(x − c S (t))/|x − c S (t)| · ( p − b S (t)).

Asymptotic Completeness for N -Body Quantum Systems

467

We here used 2c S (t) = tb S (t) by (3.5). Noting that Fε (t −2 |x − c S (t)| ≤ 2ε) = Fε (t −2 |x − c S (t)| ≤ 2ε)Fε/2 (t −2 |x − c S (t)| ≥ ε/2), we have |(ψ(t), (z − |E|t 2 /2)I1 (t)U S (t, 0)φ) L 2 (X ) | = O(1) by (3.9). Using Fε (|t −1 A − |E|| ≥ ε ) + Fε (|t −1 A − |E|| ≤ 2ε ) = Id, and noting i[(t −1 A − |E|)Fε (|t −1 A − |E|| ≥ ε ), Fε/2 (t −2 |x − c S (t)| ≥ ε/2)] = O(t −3 ), we see t (A − |E|t)Fε/2 (t −2 |x − c S (t)| ≥ ε/2)ψ(t) L 2 (X ) = O(t 3/2 ) by virtue of Proposition 3.9 and (3.9). Thus we have |(ψ(t), t (A − |E|t)I1 (t)U S (t, 0)φ) L 2 (X ) | = O(1) by (3.9) again. By the same argument as above, we see that I2 (t)Fε/2 (t −2 |x − c S (t)| ≥ ε/2)U S (t, 0)φ L 2 (X ) = O(t −3/2 ) by virtue of Theorem 3.12 and (3.9). Thus we have |(ψ(t), (z − At + |E|t 2 /2)I2 (t)U S (t, 0)φ) L 2 (X ) | = O(1). From these estimates, we see that   d   (ψ(t), (z − At + |E|t 2 /2)ψ(t)) L 2 (X )  = O(1),  dt  which implies (3.38). Since | p − b S (t)|ψ(t)2L 2 (X ) = 2(ψ(t), {H0S + |E|(z − At +

|E|t 2 /2)}ψ(t)) L 2 (X ) as seen before, we obtain

| p − b S (t)|ψ(t)2L 2 (X ) = O(t) by (3.36) and (3.38), which implies (3.10). Noting that the Heisenberg derivative of x −c S (t) is p −b S (t), we obtain the estimate    d  (U S (t, 0)∗ (x − c S (t))ψ(t))   dt by virtue of (3.9) and (3.10), which implies (3.11).

L 2 (X )

 

= O(t 1/2 )

468

T. Adachi

4. Proof of Theorem 1.1 In this section, we will prove Theorem 1.1. Throughout this section, we assume that are fulfilled. We first note that under (V )c,G (V )c,L and (V )c,G ¯ ¯ , |∂xβ Ic (t, x)| ≤ Cβ (t + x1/2 )−2(ρG +|β|) , |β| ≤ 1,

(4.1)

holds for t > 0, which is finer than (2.3). We introduce the time-dependent Hamiltonian HcG (t) as HcG (t) = Hc (t) + Ic (c(t)). ˜

(4.2)

UcG (t) denotes the propagator generated by HcG (t) such that UcG (0) = Id. We here note that Ic (c(t)) ˜ = Ic (t, c(t)) ˜

(4.3)

for t > 0, and that UcG (t) is represented as UcG (t) = Uc (t, 0)e−i

t

˜ )) dτ 0 Ic (c(τ

.

(4.4)

˜ ˜ Noticing D HcG (t) = 0, D HcG (t) ( pc − b(t)) = 0 and D HcG (t) (x − c(t)) ˜ = p − b(t), the following propagation property of UcG (t) can be proved as in the proof of Proposition 2.2. We omit the proof. (H c )

Lemma 4.1. The following estimate holds for φ ∈ D( p 2 + x 2 ) as t → ∞:    |x − c(t)|U ˜ = O(t). cG (t)φ 2 L (X )

(4.5)

By using this lemma and Proposition 2.2, we obtain the following. Proposition 4.2. The strong limit s-lim UcG (t)∗ U˜ c (t) t→∞

(4.6)

exists and is unitary on L 2 (X ). Proof. We have only to show the existence of lim UcG (t)∗ U˜ c (t)φ,

(4.7)

lim U˜ c (t)∗ UcG (t)φ

(4.8)

t→∞ t→∞

for φ ∈ D( p 2 + x 2 ). Using (4.3), we have d (UcG (t)∗ U˜ c (t)φ) = UcG (t)∗ i(Ic (t, c(t)) ˜ − Ic (t, x))U˜ c (t)φ. dt Since




1

˜ − Ic (t, x) = − Ic (t, c(t))

(∇ Ic )(t, sx + (1 − s)c(t)) ˜ · (x − c(t)) ˜ ds

0

and supx∈X |(∇ Ic )(t, x)| = O(t −2ρG −2 ) by (V )c,G ¯ , the existence of (4.7) can be proved by Proposition 2.2 and the Cook-Kuroda method, because −2ρG − 2 + 1 < −1. The existence of (4.8) can be proved quite similarly by virtue of (4.5).  

Asymptotic Completeness for N -Body Quantum Systems

469

Combining this with Theorem 2.1, we obtain the following, which is the key to the proof of Theorem 1.1: Corollary 4.3. The strong limit ΩcG = s-lim U (t, 0)∗ UcG (t)

(4.9)

t→∞

exists and is unitary on L 2 (X ). Since UcG (t, 0) = e−it H ⊗ (U¯ c (t, 0)e−i c

t

˜ )) dτ 0 Ic (c(τ

)

by (1.5), Theorem 1.1 can be proved in the same way as in [A3, AT1 and HMS2], by combining Corollary 4.3 and the following result of the asymptotic completeness for H c = −∆c /2 + V c (x c ), which is proved by Derezi´nski [D] (see also [DG1] and [Z]). So we omit the proofs: We introduce some notations. Suppose a ⊂ c. We define the cluster Hamiltonian Hac = −∆c /2 + V a on L 2 (X c ) and put c (t) = e−it Ha e−i Ua,D c

t

c 0 Ia ( pa u) du

acting on L 2 (X c ). We put X ac = X c  X a . Then we see that L 2 (X c ) is decomposed into L 2 (X a ) ⊗ L 2 (X ac ). Thus Hac is decomposed into Hac = H a ⊗ Id + Id ⊗ Tac on L 2 (X c ) = L 2 (X a ) ⊗ L 2 (X ac ), where Tac = −∆ac /2 and ∆ac is the Laplace-Beltrami operator on X ac . It follows from this that c (t) = e−it H ⊗ (e−it Ta e−i Ua,D a

c

t

c 0 Ia ( pa u) du

)

(4.10)

on L 2 (X c ) = L 2 (X a ) ⊗ L 2 (X ac ). Theorem 4.4. Assume that (V )c,L is fulfilled. Then the modified wave operators c

c (t)(P a ⊗ Id) Ωac,± = s-lim eit H Ua,D t→±∞

acting on L 2 (X c ), exist for all a ⊂ c, and are asymptotically complete  L 2 (X c ) = ⊕ Ran Ωac,± . a⊂c

5. Proof of Theorem 1.2 In this section, we prove Theorem 1.2.√Throughout this section, we assume that c = amin and that (V )c,L and (V )c,D,ρ with ( 3 − 1)/2 < ρ ≤ 1/2 are fulfilled. We first note ¯ that under (V )c,D,ρ , ¯ |∂xβ Ic (t, x)| ≤ Cβ (t + x1/2 )−(2ρ+|β|) , t > 0,

(5.1)

holds. Since the proof is quite similar to the one in Adachi-Tamura [AT2], we sketch it. We introduce the time-dependent Hamiltonians

470

T. Adachi

˜ ˜ − t b(t)), H˜ a D (t) = Ha (t) + Iac ( pa t) + Ic ( pa t + c(t) c ˜ Ha,1 (t) = Ha (t) + Ia ( pa t) + Ic (t, pa t + c(t) ˜ − t b(t)), HcSc (t) = HcSc + Ic (t, x + c(t)), ˜ Sc Ha,1 (t) = HaSc + Iac ( pa t) + Ic (t, pa t + c(t)) ˜

for a ⊂ c, where HaSc = −∆/2 + V a (x a ) acts on L 2 (X ). U˜ a D (t), Ua,1 (t), UcSc (t) Sc (t) denote the propagators generated by H ˜ a D (t), Ha,1 (t), HcSc (t) and H Sc (t), and Ua,1 a,1 Sc (T ) = Id. Since respectively, where U˜ a D (0) = Id, Ua,1 (T ) = Id, UcSc (T ) = Id and Ua,1 ˜ for a ⊂ c, U˜ a D (t) is explicitly represented by Ua (t, 0) pa Ua (t, 0)∗ = pa − b(t) U˜ a D (t) = Ua,D (t, 0)e−i

t

˜ ds 0 Ic ( pa s+c(s))

.

Then the following Avron-Herbst formula holds: Sc (t)T˜ (T )∗ . U˜ c (t) = T˜ (t)UcSc (t)T˜ (T )∗ , Ua,1 (t) = T˜ (t)Ua,1

(5.2)

By virtue of the relation (5.2), we have only to study the asymptotic behavior of UcSc (t). We now apply to UcSc (t)the result by Derezi´nski [D] on the asymptotic completeness for long-range N -body quantum systems without electric fields. √ with ( 3 − 1)/2 < ρ ≤ 1/2 are Theorem 5.1. Assume that (V )c,L and (V )c,D,ρ ¯ fulfilled. Then the modified wave operators Sc Sc Ωa,1 = s-lim UcSc (t)∗ Ua,1 (t)(P a ⊗ Id) t→∞

exist for all a ⊂ c, and are asymptotically complete  Sc ⊕ Ran Ωa,1 . L 2 (X ) = a⊂c

√

The condition 2ρ > 3 − 1 is essentially used to prove this theorem only. By virtue of the Avron-Herbst formula (5.2), the following corollary is obtained as an immediate consequence of this theorem. √ Corollary 5.2. Assume that (V )c,L and (V )c,D,ρ with ( 3 − 1)/2 < ρ ≤ 1/2 are ¯ fulfilled. Then the modified wave operators Ω˜ a,1 = s-lim U˜ c (t)∗ Ua,1 (t)(P a ⊗ Id) t→∞

exist for all a ⊂ c, and are asymptotically complete  ⊕ Ran Ω˜ a,1 . L 2 (X ) = a⊂c

˜ ˜ Let a ⊂ c. Since D H˜ a D (t) ( pa − b(t)) = D Ha,1 (t) ( pa − b(t)) = 0, we have the following propagation properties of U˜ a D (t) and Ua,1 (t).

Asymptotic Completeness for N -Body Quantum Systems

471

Lemma 5.3. The following estimates hold for φ ∈ D( p 2 + x 2 ) as t → ∞:   | pa − b(t)| ˜ U˜ a D (t)φ  2 = O(1), L (X )    | pa − b(t)|U ˜ a,1 (t)φ L 2 (X ) = O(1). Corollary 5.4. The following estimates hold for φ ∈ D( p 2 + x 2 ) as t → ∞:    Fε /2 (t −1 | pa − b(t)| ˜ ≥ ε1 /2)U˜ a D (t)φ  L 2 (X ) = O(t −1 ), 1    Fε /2 (t −1 | pa − b(t)| ˜ ≥ ε1 /2)Ua,1 (t)φ  L 2 (X ) = O(t −1 ). 1 By these estimates, we have the following. Proposition 5.5. The strong limit s-lim U˜ a D (t)∗ Ua,1 (t) t→∞

exists and is unitary on L 2 (X ). ˜ ≤ ε1 ). By Corollary 5.4, we have only to Proof. We put ηa (t) = Fε1 /2 (t −1 | pa − b(t)| prove the existence of the limits lim U˜ a D (t)∗ ηa (t)Ua,1 (t)φ,

t→∞

lim Ua,1 (t)∗ ηa (t)U˜ a D (t)φ

t→∞

for φ ∈ D( p 2 + x 2 ). Noting ˜ ˜ ˜ − t b(t))η ˜ − t b(t))η Ic ( pa t + c(t) a (t) = Ic (t, pa t + c(t) a (t), −2  −1 ˜ ˜ D Ha (t) (ηa (t)) = −t Fε /2 (t | pa − b(t)| ≤ ε1 )| pa − b(t)|, 1

we obtain the proposition by virtue of Lemma 5.3.

 

Combining Corollary 5.2 and Proposition 5.5 with Theorem 2.1, Theorem 1.2 can be obtained immediately. 6. Proof of Theorem 1.3 In this section, we prove Theorem 1.3. Throughout this section, we assume that c = amin and that (V )c,L and (V )c,D,ρ with 1/{2( j0 +1)} < ρ < 1/(2 j0 ) for some j0 ∈ N are ful¯ filled. The case where ρ = 1/(2 j0 ) can be included in the 1/{2( j0 + 1)} < ρ < 1/(2 j0 ) by making ρ slightly smaller than 1/(2 j0 ). Since the proof is quite similar to the one in Adachi-Tamura [AT2], we sketch it with minor modification. We construct an approximate solution of the Hamilton-Jacobi equation (∂t S)(t, ξ ) =

1 2 ξ − E(t) · (∇ξ S)(t, ξ ) + Ic (t, (∇ξ S)(t, ξ )) 2

(6.1)

˜ associated with H˜ c (t). Putting K (t, ξ ) = S(t, ξ + b(t)), (6.1) is translated into (∂t K )(t, ξ ) =

1 2 ˜ (ξ + b(t)) + Ic (t, (∇ξ K )(t, ξ )). 2

(6.2)

472

T. Adachi

Thus we will construct an approximate solution of (6.2). K 0 (t, ξ ) denotes the solution of 1 2 ˜ , K 0 (0, ξ ) = 0. (∂t K 0 )(t, ξ ) = (ξ + b(t)) 2 As mentioned in §1, K 0 (t, ξ ) is written by (1.9), and (1.10) holds. We further define K j (t, ξ ), 1 ≤ j ≤ j0 , for t ≥ T inductively as the solution of (∂t K j )(t, ξ ) =

1 2 ˜ (ξ + b(t)) + Ic (t, (∇ξ K j−1 )(t, ξ )), 2

K j (T, ξ ) = K j−1 (T, ξ ).

2 /2, we have ˜ Noting (∂t K 0 )(t, ξ ) = (ξ + b(t))
t Ic (τ, (∇ξ K j−1 )(τ, ξ )) dτ, t ≥ T K j (t, ξ ) = K 0 (t, ξ ) +

(6.3)

T

for 1 ≤ j ≤ j0 . We here note that β

sup |∂ξ (K j (t, ξ ) − K j−1 (t, ξ ))| = O(t 1−2 jρ )

ξ ∈X

(6.4)

holds for 1 ≤ j ≤ j0 by virtue of (5.1), which can be proved by the Faà di Bruno formula and induction in j. ˜ Putting S j (t, ξ ) = K j (t, ξ − b(t)), S j0 (t, ξ ) satisfies (∂t S j0 )(t, ξ ) =

1 2 ξ − E(t) · (∇ξ S j0 )(t, ξ ) + Ic (t, (∇ξ S j0 −1 )(t, ξ )). 2

(6.5)

We will write Ic (t, (∇ξ S j )(t, ξ )) as Ic, j (t, ξ ) below. We define the Hamiltonian Hˆ c (t) by Hˆ c (t) = Hc (t) + Ic, j0 −1 (t, p) for t ≥ T , whose definition is slightly different from the one in [AT2]. Uˆ c (t) denotes the propagator generated by Hˆ c (t) such that Uˆ c (T ) = Id. Lemma 6.1. The following estimates hold for φ ∈ D( p 2 + x 2 ) as t → ∞:   |x − (∇ξ S j −1 )(t, p)|U˜ c (t)φ  2 = O(t 1−2 j0 ρ ), 0 L (X )   |x − (∇ξ S j −1 )(t, p)|Uˆ c (t)φ  2 = O(t 1−2 j0 ρ ). 0 L (X )

(6.6) (6.7)

Proof. Let Φ(t) = x − (∇ξ S j0 )(t, p). We first note that D Hc (t) (Φ(t)) = p − (∂t ∇ξ S j0 )(t, p) − E(t) · ∇ξ2 S j0 (t, p) = −(∇ξ Ic, j0 −1 )(t, p) holds by (6.5). By virtue of this, we have D Hˆ c (t) (Φ(t)) = (∇ξ Ic, j0 −1 )(t, p) − (∇ξ Ic, j0 −1 )(t, p) = 0. Equation (6.7) is obtained by this and (6.4). We next note that H˜ c (t) = Hˆ c (t)+ Ic (t, x)− Ic, j0 −1 (t, p). We put
1 (∇ Ic )(t, θ x + (1 − θ )(∇ξ S j0 )(t, ξ )) dθ g(t, x, ξ ) = 0

Asymptotic Completeness for N -Body Quantum Systems

473

and r (t, x, ξ ) = divξ g(t, x, ξ ). Then it is known that Ic (t, x) − Ic, j0 (t, p) = g(t, x, p) · Φ(t) + r (t, x, p)

(6.8)

holds by a pseudodifferential calculus (see e.g. [DG1]). By virtue of this, we have i[Ic (t, x) − Ic, j0 (t, p), Φ(t)] = O(t −(2ρ+1) )Φ(t) + O(t −(2ρ+1) ). We here used (1.10), (5.1) and (6.4). On the other hand, Ic, j0 (t, p) − Ic, j0 −1 (t, p) = O(t −2( j0 +1)ρ ), i[Ic, j0 (t, p) − Ic, j0 −1 (t, p), Φ(t)] = O(t −2( j0 +1)ρ )

(6.9)

hold, where we used (1.10), (5.1) and (6.4) again. Hence D H˜ c (t) Φ(t) takes the form D H˜ c (t) Φ(t) = O(t −(2ρ+1) )Φ(t) + O(t −(2ρ+1) ) + O(t −2( j0 +1)ρ ). Since −(2ρ + 1) < −1 and −2( j0 + 1)ρ < −1 by assumption, we have  
t −(2ρ+1) ˜ ˜ s |Φ(s)|Uc (s)φ L 2 (X ) ds , |Φ(t)|Uc (t)φ L 2 (X ) ≤ C 1 + T

which implies |Φ(t)|U˜ c (t)φ L 2 (X ) = O(1) by virtue of Gronwall’s lemma. Thus (6.6) follows from this and (6.4).   Since g(t, x, p) = O(t −(2ρ+1) ) and r (t, x, p) = O(t −(2ρ+1) ), Ic (t, x) − Ic, j0 −1 (t, p) = O(t −(2ρ+1) )(x − (∇ξ S j0 −1 )(t, p)) + O(t −(2ρ+1) ) + O(t −2( j0 +1)ρ ) holds by virtue of (6.4), (6.8) and (6.9). By this and Lemma 6.1, the following proposition can be obtained immediately, because −(2ρ + 1) + (1 − 2 j0 ρ) = −2( j0 + 1)ρ < −1 and −(2ρ + 1) < −1 by assumption. Proposition 6.2. The strong limit s-lim Uˆ c (t)∗ U˜ c (t) t→∞

exists and is unitary on L 2 (X ). We would like to replace Uˆ c (t) by Uˇ c (t) = Uc (t, 0)e−i Uˇ c (t) = Uc (t, 0)e−i

t

t

0 Ic ((∇ξ K 0 )(τ, p)) dτ

, t ≥ 0,

T Ic ((∇ξ K j0 −1 )(τ, p)) dτ

, t ≥ T,

if j0 = 1, if j0 ≥ 2.

(6.10)

We note that Uˇ c (t) is the propagator generated by the time-dependent Hamiltonian Hˇ c (t) = Hc (t) + Ic ((∇ξ S j0 −1 )(t, p)). ˜ We here used Uc (t, 0) pUc (t, 0)∗ = p − b(t). We need the following lemma.

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T. Adachi

Lemma 6.3. The following estimates hold for φ ∈ D( p 2 + x 2 ) as t → ∞:   | p − b(t)| ˜ Uˆ c (t)φ  2 = O(1), L (X )   | p − b(t)| ˜ Uˇ c (t)φ  2 = O(1), L (X )   |(∇ξ S j −1 )(t, p) − c(t)| ˜ Uˆ c (t)φ  L 2 (X ) = O(t), 0   |(∇ξ S j −1 )(t, p) − c(t)| ˜ Uˇ c (t)φ  L 2 (X ) = O(t). 0

(6.11) (6.12) (6.13) (6.14)

˜ ˜ Proof. Equations (6.11) and (6.12) follow from D Hˆ c (t) ( p−b(t)) = D Hˇ c (t) ( p−b(t)) =0 immediately. We set Φ(t) = (∇ξ S j0 −1 )(t, p) − c(t). ˜ Noting D Hˆ c (t) (Φ(t)) = D Hˇ c (t) (Φ(t)) = D Hc (t) (Φ(t)), we calculate D Hc (t) (Φ(t)) with the convention (∇ξ Ic,−1 )(t, p) = 0 as follows: ˜ D Hc (t) (Φ(t)) = (∂t ∇ξ S j0 −1 )(t, p) + E(t) · (∇ξ2 S j0 −1 )(t, p) − b(t) ˜ + (∇ξ Ic, j0 −2 )(t, p). = p − b(t) We here used (6.5). Since (∇ξ Ic, j0 −2 )(t, p) = O(t −2ρ ), (6.13) and (6.14) follow from (6.11) and (6.12) immediately.   Corollary 6.4. The following estimates hold for φ ∈ D( p 2 + x 2 ) as t → ∞:    Fε /2 (t −2 |(∇ξ S j −1 )(t, p) − c(t)| ˜ ≥ ε1 /2)Uˆ c (t)φ  L 2 (X ) = O(t −1 ), 1 0    Fε /2 (t −2 |(∇ξ S j −1 )(t, p) − c(t)| ˜ ≥ ε1 /2)Uˇ c (t)φ  2 = O(t −1 ). 1

L (X )

0

(6.15) (6.16)

By these results, we have the following. Proposition 6.5. The strong limit s-lim Uˇ c (t)∗ Uˆ c (t) t→∞

exists and is unitary on L 2 (X ). Proof. We put η(t) = Fε1 /2 (t −2 |(∇ξ S j0 −1 )(t, p) − c(t)| ˜ ≤ ε1 ). By Corollary 6.4, we have only to prove the existence of the limits lim Uˇ c (t)∗ η(t)Uˆ c (t)φ,

t→∞

lim Uˆ c (t)∗ η(t)Uˇ c (t)φ

t→∞

˜ ≤ ε1 ) for φ ∈ D( p 2 + x 2 ). Putting a(t, ξ ) = Fε1 /2 (t −2 |(∇ξ S j0 −1 )(t, ξ ) − c(t)| ((∇ξ S j0 −1 )(t, ξ ) − c(t))/|(∇ ˜ S )(t, ξ ) − c(t)|, ˜ D (η(t)) is calculated as ξ j0 −1 Hc (t) D Hc (t) (η(t)) = a(t, p) · {−2t −3 ((∇ξ S j0 −1 )(t, p) − c(t)) ˜ −2 2 ˜ + t ((∂t ∇ξ S j0 −1 )(t, p) + E(t) · (∇ξ S j0 −1 )(t, p) − b(t))} ˜ + (∇ξ Ic, j0 −2 )(t, p))}, = a(t, p) · {−2t −3 ((∇ξ S j0 −1 )(t, p) − c(t)) ˜ + t −2 ( p − b(t) where we used (6.5). Therefore the proposition can be obtained by virtue of Lemma 6.3.  

Asymptotic Completeness for N -Body Quantum Systems

475

Combining Propositions 6.2 and 6.5 with Theorem 2.1, Theorem 1.3 can be obtained immediately. Remark 6.6. When N = 2, under the same assumptions as in this paper, the existence of the modified wave operators has been already obtained in Hasegawa [Ha] (in the case where 1/4 < ρ ≤ 1/2), Shimizu [S] (also in the case where 1/4 < ρ ≤ 1/2) and Kimura [K] (in the case where 0 < ρ ≤ 1/2). These papers propose the definition of the modified wave operators in the position representation, which was initiated by Yafaev [Y] under E(t) ≡ 0 (see also [DG2]): ± ˜ p W0,ψ = s-lim U (t, 0)∗ Uψ (t), Uψ (t) = eiψ(t,x) e−i c(t)· D(t)F , t→±∞

(6.17)

where ψ(t, x) is an approximate solution of the Hamilton-Jacobi equation in the position representation −(∂t ψ)(t, x) =

1 ((∇ψ)(t, x))2 − E(t) · x + Ic (t, x), 2

(6.18)

D(t) is the dilation operator given by (D(t)φ)(x) = (it)−dim X/2 φ(t −1 x), and F is the Fourier transform on L 2 (X ). We emphasize that by virtue of the Avron-Herbst formula, the free propagator U0 (t, 0) generated by H0 (t) is represented as 2 ˜ ˜ · x + (x − c(t)) , ˜ + b(t) U0 (t, 0) = Uψ0 (t)M(t), ψ0 (t, x) = −a(t) 2t

for t = 0, where M(t) is the multiplication by ei x /(2t) , and ψ0 (t, x) is a solution of (6.18) with Ic (t, x) ≡ 0. Noting that s-limt→±∞ M(t) = Id, we see that 2

± W0,ψ = s-lim U (t, 0)∗ J (t)U0 (t, 0), t→±∞

J (t) = ei(ψ(t,x)−ψ0 (t,x)) .

We emphasize that this modifier J (t) is the multiplication. In [S], the relationship t ± ± ˜ )) dτ has been and W0,D = s-limt→±∞ U (t, 0)∗ U0 (t, 0)e−i 0 V ( pτ +c(τ between W0,ψ 1 studied in the case where 1/4 < ρ ≤ 1/2, where ψ1 (t, x) = cvξ (x · ξ − S1 (t, ξ )) for sufficiently large |t|, which is an approximate solution of (6.18) (cf. [DG2]). However, the asymptotic completeness of modified wave operators has not been discussed until now (when E(t) ≡ E). Acknowledgement. The author is partially supported by the Grant-in-Aid for Young Scientists of MEXT #17740078. He is also grateful to the referee for valuable comments.

References [A1] [A2] [A3] [AT1] [AT2]

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Avron, J.E., Herbst, I.W.: Spectral and scattering theory of Schrödinger operators related to the Stark effect. Commun. Math. Phys. 52, 239–254 (1977) Brezis, H.: Analyse fonctionnelle, Théorie et applications. Paris: Masson, 1983 Cycon, H., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators with Application to Quantum Mechanics and Global Geometry. Berlin-Heidelberg-New York: Springer-Verlag, 1987 Derezi´nski, J.: Asymptotic completeness of long-range N -body quantum systems. Ann. of Math. 138, 427–476 (1993) Derezi´nski, J., Gérard, C.: Scattering theory of classical and quantum mechanical N -particle systems. Berlin-Heidelberg-New York: Springer-Verlag, 1997 Derezi´nski, J., Gérard, C.: Long-range scattering in the position representation. J. Math. Phys. 38, 3925–3942 (1997) Enss, V., Veseli´c, K.: Bound states and propagating states for time-dependent hamiltonians. Ann. Inst. H. Poincaré Sect. A (N.S.) 39, 159–191 (1983) Gérard, C.: Sharp propagation estimates for N -particle systems. Duke Math. J. 67, 483–515 (1992) Gérard, C., Łaba, I.: Multiparticle Quantum Scattering in Constant Magnetic Fields. Providence. RI: Amer. Math. Soc., 2002 Graf, G.M.: Phase space analysis of the charge transfer model. Helv. Phys. Acta 64, 107–138 (1990) Graf, G.M.: Asymptotic completeness for N -body short-range quantum systems: a new proof. Commun. Math. Phys. 132, 73–101 (1990) Graf, G.M.: A remark on long-range Stark scattering. Helv. Phys. Acta 64, 1167–1174 (1991) Hasegawa, T.: On Applications of the Avron-Herbst Formula. Master Thesis, Kobe University, 2005 (in Japanese) Helffer, B., Sjöstrand, J.: Equation de Schrödinger avec champ magnétique et équation de Harper. In: Lecture Notes in Physics 345, Berlin-Heidelberg-New York: Springer-Verlag, 1989, 118–197 Herbst, I., Møller, J.S., Skibsted, E.: Spectral analysis of N -body Stark Hamiltonians. Commun. Math. Phys. 174, 261–294 (1995) Herbst, I., Møller, J.S., Skibsted, E.: Asymptotic completeness for N -body Stark Hamiltonians. Commun. Math. Phys. 174, 509–535 (1996) Howland, J.S.: Scattering theory for time-dependent Hamiltonians. Math. Ann. 207, 315–335 (1974) Howland, J.S.: Scattering theory for Hamiltonians periodic in time. Indiana Univ. Math. J. 28, 471–494 (1979) Hunziker, W.: On the space-time behavior of Schrödinger wavefunctions. J. Math. Phys. 7, 300–304 (1966) Jensen, A., Ozawa, T.: Existence and non-existence results for wave operators for perturbations of the Laplacian. Rev. Math. Phys. 5, 601–629 (1993) Jensen, A., Yajima, K.: On the long-range scattering for Stark Hamiltonians. J. Reine Angew. Math. 420, 179–193 (1991) Kimura, T.: Long-range quantum scattering theory in a time-periodic electric field. Master Thesis, Kobe University, 2006 (in Japanese) Kitada, H., Yajima, K.: Scattering theory for time-dependent long-range potentials. Duke Math. J. 49, 341–376 (1982) Møller, J.S.: Two-body short-range systems in a time-periodic electric field. Duke Math. J. 105, 135–166 (2000) Møller, J.S., Skibsted, E.: Spectral theory of time-periodic many-body systems. Adv. Math. 188, 137–221 (2004) Mourre, E.: Absence of singular continuous spectrum for certain self-adjoint operators. Commun. Math. Phys. 78, 391–408 (1981) Nakamura, S.: Asymptotic completeness for three-body Schrödinger equations with time-periodic potentials. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 33, 379–402 (1986) Perry, P.A.: Scattering Theory by the Enss Method. Math. Rep. Vol. 1, Chur-London-ParisUtrecht-New York: Harwood Academic, 1983 Radin, C., Simon, B.: Invariant Domains for the Time-Dependent Schrödinger Equation. J. Differ. Eqs. 29, 289–296 (1978) Reed, M., Simon, B.: Methods of Modern Mathematical Physics I–IV, New York: Academic Press, 1972, 1975, 1977, 1978 Shimizu, Y.: Long-Range Scattering Theory for Schrödinger Operators in a Time-Periodic Electric Field. Master Thesis, Kobe University, 2006 Sigal, I.M., Soffer, A.: The N -particle scattering problem: asymptotic completeness for shortrange systems. Ann. of Math. 125, 35–108 (1987) Skibsted, E.: Propagation estimates for N -body Schroedinger Operators. Commun. Math. Phys. 142, 67–98 (1991)

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Communicated by B. Simon

Commun. Math. Phys. 275, 479–489 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0307-2

Communications in

Mathematical Physics

Stability of Relativistic Matter with Magnetic Fields for Nuclear Charges up to the Critical Value
Rupert L. Frank1 , Elliott H. Lieb2 , Robert Seiringer3 1 Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden.

E-mail: [email protected]

2 Departments of Mathematics and Physics, Princeton University, P. O. Box 708,

Princeton, NJ 08544, USA. E-mail: [email protected]

3 Department of Physics, Princeton University, P. O. Box 708, Princeton, NJ 08544, USA.

E-mail: [email protected] Received: 20 October 2006 / Accepted: 20 March 2007 Published online: 31 July 2007 – © R.L. Frank, E.H. Lieb, R. Seiringer 2007

Abstract: We give a proof of stability of relativistic matter with magnetic fields all the way up to the critical value of the nuclear charge Z α = 2/π . 1. Introduction We shall give a proof of the ‘stability of relativistic matter’ that goes further than previous proofs by permitting the inclusion of magnetic fields for values of the nuclear charge Z all the way up to Z α = 2/π , which is the well known critical value in the absence of a field. (The dimensionless number α = e2 /
c is the ‘fine-structure constant’ and equals 1/137.036 in nature.) More precisely, we shall show how to modify the earlier proof of Theorem 2 in [LY] so that an arbitrary magnetic field can be included. Reference will freely be made to items in the [LY] paper. The quantum mechanical Hamiltonian used here and in [LY], as well as the definition of stability of matter, will be given in the next section. For a detailed overview of this topic, we refer to [L1, L2]. For the present we note that stability requires a bound on α in two ways. One is the requirement, for any number of electrons, that Z α ≤ 2/π . In fact, if Z α > 2/π the Hamiltonian is not bounded below even for a single electron. The other requirement is a bound on α itself, α ≤ αc , even for arbitrarily small Z > 0, which comes into play when the number of particles is sufficiently large. It is known that αc ≤ 128/15π ; see [LY, Thm. 3] and also [DL]. For values of Z α strictly smaller than the critical value 2/π , it has been shown that stability holds with a magnetic field included. This is the content of Theorem 1 in [LY], in which the critical value of αc goes to zero as Z α approaches 2/π , however. (The result in [LY, Theorem 1] does not explicitly include a magnetic field, but the fact that the proof can easily be modified was noted in [LLoSo].) A similar result, by a different method, was proved in [LLoSi].  This paper may be reproduced, in its entirety, for non-commercial purposes.

480

R. L. Frank, E. H. Lieb, R. Seiringer

The more refined Theorem 2 in [LY] gives stability for the ‘natural’ value Z α ≤ 2/π and all α ≤ 1/94. While the true value of αc is probably closer to 1, the value 1/94 > 1/137 is sufficient for physics. The problem with the proof of [LY, Theorem 2] is that it does not allow for the inclusion of magnetic fields. Specifically, Theorems 9–11 have to be substantially modified, and doing so was an open problem for many years. This will be accomplished here at the price of decreasing αc from 1/94 to 1/133. Fortunately, this is still larger than the physical value 1/137! In a closely related paper [FLSe] we also show how to achieve a proof of stability for all Z α ≤ 2/π with an arbitrary magnetic field, but the value of αc there is very much smaller than the value obtained here. In particular, the physical value of α = 1/137 is not covered by the result in [FLSe]. The focus of [FLSe] is much broader than ‘stability of matter’, however. It is concerned with a general connection between Sobolev and Lieb-Thirring type inequalities, and includes as a special case Theorem 4.5 of this paper. The proof of the general result in [FLSe] is much more involved than the one of the special case presented here, and yields a worse bound on the relevant constant. 2. Definitions and Main Theorem We consider N electrons of mass m ≥ 0 with q spin states (q = 2 for real electrons) and K fixed nuclei with (distinct) coordinates R1 , . . . , R K ∈ R3 and charges Z 1 , . . . , Z K > 0. The electrons interact with an external, spatially dependent magnetic field B(x), which is given in terms of the magnetic vector potential A(x) by B = curl A. A pseudo-relativistic description of the corresponding quantum-mechanical system is given by the Hamiltonian H N ,K :=

 N 
( p j + A(x j ))2 + m 2 − m + αVN ,K (x1 , . . . , x N ; R1 , . . . , R K ) . j=1

(2.1) The Pauli exclusion principle for fermions dictates that H N ,K acts on functions in the anti-symmetric N -fold tensor product ∧ N L 2 (R3 ; Cq ). We use units in which
= c = 1, α > 0 is the fine structure constant, and


VN ,K (x1 , . . . , x N ; R1 , . . . R K ) :=

|xi − x j |−1 −

1≤i< j≤N

+




N
K


Z k |x j − Rk |−1

j=1 k=1

Z k Z l |Rk − Rl |−1

(2.2)

1≤k
is the Coulomb potential (electron-electron, electron-nuclei, nuclei-nuclei, respectively). In this model there is no interaction√of the electron spin with the magnetic field. Note that we absorb √ the electron charge α into the vector potential A, i.e., we write A(x) instead of α A(x) in (2.1). Since A is arbitrary and our bounds are independent of A, this does not affect our results. Stability of matter means that H N ,K is bounded from below by a constant times (N + K ), independently of the positions Rk of the nuclei and of A. For a thorough discussion see [L1, L2]. By scaling all spatial coordinates it is easy to see that either inf Rk ,A (inf spec H N ,K ) ≥ −m N or = −∞. We shall prove the following.

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Theorem 2.1 (Stability of relativistic matter with magnetic fields). For qα ≤ 1/66.5 and α Z k ≤ 2/π for all k, H N ,K ≥ −m N for all N , K , R1 , . . . , R K and A. For electrons q = 2 and hence our proof works up to α=

1 1 > . 133 137

The rest of this paper contains the proof of Theorem 2.1, but let us first state an obvious fact. Corollary 2.2. As a multiplication operator on ∧ N L 2 (R3 ; Cq ), VN ,K (x1 , . . . , x N ; R1 , . . . R K ) ≥ − max{66.5 q, π Z k /2}

N


| p j + A(x j )| (2.3)

j=1

for all A. This, of course, is just a rewording of Theorem 2.1, but the point is that it provides a lower bound for the Coulomb potential of interacting particles in terms of a onebody operator | p + A(x)|. This operator is dominated by the nonrelativistic operator | p + A(x)|2 and, therefore, (2.3) is useful in certain nonrelativistic problems. For example, an inequality of this type was used in [LLoSo] to prove stability of matter with the Pauli operator | p + A(x)|2 + σ · B(x) in place of | p + A(x)|2 . It was also used in [LSiSo] to control the no-pair Brown-Ravenhall relativistic model. An examination of the proof of Theorem 2 in [LY] shows that there are two places that do not permit the inclusion of a magnetic vector potential A. These are Theorem 9 (Localization of kinetic energy – general form) and Theorem 11 (Lower bound to the short-range energy in a ball). Our Theorem 3.1 is precisely the extension of Theorem 9 to the magnetic case. It may be regarded as a diamagnetic inequality on the localization error. It implies that Theorem 10 in [LY] holds also in the magnetic case, without change except for replacing | p| by | p + A|; see Theorem 3.2 below. A substitute for Theorem 11 in [LY] will be given in Theorem 4.5 below. It is based on the observation that an estimate on eigenvalue sums of a non-magnetic operator with discrete spectrum implies a similar estimate (with a modified constant) for the corresponding magnetic operator. This is not completely obvious, since there is no diamagnetic inequality for sums of eigenvalues. (In fact, a conjectured diamagnetic inequality actually fails for fermions on a lattice and leads to the ‘flux phase’ [L3].) It is for the different constants in Theorem 11 in [LY] and in our Theorem 4.5 that our bound on αc becomes worse than the one in [LY]. As should be clear from the above discussion, our main tool will be a diamagnetic inequality for single functions. The one we use is the diamagnetic inequality for the heat kernel. In the relativistic case it states that for any A ∈ L 2loc (R3 ; R3 ) and u ∈ L 2 (R3 ) one has       exp(−t| p + A|)u (x) ≤ exp(−t| p|)|u| (x), x ∈ R3 . (2.4)

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This follows with the help of the subordination formula  ∞ dt 2 −|ξ | e = e−t−|ξ | /(4t) √ πt 0 from the ‘usual’ (nonrelativistic) diamagnetic inequality for the semigroup exp(−t| p + A|2 ); see, e.g., [S3]. The heat kernel is not prominent in [LY], and our reformulation of some of the key estimates in [LY] in terms of the heat kernel is the principal novel feature of this paper. 3. Localization of the Kinetic Energy with Magnetic Fields 3.1. Relativistic IMS formula. In this subsection we establish the analogue of Theorem 9 in [LY] in the general case A = 0. First, recall that the IMS formula in the nonrelativistic case says that for any u and A,  R3

|( p + A)u|2 d x =



n 
j=0

R3

|( p + A)(χ j u)|2 d x −

R3

n


|∇χ j |2 |u|2 d x,

j=0

whenever χ j are real functions with nj=1 χ 2j ≡ 1. In this case the localization error n 2 j=0 |∇χ j | is local and independent of A. The analogue in the relativistic case is the following special case of [FLSe, Lemma B.1]. For the sake of completeness, we include its proof here. Theorem 3.1 (Localization of kinetic energy – general form). Let A ∈ L 2loc (R3 ; R3 ). If χ0 , . . . , χn are real Lipschitz continuous functions on R3 satisfying nj=0 χ 2j ≡ 1, then one has (u, | p + A|u) =

n


(χ j u, | p + A|χ j u) − (u, L A u) .

(3.1)

j=0

Here L A is a bounded operator with integral kernel L A (x, y) := k A (x, y)

n
(χ j (x) − χ j (y))2 , j=0

where k A (x, y) := limt↑0 t −1 exp(−t| p + A|)(x, y) for a.e. x, y ∈ R3 and |k A (x, y)| ≤

1 . − y|4

2π 2 |x

(3.2)

Note that (3.2) says that n
1 |L A (x, y)| ≤ L(x, y) := (χ j (x) − χ j (y))2 . 2π 2 |x − y|4

(3.3)

j=0

Here, L(x, y) is the same as in [LY, Eq. (3.7)]. Therefore, (3.2) is a diamagnetic inequality for the localization error.

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Proof. We write k A (x, y, t) := exp(−t| p + A|)(x, y) for the heat kernel and find n
(χ j u, (1 − exp(−t| p + A|))χ j u) = (u, (1 − exp(−t| p + A|))u) j=0

1
+ 2 n

 k A (x, y, t)(χ j (x) − χ j (y))2 u(x)u(y) d x d y.

j=0

(This is proved simply by writing out both sides in terms of k A (x, y, t) and using χ 2j ≡ n 1.) Now we divide by t and let t → 0. The left side converges to j=0 (χ j u, | p+ A|χ j u). Similarly, the first term on the right side divided by t converges to (u, | p + A|u). Hence the last term divided by t converges to some limit (u, L A u). The diamagnetic inequality (2.4) says that t |k A (x, y, t)| ≤ exp(−t| p|)(x, y) =  2 π 2 |x − y|2 + t 2 (see [LLo, Eq. 7.11(9)]). This implies, in particular, that L A is a bounded operator. Now it is easy to check that L A is an integral operator and that the absolute value of its kernel is bounded pointwise by the one of L in (3.3).

3.2. Localization of the kinetic energy. In this subsection we will bound the localization error L A by a potential energy correction and an additive constant. This is the extension of Theorem 10 in [LY] to the case A = 0. It is important that both error terms in our bound can be chosen independently of A. First we need to introduce some notation. We write B R := {x : |x| < R} for the ball of radius R and χB R for its characteristic function. If R = 1, we omit the index in the notation. We fix a constant 0 < σ < 1 and Lipschitz continuous functions χ0 , χ1 with χ02 + χ12 ≡ 1 such that supp χ1 ⊂ B1−σ . With these we define L as in (3.3) with n = 1. We decompose L in a short-range part L 0 and a long-range part L 1 given by the kernels L 1 (x, y) := L(x, y)χB (x)χB (y)χBσ (x − y),

L 0 (x, y) := L(x, y) − L 1 (x, y). (3.4)

Define

1 0 2 Tr L 2 and, for an arbitrary positive function h on B,   −1 1 −1 θ (x) := h (x) L (x, y)h(y) dy = h (x)χB (x)  :=

B

(3.5)

|y|<1, |x−y|<σ

L(x, y)h(y) dy .

Finally, for ε > 0 we define the function Uε∗ := εχB1−σ + θ and note that Uε∗ is supported in B.

(3.6)

484

R. L. Frank, E. H. Lieb, R. Seiringer

Theorem 3.2 (Localization of kinetic energy – explicit bound in the one-center case). For any ε > 0 and any non-negative trace-class operator γ one has Tr γ | p + A| ≥

1


Tr χ j γ χ j (| p + A| − Uε∗ ) − ε−1 γ .

(3.7)

j=0

For A = 0 this is exactly Theorem 10 in [LY]. As explained there, Uε∗ is a potential energy correction with only slightly larger support than χ1 . The last term in (3.7) is due to the long range nature of | p + A|. It depends on γ through its norm γ  but not through its trace. We emphasize again that both error terms in the inequality (3.7) are independent of A. Proof. The localization formula (3.1) yields Tr γ | p + A| =

1


Tr χ j γ χ j | p + A| − Tr γ L A ,

j=0

so we only have to find an upper bound for Tr γ L A . We decompose L A = L 0A + L 1A in the manner of (3.4) and, following the proof of Theorem 10 in [LY] word by word, we obtain

2 Tr γ L 1A ≤ Tr γ θ A . Tr γ L 0A ≤ ε Tr γ χB1−σ + (2ε)−1 γ  Tr L 0A , Here θ A (x) := 0 if x ∈ B and, if x ∈ B, θ A (x) := h

−1

(x)

 B

|L 1A (x, y)|h(y) dy.

 2 The estimate |L A (x, y)| ≤ L(x, y) from Theorem 3.1 implies that Tr L 0A ≤ 2 and that θ A ≤ θ . This leads to the stated lower bound.

4. Bounds on Eigenvalues in Balls So far we have considered | p + A| and its heat kernel. Now we address | p + A|−2/(π |x|) and its heat kernel. First of all, let us recall Kato’s inequality [Ka, Eq. (V.5.33)] (u, | p|u) ≥ (2/π )(u, |x|−1 u) .

(4.1)

(See also [H, W, KPS].) Now let  ⊂ R3 be an open set (we shall be interested in the case where  is a ball) and consider the quadratic form given by Q  (u) = (u, (| p| − 2/π |x|)u), restricted to those functions u ∈ L 2 (R3 ) that satisfy u = 0 on  c , the complement of . Of course, we also require u to be in the quadratic form domain of | p| − 2/π |x|. The quadratic form Q  is non-negative by (4.1) and it is closed (because the form | p| − 2/π |x| is closed on L 2 (R3 ) and limits of functions that are zero on  c are zero on  c ). From this it follows that there is a self-adjoint operator H on some domain in L 2 () such that Q  (u) = (u, H u). With this operator, we can define the ‘heat kernel’ exp (−t H ) on L 2 () and its trace. (The fact that the trace is finite when the volume of  is finite follows from subsequent considerations.)

Stability of Relativistic Matter

485

Similarly, for a magnetic vector potential A ∈ L 2loc (R3 ; R3 ), we define the operator in L 2 () using the quadratic form (u, (| p + A| − 2/π |x|)u). Note that (2.4) implies that HA

(u, | p + A|u) ≥ (|u|, | p||u|) .

(4.2)

This, together with (4.1), shows that (u, (| p + A| − 2/π |x|)u) is non-negative. Lemma 4.1 (Heat kernel diamagnetic inequality). Let ⊂ R3 and let A ∈ L 2loc (R3 ; R3 ). Then, for any t > 0, 

Tr L 2 () exp −t HA ≤ Tr L 2 () exp (−t H ) .

(4.3)

Proof. For n = 0, 1, 2, . . . , let h n := | p| − 2/(π |x|) + nχ c in L 2 (R3 ), where χ c denotes the characteristic function of the complement of . Similarly, let h nA := | p + A| − 2/(π |x|) + nχ c . The diamagnetic inequality (2.4) and standard approximation arguments using Trotter’s product formula imply that, for any u ∈ L 2 (R3 ),       exp(−th A )u (x) ≤ exp(−th n )|u| (x). n

(See [FLSe, Sect. 6.2] for details of the argument.) By the monotone convergence theorem [S1, Thm. 4.1], exp(−th n ) converges strongly to exp(−t H ) on the subspace L 2 (), and similarly for h nA . It follows that, for any u ∈ L 2 (),       exp(−t H A )u (x) ≤ exp(−t H )|u| (x).  Theorem 2.13 in [S3] yields the inequality  exp(−t HA )2 ≤  exp(−t H )2 for the Hilbert-Schmidt norm, and hence  exp(−2t HA )1 ≤  exp(−2t H )1 for the trace norm by the semigroup property. This holds for all t > 0, and hence proves (4.3).

We use the notation (x)− = max{0, −x} for the negative part of x ∈ R in the following. Lemma 4.2. Assume that there is constant M > 0 such that Tr L 2 () (H − )− ≤ M 4

(4.4)

 6e3 Tr L 2 () HA − ≤ 3 M 4 − 4

(4.5)

for all ≥ 0. Then

for all ≥ 0. We note the the numerical factor in (4.5) equals 6(e/4)3 ≈ 1.883. This factor is the price we have to pay, using our methods, to include an arbitrary magnetic field. It is the reason of the decrease of αc from 1/94 to 1/133.

486

R. L. Frank, E. H. Lieb, R. Seiringer

Proof. Since (x)− ≤ e−x−1 , we have

 

et Tr L 2 () exp −t HA Tr L 2 () HA − ≤ − te for any t > 0. Using the diamagnetic inequality (4.3), 

Tr L 2 () exp −t HA ≤ Tr L 2 () exp (−t H ). Moreover, integrating by parts twice, e−t x = t 2  Tr L 2 () exp (−t H ) = t 2

∞

0

∞ 0

e−tλ (x − λ)− dλ, and hence

e−tλ Tr L 2 () (H − λ)− dλ.

Using the assumption (4.4), we thus obtain

Tr L 2 ()

HA

 ∞  tet et M − ≤ e−tλ λ4 dλ = 24 4 M. − e t e 0

To minimize the right side, the optimal choice of t is t = 4/ . This yields (4.5).



In [LY, Thm. 11] it is shown that (4.4) holds for  = B R a ball of radius R centered at the origin. More precisely, the following proposition holds. Proposition 4.3. For any R > 0 and ≥ 0,   Tr L 2 (B R ) HB R − − ≤ 4.4827 R 3 4 . Proposition 4.3 follows from Theorem 11 in [LY] by choosing χ to be the characteristic function of the ball B R , q = 1 and γ to be the projection onto the negative spectral subspace of HB R − . Remark 4.4. It is illustrative to compare Proposition 4.3 with the Berezin-Li-Yau [B, LiY] type bound Tr L 2 () (| p| − )− ≤

  1 1 4 || . (|ξ | − )− d x dξ = (2π )3 R3  24π 2

(4.6)

(This can be proved in the same way as [LLo, Thm. 12.3].) The right side of (4.6) is the semi-classical phase-space integral. The operator | p| is defined as H above, but without the Hardy-term 2/(π |x|). If the Hardy term were added, the phase-space integral would diverge (provided  contains the origin), but Proposition 4.3 says that a bound of the form (4.6) still holds. (An examination of the proof in [LY] shows that Proposition 4.3 actually holds for any open set  of finite measure.) Combining Lemma 4.2 and Proposition 4.3 we obtain the following theorem, which replaces [LY, Thm. 11] in the magnetic case.

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487

Theorem 4.5 (Lower bound on the short-range energy in a ball). Let C > 0 and R > 0 and let HCAR := | p + A| −

C 2 − π |x| R

be defined on L 2 (R3 ) as a quadratic form. Let 0 ≤ γ ≤ q be a density matrix (i.e., a positive trace-class operator) and let χ by any bounded function with support in B R . Then Tr χ γ χ HCAR ≥ −8.4411

qC 4 χ 2∞ . R

(4.7)

As compared with [LY, Thm. 11], the constant has been multiplied by 6(e/4)3 , and χ 2∞ appears instead of |B R |−1 χ 22 . Proof. Note that

 Tr χγ χ HCAR = Tr χ γ χ HBAR − C/R ≥ −χ γ χ ∞ Tr L 2 (B R ) (HBAR − C/R)− .

The assertion follows from Lemma 4.2 and Proposition 4.3, observing that χ γ χ ∞ ≤ qχ 2∞ .

5. Proof of Theorem 2.1 We assume that the reader is familiar with the proof of Theorem 2 in [LY]. We shall only emphasize changes in their argument. The main idea is to replace Theorems 10 and 11 in [LY] by our Theorems 3.2 and 4.5, respectively.

There are some immediate simplifications. First, in view of the simple inequality | p|2 + m 2 ≥ | p| it is enough to prove Theorem 2.1 for m = 0. Moreover, by the convexity argument of [DL] it suffices to treat the case Z 1 = . . . = Z K =: z and αz = 2/π . So henceforth we assume m = 0, Z 1 = . . . = Z K = z and αz = 2/π . Let Dk := min{|Rk − Rl | : l = k} and define the Voronoi cell k := {x ∈ R3 : |x − Rk | < |x − Rl | for all l = k}. Fix 0 < λ < 1 and define a function W := G + F in each Voronoi cell by ˜ F(x) := Dk−1 F(|x − Rk |/Dk ),

G(x) := z|x − Rk |−1 , where ˜ F(t) :=



x ∈ k ,

−1 (1 − t 2 )−1 if t ≤ λ, 2√ ( 2z + 21 )t −1 if t > λ.

By the electrostatic inequality in [LY, Sect. III, Step A] our Theorem 2.1 will follow if we can prove that Tr γ (| p + A| − αW ) ≥ −

K z 2 α
−1 Dk 8

(5.1)

k=1

for some 0 < λ < 1 and all density matrices γ with 0 ≤ γ ≤ q). Note that (5.1) is an inequality for a one-particle operator.

488

R. L. Frank, E. H. Lieb, R. Seiringer

For fixed 0 < σ < 1/3 we choose χ , h as in (3.22), (3.24) in [LY]. Note that supp χ ⊂ B1−σ . Let χk (x) := χ (|x − Rk |/Dk ),

h k (x) := h(|x − Rk |/Dk ).

After scaling and translation, Proposition 3.2 yields that for any 0 ≤ γ ≤ q, ∗ − αW ) Tr γ (| p + A| − αW ) ≥ Tr χ1 γ χ1 (| p + A| − U1,ε ∗ + Tr(1 − χ12 )1/2 γ (1 − χ12 )1/2 (| p + A| − U1,ε − αW )

−ε−1 q/D1 .

(5.2)

∗ (x) := D −1 U ∗ ((x − R )/D ) and , U ∗ were defined in (3.5), (3.6). (Note Here, U1,ε 1 1 ε ε 1 that our  is denoted by 1 in [LY]). Recall that Uε∗ and  are independent of A. We turn to the first term on the right side of (5.2). Let C be a constant such that

 ˜ C ≥ (1 − σ ) α F(|x|) + Uε∗ (x) for |x| ≤ 1 − σ. (5.3)

Note that χ1 is supported on a ball of radius (1 − σ )D1 centered at R1 . Hence αW (x) = ˜ − R1 |/D1 ) on the support of χ1 and we can apply (2/π )|x − R1 |−1 + D1−1 F(|x Theorem 4.5 to obtain the lower bound   2 C ∗ − Tr χ1 γ χ1 (|D − A| − U1,ε − αW ) ≥ Tr χ1 γ χ1 | p + A|− π |x − R1 | (1−σ )D1 ≥ −8.4411

qC 4 . (1 − σ )D1

(5.4)

We used also that |χ1 | ≤ 1. Inserting (5.4) into (5.2) we find Tr γ (| p + A| − αW ) ≥ −q D1−1 A˜ ∗ − αW ) + Tr(1 − χ12 )1/2 γ (1 − χ12 )1/2 (| p + A| − U1,ε

with C4  + 8.4411 . A˜ := ε (1 − σ ) This estimate is exactly of the form (3.26) in [LY], except for the value of the constant in A˜ (which is called A in [LY]). Starting from there one can continue along the lines of their proof. We need only note that in order to bound the last term in (3.29) in [LY] one uses the Daubechies inequality [D], which holds with the same constant in the presence of a magnetic field. (This is explained, for instance, in [LLoSi, Sect. 5].) We conclude that stability holds as long as αq( A˜ + J ) ≤ where, as in [LY, Eq. (3.31)],  J := 0.0258

|x|≥1−3σ



1 , 2π 2

2 ˜ + α F(|x|) + Uε∗ (x) π |x|

(5.5) 4 d x.

This completes our proof of Theorem 2.1, except for our bound on the critical α, which we justify now.

Stability of Relativistic Matter

489

As in [LY], we choose σ = 0.3, ε = 0.2077 and λ = 0.97. Our goal is to prove stability when qα ≤ 1/66.5. We may assume α < 1/47, which is the assumption used in [LY]. Hence we can use the estimate J ≤ 1.64 from [LY, Eq. (3.40)]. ˜ note that ε−1  = 0.5571 as in [LY, Eq. (3.30)]. It remains to choose an To bound A, ˜ appropriate C satisfying (5.3). For |x| ≤ 0.7 we have | F(|x|)| ≤ 1/1.02. Moreover, for ∗ ∗ Uε we use the same estimate as in [LY], namely Uε (x) ≤ 0.2077 + 0.5751 = 0.7828. Using α ≤ 1/(66.5 q) ≤ 1/66.5, (5.3) therefore holds with 0.7 (1/(66.5 · 1.02) + 0.7828) < 0.5583 =: C. This leads to a value of A˜ = 1.7287. Hence (5.5) holds for qα ≤ 1/66.5. Acknowledgements. We thank Heinz Siedentop for helpful remarks. This work was partially supported by the Swedish Foundation for International Cooperation in Research and Higher Education (STINT) (R.F.), by U.S. National Science Foundation grants PHY 01 39984 (E.L.) and PHY 03 53181 (R.S.), and by an A.P. Sloan Fellowship (R.S.).

References [B]

Berezin, F.A.: Covariant and contravariant symbols of operators. (English Transl.), Math. USSR. Izv. 6, 1117–1151 (1972) [D] Daubechies, I.: An uncertainty principle for fermions with generalized kinetic energy. Commun. Math. Phys. 90, 511–520 (1983) [DL] Daubechies, I., Lieb, E.H.: One electron relativistic molecules with coulomb interaction. Commun. Math. Phys. 90, 497–510 (1983) [FLSe] Frank, R.L., Lieb, E.H., Seiringer, R.: Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators. http://arxiv.org/list/math.SP/0610593, 2006 [H] Herbst, I.W.: Spectral theory of the operator ( p 2 + m 2 )1/2 − ze2 /r . Commun. Math. Phys. 53, 285–294 (1977) [Ka] Kato, T.: Perturbation theory for linear operators. Berlin-Heidelberg-New York: Springer, 1976 l/2 [KPS] Kovalenko, V., Perelmuter, M., Semenov, Ya.: Schrödinger operators with L w (Rl/2 ) potentials. J. Math. Phys. 22, 1033–1044 (1981) [LiY] Li, P., Yau, S.-T.: On the Schrödinger equation and the eigenvalue problem. Commun. Math. Phys. 88, 309–318 (1983) [L1] Lieb, E.H.: The stability of matter: from atoms to stars. Bull. Amer. Math. Soc. (N.S.) 22, 1–49 (1990) [L2] Lieb, E.H.: The stability of matter and quantum electrodynamics. In: Proceedings of the Heisenberg symposium (Munich, Dec. 2001), Fundamental physics–Heisenberg and beyond, G. Buschhorn, J. Wess, eds., Berlin-Heidelberg-New York: Springer 2004, pp. 53–68, A modified version appears in the Milan J. Math. 71, 199–217 (2003) [L3] Lieb, E.H.: Flux phase of the half-filled band. Phys. Rev. Lett. 73, 2158–2161 (1994) [LLo] Lieb, E.H., Loss, M.: Analysis. Second edition. Graduate Studies in Mathematics 14, Providence, RI: Amer. Math. Soc., 2001 [LLoSi] Lieb, E.H., Loss, M., Siedentop, H.: Stability of relativistic matter via Thomas-Fermi theory. Helv. Phys. Acta 69, 974–984 (1996) [LLoSo] Lieb, E.H., Loss, M., Solovej, J.P.: Stability of matter in magnetic fields. Phys. Rev. Lett. 75, 985–989 (1995) [LSiSo] Lieb, E.H., Siedentop, H., Solovej, J.P.: Stability and instability of relativistic electrons in magnetic fields. J. Stat. Phys. 89, 37–59 (1997) [LY] Lieb, E.H., Yau, H.-T.: The stability and instability of relativistic matter. Commun. Math. Phys. 118, 177–213 (1988) [S1] Simon, B.: A canonical decomposition for quadratic forms with applications to monotone convergence theorems. J. Funct. Anal. 28, 377–385 (1978) [S2] Simon, B.: Maximal and minimal Schrödinger forms. J. Operator Theory 1, 37–47 (1979) [S3] Simon, B.: Trace ideals and their applications. Second edition, Mathematical Surveys and Monographs 120, Providence, RI: Amer. Math. Soc., 2005 [W] Weder, R.A.: Spectral analysis of pseudodifferential operators. J. Funct. Anal. 20, 319–337 (1975) Communicated by H.-T. Yau

Commun. Math. Phys. 275, 491–527 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0309-0

Communications in

Mathematical Physics

Poincaré Duality and G +++ Algebras Arjan Keurentjes
Theoretische Natuurkunde, Vrije Universiteit Brussel, and The International Solvay Institutes, Pleinlaan 2, B-1050 Brussels, Belgium. E-mail: [email protected] Received: 7 November 2006 / Accepted: 26 February 2007 Published online: 24 July 2007 – © Springer-Verlag 2007

Abstract: Theories with General Relativity as a sub-sector exhibit enhanced symmetries upon dimensional reduction, which is suggestive of “exotic dualities”. Upon inclusion of time-like directions in the reductions one can dualize to theories in different space-time signatures. We clarify the nature of these dualities and show that they are well captured by the properties of infinite-dimensional symmetry algebras (G +++ algebras), but only after taking into account that the realization of Poincaré duality leads to restrictions on the denominator subalgebra appearing in the non-linear realization. The correct realization of Poincaré duality can be encoded in a simple algebraic constraint, that is invariant under the Weyl-group of the G +++ -algebra, and therefore independent of the detailed realization of the theory under consideration. We also construct other Weyl-invariant quantities that can be used to extract information from the G +++ -algebra without fixing a level decomposition. 1. Introduction One of the significant features of string theory is T-duality: Strings compactified on a large circle exhibit the same spectrum as strings compactified on a small circle. There are several arguments indicating that this is an exact symmetry of the full non-perturbative theory (for reviews and original references see [1]). A remnant of T-duality is present at the level of the low-energy effective action, of the reduced theory, where all the massive string modes have been thrown away. It manifests itself as an ambiguity: Given the low-energy effective action, there are multiple, inequivalent ways of interpreting it as a reduction of a higher dimensional theory. Applying these ambiguities to solutions of the reduced theory allows for solution-generating techniques. T-duality (and U-duality) however goes one step further, and claims that some of the solutions related by these “accidental” symmetries actually are equivalent.
Post-doctoraal onderzoeker van het Fonds voor Wetenschappelijk Onderzoek, Vlaanderen.

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The (bosonic sector of the) dimensionally reduced low-energy action describes a theory of General relativity coupled to various form fields. One expects the appearance of symmetries that mix the directions over which one is reducing, but the theory may exhibit enhanced, previously “hidden” symmetries [2–4]. Such enhanced symmetries render the assignment of a subgroup of “geometrical” symmetries ambiguous, and are therefore a clear indication of the existence of dual formulations. They are however not an exclusive feature of the low-energy actions of string theories and M-theories; they occur in a much wider class of theories, of which many seem to have no direct relation with string- or M-theory [5–8]. Most notable, among these theories are the “bare” theories of General Relativity, as first discovered by Ehlers. A natural question is to what extent one can make sense of these duality symmetries. A simple observation is that, if a theory of form fields is coupled to gravity, these exhibit p-brane solutions describing the gravitational back-reaction to some extended object (see [9] for a pedagogical review and an extensive reference list). Compactifying such an extended object on a torus of suitable dimension, one finds an effective theory describing strings, which possibly should exhibit T-dualities. This argument is admittedly very naïve, as it is well known that the higher dimensional action should also contain suitable couplings to dilatons and Chern-Simons terms if the reduced theory is to be regarded as a string theory. Even if we fine-tune for these, the resulting theory will in general be a theory of which it is unclear whether there exists a consistent, non-perturbative quantum extension. We will however ignore such questions and focus on necessary conditions for the presence of T-dualities from the classical theory. It will be clear then that non-linearly realized, enhanced symmetries and p-brane solutions are vital ingredients. This discussion even applies to (higher-dimensional) General Relativity, where there is at least one category of suitable solutions: The extremal Kaluza-Klein monopoles are for theories in 5 or more dimensions extended objects, and can be compactified to 5 dimensions to make string-like objects. An interesting development was the application of T-duality to time-like circles [10–12]. Under such dualities, it is not at all guaranteed that a space-like circle gives rise to a space-like circle in a dual picture, or that a time-like circle will dualize to a time-like circle. In these circumstances the space-time signature is no longer a fixed quantity, but is dependent on the various dual points of view. Again these dual points of view should be reflected in the symmetries of the low-energy effective action [13, 14, 11, 15]. For the ultimate unified point of view one would like to have a formulation with a symmetry that unifies all the possible space-like and time-like reductions, and the possible dualities. For the special situation where the reduction to 3-dimensions of the theory exhibits a non-linearly realized symmetry G [5–8], it has been proposed that candidate symmetry algebras should contain a “triple extended” Lie algebra G +++ [16–20]. The extended algebra G +++ is built (with the standard rules [21]) from its Dynkin diagram, which is basically the Dynkin diagram of G extended with 3 more nodes. According to the conjectures of [18–20], the degrees of freedom of the theory should be described by G +++ /K (G +++ ), where K (G +++ ) is generated by a sub-algebra of G +++ invariant under a particular involution. The more conventional formulation of the theory should be retrieved by a kind of level-decomposition, where a “gravitational sub-algebra” is singled out. It was soon realized that the space-time signature is encoded in (a subgroup of) K (G +++ ) [22–26]. It however turns out that the space-time signature in these theories is not fixed, but depends on the choice of gravitational sub-algebra [26]. Also outside the Lorentz-subalgebra of K (G +++ ) there may be additional signs that should be reflected

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in the kinetic terms of the form fields [26]. This is actually the algebraic translation of the time-like T-dualities that have subsequently been studied in this algebraic context [27–29]. In the present paper we would like to point out that there is one more aspect that should play a role in the analysis. A priori it seems as if G +++ algebras can accommodate for any space-time signature and many possibilities for signs of the gauge-fields [26]. On second thought, this should be a cause of concern: In G +++ algebras forms and their duals occur independently, though the signs of forms and their duals should not be independent; and theories with (anti-)self dual tensors can only exist in spacetimes of appropriate signature. Both issues are related to Poincaré duality, which is not, but should be implemented somehow on these algebras. In the present paper we find a simple criterion which as we will prove implements Poincaré duality correctly. The same ingredient is needed crucially to ensure the correct transformation properties of the theory under T-duality. In Sect. 2 we will collect the relevant facts on Poincaré duality and T-duality. The allowed signature changes actually follow a very simple pattern that can be analyzed without any reference to G +++ algebras or string theory. This is independent of, but should be implemented correctly on the G +++ -algebras. Section 3 recalls and explains why Weyl-reflections and other diagram automorphisms correspond to dualities. We will also sketch two, in our view, important complications in the treatment of G/H sigma models which restrict explicit computations, but can be avoided in an abstract approach. In Sect. 4 we will briefly review the implementation of space-time signature and other signs in the context of G +++ -theories. Then we will derive a simple but universal criterion that ensures that the theory has the correct transformation properties under Poincaré duality. The criterion is formulated as a constraint on a quantity that is invariant under Weyl reflections and outer automorphisms, and hence in one go restricts all dual formulations of the theory. For specific G +++ algebras there is a larger class of such invariants (though not necessarily invariant under outer automorphisms), for which we give a construction recipe. Section 5 discusses the specific theories with their invariants. In the concluding Sect. 6 we will summarize and discuss our results. 2. Duality from Dimensional Reduction and Oxidation The aim of the present section is to give an intuitive feeling what the meaning of T-duality and in particular time-like T-duality is outside the context of string theory. For this we will (ab-)use the low energy effective action. This section will be rather sketchy in nature, but we will not need much detail to derive precise statements about which signature changes should be allowed. 2.1. Sources of signs. The discussion that follows will be essentially one about minus signs. In the context of dimensional reduction and oxidation, such minus signs are essentially contributed by 3 sources: 1. The space-time signature, that we will leave arbitrary unless indicated otherwise. This assigns different signs to timelike and space-like directions in the Lorentzinvariant inner product. After fixing a space-time signature, the Lorentz-invariant inner product on the tangent space assigns definite signs to contractions of tensors. Our convention will be to assign to time-like directions a minus sign in the Lorentz-invariant inner product, and to space-like directions a plus sign.

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2. Explicit signs. The choice of space-time signature fixes the implicit signs present in the contraction of field strengths for the various gauge fields to a Lorentz-invariant combination, but not the overall sign. Though it is customary to choose a particular sign to avoid “ghosts” (unphysical modes) in quantum theory, we will leave also these arbitrary unless indicated otherwise. 3. Poincaré duality. The non-linear symmetries present in dimensionally reduced theories of General Relativity coupled to various form-fields, mix fields and their Poincaré duals. It is a simple exercise to derive that (in the absence of Chern-Simons terms and other complicating factors) [14]

d D x (F(n) )2 → (−)T −1 d D x (F(D−n) )2 . (1) Hence if the number of time directions is even, either the form or its dual carries a kinetic term with the “wrong” sign. Another obvious but important point is that Poincaré duality is consistent with dimensional reduction and oxidation. Reducing over timelike directions, the forms may obtain additional minus-signs, but then the number T also changes, and the Poincaré duality relation remains intact under dimensional reduction. In the dimensional reduction of a theory we find many form fields. Poincaré duality imposes relations between the signs of these form fields, and in this way also restricts us in our possibilities in assigning signs to various form fields. After having fixed a space-time signature and a definite sign for a form field, the sign for the dual form field is fixed. In the algebraic approach to these theories using G +++ algebras many signs cannot be chosen arbitrarily because the algebraic relations relate them. It turns out however that this is not the case for Poincaré duality: This is not automatically built in in G +++ algebras, but has to be imposed as a separate constraint. We will derive the precise form of the constraint in Sect. 4.

2.2. T-Duality. From the effective field theory point of view, T-duality is a reinterpretation of the spectrum of scalar charges. Such charges couple to 1-form vector fields, so we focus on the sector of the effective field theory containing the 1-form vector fields. One of the sectors contributing to the 1-form sector is the gravity-sector. We reduce  D-dimensional general relativity to D dimensions, and allow for the possibility that the  − D directions over which we are reducing may be timelike. The part of the n = D Lagrangian describing the Kaluza-Klein-vectors is LK K =

n 

 2 ηi f i (φk )−2 Fµν,i .

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i=1

The symbols ηi = ±1 indicate whether the direction corresponding to i was a time-like (ηi = −1) or a space-like (ηi = 1) one. The prefactor f i (φk ) depends on i and on the dilatonic scalars φk in the theory. We also define the overall signature of the reduced dimensions η (T is the number of time directions among the directions over which one is reducing)  η= η j = (−)T . (3) j

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In the most familiar reduction schemes [9, 6] (using “upper triangular” or “Borel” gauges) f i (φk ) is of the form expλi , φ, but for reasons to be discussed later we will allow more general forms. The Fµν,i obey Bianchi identities of the form dFi = derivatives of scalars ∧ F j .

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Again, the exact expressions are depending on the dimensional reduction scheme, but will not be of much importance to us. Equally important, when given a sector in the theory with Lagrangian (2), then, under suitable conditions, we can reinterpret it as representing reduction over a number of directions with appropriate signature, a procedure that is known as oxidation. Now assume that the Lagrangian contains a second part, coming from the reduction of the dynamics of a space-time (n +1)-form Cµ1 ,...,µn+1 . We will generalize slightly, and actually allow for a multiplet of p such n + 1-forms. The reduction of such a p-plet of forms over n dimensions gives rise to another sector in the reduced theory with 1-forms, with Lagrangian Ln = η

p 

 2 θi gi (φk )−2 G µν;i1 ...in ;i .

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i=1

The signs θi = ±1 are the signs appearing in front of the kinetic term for the C(n+1),i -field before reduction. Provided the gi and f i take similar forms (as they do in available dimensional reduction schemes), the Lagrangians (2) and (5) are similar in form. They arose from the reduction of a p-plet of n-forms over n-dimensions, coupled to gravity, but it is tempting to explore the possibility that they can also be interpreted as coming from the reduction of an n-plet of p-forms, reduced over p dimensions. The signature of the dual theory is set by the p signs ηθi , while the n signs ηi will appear in front of the kinetic terms for np-forms. We also note that the existence of an (n + 1)-form Cµ1 ,...,µn+1 coupled to gravity allows for an extremal n-brane solution. It is to be expected that the above duality exchanges a p-plet of n-branes in one theory, for an n-plet of p-branes in the other theory. An immediate condition that has to be met for this reinterpretation, is the presence of a subsector of the scalar sector that can be interpreted as an S L(n)/S O(n − q, q) × S L( p)/S O( p − r, r ) sigma-model [3]. The S L(n)/S O(n − q, q) factor arises automatically in the reduction of the gravity sector, where q is the number of ηi = −1 in (2), encoding the number of time-like directions involved in the reduction. Similarly, r is the number of θi = −1 in (5). The condition on the presence of the relevant sigma models however is only a strong one if p > 1, because when p = 1 the S L( p) factor trivializes. The possible presence of enhanced symmetries is however a significant point. Further dimensional reduction will not destroy these enhanced symmetries, and hence they can be equally well studied in theories that have been reduced further. In [7, 8] it was demonstrated how from enhanced symmetries in 3 dimensions the bosonic sector of the higher-dimensional theory can be reconstructed; it turns out no additional conditions are needed. These considerations give a potential duality between two, a priori different theories. In general the two dual theories do not even live in a space-time of the same dimension. We should stress however the “a priori”: As we are focusing on a subsector of a theory to start with, the dual description also involves only a sector of a theory and extending to the full theory we may be studying different truncations of the same theory.

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We illustrate these considerations with some familiar examples. The membrane of 11-dimensional M-theory dualizes to a doublet of strings in a dual 10 dimensional theory; of course this is IIB -supergravity. We reduce on a 2 torus, that can be of signature (0,2) or (1,1). With signature (0,2) η1 = η2 = η = θ , and the dual theory has signature (1,9) with two standard 2-form terms. With signature (1,1) we have η = −1. Together with the standard sign of the M-theory 3-form kinetic term, this means that we get a time-like direction back for the T 1,1 we have sacrificed, but the different signs of the ηi imply that the 2-form field kinetic terms have opposite signs [10, 11]. Also the other relations between M, M  , M ∗ and I I B, I I B  , I I B ∗ theories can be analyzed along these lines. A 3-brane in a space-time of signature (1, 9), as relevant to IIB-theory, but also to the theory with conjectured E 7+++ symmetry, dualizes to a triplet of strings in 8 dimensions. With the various options for the signature, the space-time signature of the dual space is always (1,7), but the 2-forms coupling to the strings have kinetic terms with signs (+ + +) or (− + +) depending on the choice of signature on the 3-torus. It is easy to see that they are mixed by S L(3, R)/S O(3) respectively S L(3, R)/S O(1, 2). Notice that for IIB-theory, the invariance of the 3-brane under S L(2, R)/S O(2) implies that the dual strings should be inert under the S L(2, R) factor in the S L(2, R) × S L(3, R) U-duality group in 7 dimensions, as they are. Another simple example is given by the doublet of 5-branes in IIB-theory. According to the above rules, they should give rise to a quintuplet of membranes in 7 dimensions, transforming as a quintuplet under S L(5) and under S O(4, 1) or S O(5) [11, 15]. The M-theory 5-brane dualizes to the 5 strings in 7 dimensions that are the electro-magnetic duals to the membranes. As a final cautionary remark we stress that one should verify that all necessary conditions are really met. As an absurd example of too naive analysis: 7-dimensional M-theory has a decouplet of 3-branes, but this does not imply the existence of a triplet of 10 branes in a 14-dimensional theory; there is no underlying S O(10)-symmetry. 2.3. T-symmetry. Upon invoking an extra dimension in the reduction, one can return to the original theory without returning to the original parameterization: First one reduces over n dimensions, and replaces these by the p dimensions of the dual system; with the extra circle we now have p + 1 circles, from which we choose another set of p coordinates, and replace these by n coordinates of the original theory. From the effective field theory point of view, this involves the following Lagrangian. ˜ It is obtained by focusing on the subsector of 1-forms of the theory of D-dimensional General Relativity, coupled to an n-form, and reduced over n-dimensions. L=

p+1 

 2 ηi f i (φ)−2 Fµν,i

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i=1

+θG η

p+1 

ηi h i (φ)−2 Hµν,i ,

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i=1

where the Fµν,i are the p + 1 field strengths corresponding to the Kaluza-Klein vector bosons, whereas the Hµν,i = ( p !)−1 εii1 ...i p G µν,i1 ...i p correspond to the reduced form field modes. The sign θG encodes the sign in front of the kinetic terms of the C-field before reduction,  and η is again the product of the signs of the compact time-directions. = p + 1 ways of reducing the form, there are just as many G µν,i as As there are p+1 p

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Fµν,i . Provided suitable conditions on the f i and h i are met, there is a duality exchanging the F’s and the G’s. The overall signature of the F’s and G’s are exactly identical, up to an overall sign set by θG η. Hence, if θG η = −1, the duality exchanging the F’s with the G’s is a signature changing one. We conclude that the dual theory has ηi = ηθG ηi ;

η  = θG η p ; p+1

θG = θG η p+1 . p

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It is a fact is that the existence of extremal p-brane solutions coupling to the G-fields requires them to have a world volume signature −θG : If θG = 1 then the number of timelike directions incorporated will be 1 mod 2, whereas when θG = −1 the number of timelike directions is 0 mod 2 [12]. Hence, if we choose η to be in correspondence with the world volume signature of extremal p-branes the above duality will always result in some dimensions changing signature. But this then immediately implies that the corresponding p-brane can wrap the cycle over which we are reducing. In string theory it is often argued that the fact that a brane can wrap a cycle softens the potential singularity that arises if such a cycle shrinks to zero size. It would of course be most interesting to study time-like T-duality for more complicated reductions to see if this is a property that is general. The charges coupling to the F-fields are the momenta of gravitational waves, while the charges of the G-fields are winding modes (where it is understood that momenta and winding can be space-like and time-like). The spacing of the momenta is small for large volumes, while the spacing of winding modes is large (and vice versa for small volumes). As the hypothetical duality interchanges these, we have a standard “large volume-small volume” duality. For theories which are conjectured to be described by non-linear realizations of G +++ -algebras, exact formulas for the volumes can (in principle) be extracted with the techniques used in [19]. With a little manipulation we see that θG = θG (ηη ).

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The kinetic term will therefore change sign if the number of compact time dimensions changes from even to odd or vice-versa under the duality. As illustrative examples we recall that 11-dimensional M-theory has a membrane solution with world volume signature (T, S) = (1, 2). We then partition the spacetime signature (1, 10) into parts corresponding to the membrane world volume and a transversal part, and T-dualize on a 3–torus with signature (1, 2): (1, 10) = (1, 2) + (0, 8) → (2, 1) + (0, 8) = (2, 9).

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Under this duality the number of compact time-directions changes from 1 to 2, so we should expect a sign change in front of the kinetic terms for the 3-form. M-theory also has a 5-brane solution, with world volume signature (1, 5). Now the above duality applied to the 5-brane implies (1, 10) = (1, 5) + (0, 5) → (5, 1) + (0, 5) = (5, 6)

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and the sign in front of the 6-form kinetic terms does not change (and, by Poincaré duality, neither does the sign for the 3-form term). We have to pay special attention to the situation in 4 dimensions, where the KaluzaKlein gauge fields have duals that are also vector fields. We can set up the same Lagrangian, but now G is essentially ∗ F. If we ignore the complications caused by Chern-Simons terms, we could formally Poincaré dualize, to derive that θG = (−)T −1 ,

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where T is now the number of time directions in the non-compact directions. Hence a signature changing duality requires T to be even, that is 0, 2, 4. For General Relativity in signature ( p, q) there exist extremal solutions of the form R p−4,q × T N 4,0 , R p−2,q−2 × T N 2,2 R p−2,q−2 × −T N 2,2 R p,q−4 × T N 0,4 , with Ra,b flat space of signature (a, b), and T N a,b a Taub-NUT-solution of signature (a, b) [30, 12]. Again we are in the situation that for a signature changing duality we always have an extremal brane-solution that can wrap the compact directions. In this case a signature changing duality gives the dual signature (1, 10) = (1, 6) + (0, 4) → (6, 1) + (0, 4) = (6, 5).

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It is tempting to speculate that the existence of M-theory variants with signature (9, 2) and (10, 1) is evidence for objects with 10 and 11 dimensional world volumes, but of course the present methods are not suitable for discussing these (dimensional reductions to 2 and 1 dimension would be needed, in which case we never would have vectors at our disposal). A candidate 10-dimensional object (the “M9”–brane) has appeared at various places in the literature. 3. Sigma Models and Non-linear Realizations In the previous section we found suggestive dualities on the basis of an analysis of the spectrum of charges, that couple to vector-fields. Compactifying and reducing over an additional dimension, the symmetries of the vector fields become reflected in the set of scalar fields in the theory. These typically form sigma-models on symmetric spaces of the form G/H , where G is a non-compact group, and H is a real form of the complexification of the maximal compact subgroup in G. As dualities from the effective field theory point of view are nothing but ambiguities in the oxidation procedure, we should now analyze the oxidation of the sigma-model. Here we have powerful methods at our disposal, rooted in the representation theory of G. For finite dimensional G the oxidation procedure is mathematically rigorous [7, 8], for infinite dimensional G a level expansion gives the correct result at lower levels, but is not understood at higher levels [33, 18, 34, 20]. 3.1. Ambiguities from oxidation of the scalar sector. The Lagrangian for a coset sigma model on G/H can be written in the following form:
1 dd x Tr (F F − Fω(F)) . (13) 2 Here F is the Lie(G)-valued (Lie(G) is the Lie-algebra of G), left-invariant one-form (of the form V −1 (dV ) with V ∈ G), and ω is the involution that leaves Lie(H ) invariant. For non-simple G the Lagrangian decomposes in several terms, each of which has the same structure. For definiteness we will assume G to be simple in the following. If G is finite-dimensional, then Tr(F F) is the standard bilinear form on Lie(G) and hence positive definite. If in addition ω is a Cartan-involution, then by definition the bilinear form −Tr(F ω(F)) is positive definite [31], and hence the group structure does not contribute minus signs to the Lagrangian. As ω(h) = h for h ∈ Lie(H ), and the definition of Cartan involution implies that Tr(hh) = Tr(hω(h)) < 0 for h = 0, H will be compact by Cartan’s first criterion. In all other cases, H is non-compact, and the trace Tr over the group indices is not positive definite.

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The one-form F transforms as a connection under G transformations F → g −1 Fg + g −1 (dg), but the Lagrangian is only invariant if g ∈ H . Hence the theory is invariant under global G-transformations V → gV (as these leave V −1 (dV ) invariant), and local H transformations V → V h(x), and describes a sigma-model on the coset G/H . This form of the Lagrangian suffices for discussions in d-dimensions, but we actually need to relate this Lagrangian to higher dimensional theories. To make the link, one makes the field-content of the Lagrangian visible in a two-step procedure. The first step consists of fixing a way to represent Lie(G). This is necessary to do actual computations with the Lagrangian (13), and more particularly, to specify the action of ω. As an example, one can describe an S L( p + q, R)/S O( p, q) by representing S L( p + q, R) by ( p + q)-dimensional square matrices, and ω by ω(F) = −ηF T η, with η an S O( p, q) invariant metric and F T the matrix-transpose of F. However, such a representation is only convenient if not only S O( p, q), but also S L( p + q, R) can be kept manifest in the oxidation. To also be able to describe the cosets for arbitrary G, we need a general prescription. It is then more convenient to work with the adjoint representation. We will fix a triangular decomposition Lie(G) = H ⊕ n − ⊕ n + , where H is a fully reducible abelian subalgebra (a Cartan sub-algebra) and n ± are composed of ladder operators associated to positive, and negative roots respectively [21], and work with this. The second step consists of (partially) fixing the H -invariance of the theory. Essentially one is picking a set of variables to describe the coset G/H , after which (some) redundant coordinates are eliminated, but also (part of the) H -invariance is no longer manifest. This second step is needed to relate to the outcome of some dimensional reduction procedure, or in the case of the conjectures relating to G ++ and G +++ -algebras [33, 18, 34, 20], to some reconstruction algorithm. One can leave some, or even all of the H -invariance intact, provided one relates to a formulation of the higher dimensional theory with manifest H -invariance, and uses a dimensional reduction algorithm that respects these. One can for example reduce a theory from D to D  ≥ 3 dimensions, while keeping the local S O(D − D  ) covariance of the vielbein-formalism manifest. We also note the existence of formulations of maximally supersymmetric theories with local H -invariance [32]. We are emphasizing these two distinct steps, because when H is non-compact, and hence when time-like T-dualities are possible, there are subtleties concerning the second step, the fixing of the H -invariance. But as H is a local (gauge) symmetry, nothing can depend on the details of how we treat the H -invariance. We will nevertheless clarify the nature of the subtleties in our next subsection. For the discussion of possible ambiguities in the interpretation of the subsector of the Lagrangian (13) as coming from a higher dimensional theory, we only need to discuss Lie(G). A first ambiguity arises because we have to fix a “gravitational subalgebra” of the form An , and there may be multiple non-equivalent embeddings (not related by conjugation with group elements) of these. This results in general in higher dimensional theories with different matter content, and lies at the heart of the algebraic description of T-duality. Within a class of equivalent embeddings there are still ambiguities. The triangular decomposition is not invariant under the action of G, but there is a discrete subgroup of G, effectively the lift to G of the Weyl group of Lie(G), that normalizes the Cartan subalgebra, and also n + ⊕ n − , the set of ladder operators. It therefore permutes the fields

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associated to particular (combinations of) generators, and the oxidation of the theory to higher dimensions is ambiguous because of these permutations. This is what has been called T-symmetry. We note that, apart from the Weyl group, outer automorphisms of the algebra permuting the vectors of the root lattice will also result in ambiguities in the oxidation. These result however in gravitational sub-algebras that are not conjugate (as the automorphism is outer) and hence this belongs to T-duality. Even if part of the G-symmetry is kept, similar arguments apply. For example, in the example sketched in the above where S L( p + q, R) (and S O( p, q)) is kept, this group acts (in the fundamental, respectively vector- representation) on some carrier space for the representation. The assumption that this space is a ( p + q)-torus, makes backgrounds related by S L( p + q, R)-transformations inequivalent, but there is a subgroup of S L( p + q, R) with a rather trivial action: it only permutes the basis-elements of the carrier space. This permutation action is generated by the Weyl group of S L( p, q), and hence we are back in the previous situation.

3.2. Subtleties. For cosets G/H where H is non-compact, there are a number of subtleties compared to the case where H is compact. We will highlight two issues that may seem confusing, but should not distract the reader from the main point. We think however that especially the first point deserves more attention than it has received thus far. 3.2.1. The Borel gauge is a “bad” gauge for non-compact H. In the dimensionally reduced theories sigma-models on symmetric spaces make their appearance. These symmetric spaces are of the form G/H , with G a non-compact Lie-group. For the cases where the dimensional reduction/compactification includes only space-like circles, the group H = Hc is the maximal compact subgroup of G; in the case time-like directions are included the compact group Hc has to be replaced by a non-compact Hnc with the same complexification as Hc . For the sake of concreteness, we will work with G that are split real forms, though all remarks generalize to G’s in other non-compact real forms. The Lie algebra LG associated to G allows a triangular decomposition in terms of a Cartan-subalgebra with generators h i , and raising/lowering operators eα . The compact subgroup H is now generated by generators of the form eα − e−α . This description of the group H makes a very explicit use of the triangular decomposition. It allows the well-known action of the Weyl group on G, that will automatically act on H . To describe the coset G/H , one chooses a single representative of each orbit of H on G. In physical language, this amounts to fixing the gauge. If we then act with the Weyl group on G/H the gauge that we have chosen is typically not respected. Because the sigma-model is invariant under the left action of H one can allow for an H -transformation that restores the gauge. The combination of a Weyl-reflection together with its gauge-restoring H -transformation then gives the symmetry transformation of various fields. The most popular gauge-choice is the so-called Borel gauge, which relies on the Iwasawa-decomposition. The Iwasawa decomposition theorem states that any element g of a non-compact group G can be written as g = k · a · n,

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where k is an element of the maximal compact subgroup in G, a is an element of the Abelian group obtained by exponentiating (a subset of) the generators of the Cartansubalgebra (generating non-compact symmetries), and n an element of a unipotent group. In the special case that G is a so-called split real form, the elements a come from exponentiating all of the Cartan-subalgebra, and the elements n are obtained by exponentiating elements from the step-generators associated to positive roots. It is clear that this decomposition is of a great value for the description of cosets G/H , where H is the maximal compact subgroup of G. In that case the elements of G/H can be written as

   g = exp Cα eα , (15) φi h i exp α∈ +

+ being the set of positive roots. In this gauge the action of H , generated by the algebra-elements eα − e−α is completely fixed. A caveat in the application of the Iwasawa decomposition theorem is, in the context of the applications discussed here, that it is proven for finite-dimensional G only. The crucial thing to establish is that any element of G can be written as in the decomposition, and it seems that this needs the relation between the Lie-algebra, and Lie-group, which is not well understood for general (not necessarily finite-dimensional) Kac-Moody algebras. It is however generally assumed that an infinite-dimensional analogue of the Iwasawa decomposition exists [33]. More relevant in the situations of interest to us is that H is not always compact. It was observed in [14] that the relevant H is always a real form of the complexification of the maximal compact subgroup H , but that it is not necessarily the compact real form. These forms can be described by replacing some of the antihermitian generators eα − e−α by hermitian generators eα + e−α [22–25]. Demanding closure of the algebra leads to tools that allow to classify the relevant possible denominator subgroups [15]. By replacing a compact H by a non-compact one, we are however outside the scope of the Iwasawa decomposition theorem. If one nevertheless insists on the usage of elements of the form (15) we have to pay a price: of course these elements are contained in the coset, but the reverse is no longer true, not every element in the coset can be written in the form (15). This can be immediately seen in examples, but also directly. The coset G/H , where H is not a maximal compact subgroup, must contain compact cycles (because there must be compact orbits in G that cannot be compensated with H -transformations). But the ranges of the variables in (15) are non-compact, and consequently, the variables (15) describe only an open proper subset of G/H . The prototype example is found in the space S L(2, R)/S O(1, 1). The Borel gauge essentially amounts to the “upper triangular gauge”; elements in this gauge have the lower left entry set to zero. Consider now the element  cos φ − sin φ ∈ S L(2, R). (16) sin φ cos φ If we try to bring this to upper triangular form by multiplying on the left by an element of S O(1, 1), it requires essentially solving the equation tan φ = ± tanh ω

(17)

for some ω. But it is immediately clear that this cannot have solutions when | tan φ| ≥ 1. This is in a nutshell the problem for any compact cycle in G/H .

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A. Keurentjes

Of course one can define the set S of elements k · a · n for all k, a, n, with k ∈ H . This set S is in general not a group, because it will not close under group multiplication. Instead S is a proper subset of G. Hence the set of all representatives a · n will only cover part of G/H . Whether this is a serious problem depends on the application we have in mind. We argued that T-dualities are generated by a discrete set of transformations on G. Now if we start with S instead of G, then the image of S under such transformations is generically not contained in S. In explicit language, the transformation that brings us back to our gauge choice does not exist! Hence in the end we can only do an analysis on

Imw (S), (18) w∈W

where W contains all the transformations (Weyl reflections, outer automorphisms) we are interested in. For an extension to all of G/H one has to rely on some sort of continuation procedure. It would be much simpler to not rely on the Borel gauge for this kind of computation at all, but it seems however that a general “good gauge choice” (a gauge that can always be imposed) is unavailable even for finite-dimensional G/H , not even mentioning infinite-dimensional G’s 1 . Luckily, as far as T-dualities are concerned, we do not need to rely on any gauge choice. Moreover, we will argue later that in the context of G +++ algebras the use of compact subgroups in the coset construction is inconsistent with Poincaré duality. But then the Borel gauge, used throughout the literature thus far, misses part of the compact cycles in G +++ /K (G +++ ), and there are as a matter of fact infinitely many independent such cycles if K (G +++ ) is not compact! 3.2.2. The action is not invariant under the Weyl-group. A less important problem, which may nevertheless be confusing is the fact that the action (13) is generically not invariant under the transformations we are considering. This can be seen easily already if we restrict to those duality transformations that are generated by Weyl reflections. These can be implemented by conjugation with an element W , acting as W exp gW −1 = exp w(g)

g ∈ Lie(G).

(19)

The element W is an element of the maximal compact subgroup in G. In the action (13) the one form F transforms as F → W F W −1 . The action is then invariant if ωW (F) ≡ ω(W F W −1 ) = W ω(F)W −1 ∀W, F.

(20)

We may analyze this equation as follows: To every element g of the algebra there is a one parameter-subgroup of G given by exp tg. The Lie algebra homomorphism ω lifts to a group homomorphism  by defining (exp tg) = exp(tω(g)). Though not every element of G can be written as exp tg, any element can be written as a product of such 1 Good gauge choices are known in specific situations. For example one can prove that for S L(2, R)/ S O(1, 1), with the S L(2, R) elements written as



a b c d

,

a, b, c, d ∈ R, ad − bc = 1,

the gauge condition ac+bd = 0 can always be established. But we are unaware of a general, good (and useful !) gauge fixing procedure for general G/H .

Poincaré Duality and G +++ Algebras

503

elements, and then the homomorphism property guarantees that  can be extended to the whole group. We then compute eω(W F W

−1 )

−1

= (e W F W ) = (W e F W −1 ) = (W )eω(F) (W −1 ).

The condition (20) can then be rewritten to  −1    W (W ) eω(F) = eω(F) W −1 (W ) .

(21)

(22)

Because this has to be valid for any F, this implies that W −1 (W ) is proportional to the identity (by Schurs Lemma). This is trivially the case if (W ) = W , that is if W ∈ H . However W is in the maximal compact subgroup of G, and this does not coincide with H if H is non-compact. Because W is in the compact subgroup of G, we could also have W −1 (W ) = λ, with λ a phase , but this is generically not the case. So for non-compact H , the action is generically not invariant under Weyl-reflections. This should not confuse the reader. It will be clear that also ωW (F) defines a formulation of the sigma-model on G/H . Different ωW do not in any way affect G, but only change the way we embed H in G. In view of the ambiguities we are interested in, we should not view models defined by different ωW (F) as different models, but as different realizations of the same model. If one could encode ω in a way that would not rely on a specific basis for the algebra or a specific realization, one might avoid the changes in the Lagrangian under Weyl reflections; such invariant Lagrangians appear not to have been constructed, and it is not even clear (to the author) that they exist. 4. G +++ Algebras: General Theory We now explicitly turn to G +++ -algebras. The present section is devoted to G +++ -algebras in general, in the next section we will do computations with explicit G +++ algebras. 4.1. Defining G +++ and K (G +++ ). The G +++ -algebras are essentially defined by their Dynkin diagrams. The Dynkin diagram for a simple group G consists of n nodes, where n is the rank, which we label by an integer i ranging from 1 to n. To this diagram we add three more nodes, labelled by 0, −1 and −2. The node labeled by 0 is connected to the Dynkin diagram to obtain the extended Dynkin diagram, also known as the Dynkin diagram of the untwisted affine Lie algebra G + associated to G. We then connect the node labeled by −1 with a single line to the node 0, to obtain the diagram of the overextension or canonical hyperbolical extension G ++ of G. Finally, the node −2 is connected by a single line to the node −1 to obtain the diagram of the triple extended algebra G +++ . Via this node by node extension, there is a natural embedding G +++ ⊃ G ++ ⊃ G + ⊃ G.

(23)

We will make use of some results for G + and G later. From this diagram the Cartan matrix A = (ai j ), with i, j in the index set I ≡ {0, 1, . . . 10}, may be reconstructed by the standard procedure [21]. The construction of the algebra G +++ starts by choosing a real vector space H of dimension n + 3 and linearly independent sets = {α−2 , . . . , αn } ⊂ H∗ (with H∗ the space dual to H) and ∨ , . . . , α ∨ } ⊂ H, obeying α (α ∨ ) = a . The α are called the simple roots ∨ = {α−2 j i ij i n +++ of G , and αi∨ are the associated simple coroots.

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From the Cartan matrix the algebra G +++ can be constructed. The generators of the algebra consist of n + 3 basis elements h i of the Cartan sub-algebra H together with 22 generators eαi and e−αi (i ∈ I ), and of algebra elements obtained by taking multiple commutators of these. These mutual commutators are restricted by the algebraic relations: [h i , h j ] = 0 [h i , eα j ] = α j (h i )eα j [h i , f −α j ] = −α j (h i )e−α j [eαi , e−α j ] = δi j αi∨ h i ; (24) and the Serre relations ad(eαi )1−ai j eα j = 0

ad(e−αi )1−ai j e−α j = 0.

(25)

There is the root space decomposition with respect to the Cartan subalgebra, G +++ = ⊕α∈H∗ gα , with gα = {x ∈ G +++ : [h, x] = α(h)x ∀ h ∈ H}.

(26)

The set of roots of the algebra, , are defined by

= {α ∈ H : gα = 0, α = 0}.

(27)

The root lattice P(G +++ ) consists of all linear combinations of roots with integer coefficients. The roots of the algebra form a subset of the root lattice ⊂ P. We will denote the generators of gα by eαk , where k is a degeneracy index, taking into account that the dimension of gα may be bigger than 1. If dim(gα ) = 1, we will drop the degeneracy index, and write eα for the generator. This is in accordance with previous notation, as dim(gαi ) = 1 for αi a simple root. By using the Jacobi identity, one can easily prove j that [eαi , eβ ] ∈ gα+β , if this commutator is different from zero. A particular real form of the G +++ algebra is the split real form. The Lie-algebra of the split real form consists of linear combinations of the generators constructed thus far with real coefficients. Almost all of the present literature on G +++ -algebras is devoted to split real forms. Other real forms can be constructed by defining realizations for the Cartan involution. It is easy to see that for 3-dimensional coset sigma models involving non-compact real forms of G (see [8] for an exhaustive list) there exist corresponding G +++ algebras defined by extending the Satake-Tits diagram in the obvious way. We will however also restrict mostly to split real forms. Using the standard bilinear form on the root space [21] we define the simple coweights (which are elements of H) by αi , ω j  = δi j .

(28)

The coweight lattice, which we call Q(G +++ ) consists of linear combinations of the fundamental coweights with coefficients in Z, and is the lattice dual to the root lattice P(G +++ ). The Weyl group W (G +++ ) of G +++ is the group generated by the Weyl reflections in the simple roots wi (β) = β − 2

αi , β αi . αi , αi 

(29)

Poincaré Duality and G +++ Algebras

505

The Weyl group leaves the inner product invariant w(α), w(β) = α, β,

w ∈ W (G +++ ).

(30)

K (G +++ )

demands that In [26, 15] we demonstrated that the closure of the group the hermiticity properties of its generators can be conveniently encoded in an element f ∈ Q(G +++ ). The group K (G +++ ) is then generated by the generators k eβk − eiπ β, f  e−β

∈ G +++ .

(31)

From the definitions it is straightforward to derive that f and f + 2q define the same group for q ∈ Q(G +++ ). As a consequence f can be expanded in the fundamental coweights as f =

n 

pi ωi ,

pi ∈ Z2 = {0, 1}.

(32)

i=−2

These definitions reduce the characterization of the real form of the infinite-dimensional group K (G +++ ) to the specification of the n + 3 Z2 -valued coefficients pi . The coset G +++ /K (G +++ ) encodes the properties of a class of physical theories, along with all their reductions and oxidations. To recover the field content of a D-dimensional theory one chooses a regular gravitational subalgebra S L(D, R) and decomposes under this algebra. The representation content at lower levels consist of the gravitational sector, a scalar sector transforming under the centralizer of S L(D, R) in G +++ , and a bunch of form fields; the higher levels are not understood [33, 18, 34, 20]. The space-time signature in turn is encoded in how the S L(D, R) intersects with G +++ , as its intersection is the algebra so( p, D − p) [26]. This so( p, D − p) algebra at level 0 implies that the form fields at higher levels transform as so( p, D − p)-tensors. There is however the possibility of additional signs (because G +++ is strictly bigger than so( p, D − p)) which was interpreted in [26] as a consequence of the freedom to multiply the kinetic terms of the tensor fields with additional signs. We have argued thus far that Poincaré duality restricts us in our ability to add signs to the algebra in a consistent way. In the next section we will derive how the restrictions from Poincaré duality should be consistently built into the above framework. 4.2. Poincaré duality and the i 0 criterion. At first it may seem that for general G +++ we can only do computations on a case-by-case basis, as the matter content depends sensitively on the algebra. All of the theories described by G +++ algebras however have a universal feature: if reduced to 3 dimensions, they describe a scalar sigma model with global G symmetry. The duals to the axionic scalars are vectors in 3 dimensions, which should also transform under G. These matter fields can be recovered from the decomposition relevant for 3 dimensions: G +++ → S L(3) × G.

(33)

This decomposition automatically implies a level expansion, where “level” is defined with respect to the α0 -node. It is relatively easy to verify that the first two levels with respect to this decomposition contain the representations level 0 1

sl(3) ⊕ G irrep (8, 1) ⊕ (1, dim(G)) (3, dim(G))

(34)

506

A. Keurentjes

The interpretation of the matter spectrum is straightforward. At level 0 the (8, 1) is the adjoint of S L(3), which in turn is part of the G L(3) of 3-dimensional General Relativity; these are singlets with respect to G. The irrep (1, dim(G)) however forms a singlet under the S L(3) of General Relativity, hence these are scalars, and there are dim(G) of them, enough to fill the adjoint of G 2 . The first level consists of vectors (the 3-dimensional irrep of S L(3)), formally dual to the axions of the scalar sector, and hence also transforming in the adjoint of G. Hence, on the levels 0 and 1, we have 4 copies of the root system of G, a singlet of S L(3, R) at level 0 for the scalar sector, and a triplet of S L(3) at level 1, for the vectors. If we denote by α H the highest root of G, embedded in the obvious way in the G +++ root system (by identifying αi with i > 0 as the simple roots of G), the corresponding roots can be written as level 0 1

G +++ -roots 0 ± positive roots of G α0 + α H ± positive roots of G α−1 + α0 + α H ± positive roots of G α−2 + α−1 + α0 + α H ± positive roots of G

(35)

Now we turn back to the physical 3-dimensional theory. If the space-time signature of the 3-dimensional space-time is (+ + +) (3 spatial dimensions) then the overall signs of the 3 components of the vector should be (− − −); Poincaré duality requires an overall minus sign in front of the vector. If the space-time signature is (− − −) (3 time-dimensions), then the signs again have to be (− − −), because now Poincaré duality tells us that there is no extra sign for an odd number of time dimensions. Similarly, for signature (− + +) the vector has (− + +) while for signature (− − +) the vector has (+ + −). Summarizing, the number of relative minus signs appearing in the vector is always odd, regardless of the signature. We can read off the relative signs for the vector from the algebra, by computing: α0 + α H , f mod 2, α−1 + α0 + α H , f mod 2, α−2 + α−1 + α0 + α H , f mod 2. (36) We have just derived that Poincaré duality requires that there is an odd number of minus signs among these 3 signs. This can be easily checked by summing the 3 relative signs modulo 2. We should therefore demand the following Consistency criterion. The function f specifies a K (G +++ ) consistent with Poincaré duality if and only if i 0 ( f ) ≡ α−2 + α0 + α H , f  mod 2 = 1.

(37)

With this criterion, the signs of the vector become α−2 , f  + 1 mod 2;

α−2 + α−1 , f  + 1 mod 2;

α−1 , f  + 1 mod 2. (38)

As the Lorentz group is encoded in the generators as eβ − exp iπ β, f e−β ,

β = α−1 , α−2 , α−1 + α−2 ,

(39)

2 When fixing a gauge, almost half of these are eliminated. Remember however that we have chosen not to fix the gauge.

Poincaré Duality and G +++ Algebras

507

it is straightforward to check that the requirement (37) is sufficient to obtain the right signs. The derivation of the criterion was done for the 3-dimensional theory. We noted however that Poincaré duality is consistent with reduction and oxidation, which are automatically implemented in the algebra. The criterion (37) should therefore be the correct criterion in any dimension for space-time, even though it is much less straightforward to derive it for the more general case. The above expression for i 0 ( f ) is a universal one, but when we wish to apply it to a specific G +++ algebra, we need the algebra dependent expression for α H . For any G α H is the so-called “highest root”, and allows an expansion in the simple roots as αH =

r 

ai αi .

(40)

i=1

The integer numbers ai are known as the root integers, and because of the mod 2 nature of the i 0 criterion, we only need their values modulo 2. The reader can find them for example in [21], Table Aff 1 in Chap. 4. Later we will discuss the separate G +++ -algebras explicitly, and give the explicit expressions for i 0 ( f ) for all the cases. 4.3. Properties of i 0 . The quantity i 0 ( f ) enjoys the following two important properties: ∀w ∈ W (G +++ ); i 0 ( f ) = i 0 (w( f )) i 0 ( f ) = i 0 ( f + 2q) ∀q ∈ Q(G +++ ).

(41) (42)

We will call i 0 ( f ) a “mod 2 index” of K (G +++ ). Although i 0 ( f ) is formulated in terms of f , the above statements indicate that i 0 not so much characterizes f , but rather the equivalence classes of f under the Weyl group and shifts over twice the coweight lattice. These transformations do not alter K (G +++ ), but only its embedding in G +++ , and in that sense i 0 is a quantity that contains information on K (G +++ ). We will now prove the invariance statements (41) and (42). The statement (41) states that i 0 ( f ) takes the same values on f ’s related by Weylreflections. It is convenient to use the orthogonality properties of the Weyl reflections with respect to the bilinear form β, w( f ) = w −1 (β), f .

(43)

We then prove (41) as follows: α0 + α H is an imaginary root of G + . Imaginary roots of affine Kac-Moody algebras are all null |α0 + α H | = 0 [21], and orthogonal to any real root of the algebra, which in this case are the roots α0 , . . . , αr . Together with the fact that inspection of the Dynkin diagram teaches us that α−2 is orthogonal to α0 , . . . , αr (by the defining properties of the Dynkin diagram), this shows that (37) is invariant under Weyl reflections in α0 , . . . , αr . It remains to check for the Weyl reflections in α−1 and α−2 . On the basis of elementary roots these have the following effect: wα−1 : α−2 → α−2 + α−1 ; α−1 → −α−1 ; α0 → α0 + α−1 ; wα−2 : α−2 → −α−2 ; α−1 → α−1 + α−2 ;

(44)

and roots that are not listed remain invariant. It is immediately verified that wα−1 and wα−2 also leave i 0 ( f ) invariant, hence all generators of the Weyl group leave i 0 ( f )

508

A. Keurentjes

invariant. This last statement implies that any element of the Weyl group leaves i 0 ( f ) invariant. If the diagram of G +++ admits a diagram automorphism, the algebra of G +++ admits an outer automorphism (that actually is an extension of the outer automorphism of G). +++ +++ In the cases where these appear (A+++ n , Dn , E 6 ), it can be verified (see the explicit expressions given later) that the index i 0 ( f ) is not only invariant under the Weyl group, but also under the outer automorphism. In essence this can be directly traced to the fact that the set of root integers ai is invariant under outer automorphisms [21]. The second statement, that i 0 ( f ) takes the same value on f and f +2Q(G +++ ) follows trivially from the mod 2 property of i 0 ( f ) and the definition of Q. These properties are crucial for the consistency of the framework. We have claimed that Weyl reflections and outer automorphisms in the G +++ algebra correspond to a kind of duality group, acting on the physical theories constructed from the algebra, and that theories not satisfying the i 0 -criterion are incompatible with Poincaré duality. It is important then that Weyl reflections should respect Poincaré duality. The properties we have just verified ensure that they indeed do. An immediate consequence of the criterion (37) is that it is inconsistent with the use of f = 0, that is with a “compact” subalgebra. This means that within this framework, even if we decided to work with Euclidean theories, we will always encounter minus signs and hence non-compact denominator subgroups! Also Wick rotations must be performed with care, it seems that at least for some applications we should demand consistency with the i 0 criterion. Then, although the denominator subgroup K (G +++ ) can be altered by such a procedure, we can never change it to the compact algebra. Of course this could have been anticipated: For a Euclidean theory reduced to 3 dimensions, Poincaré duality implies that there should be a relative minus sign between 3-dimensional scalars and the dual vectors, as has been known for long [35]. For theories in which (anti-)self-dual tensors occur, we have to require specific signatures for space-time. If we do not, there is a set of tensors in the level decomposition of the G +++ - algebra that does not have duals (namely the would-be self-dual tensors that cannot be self-dual because it is inconsistent with the space-time signature). Imposing the i 0 criterion, Poincaré duality is implemented correctly, and this problem cannot occur. We will check later in explicit examples that everything works out as it should. 4.4. Other “mod 2 indices”. The existence of the Weyl-invariant index i 0 ( f ) immediately evokes the question: Do there exist other mod 2, Weyl-invariant quantities? It is clear that if they do, these could be used to distinguish between the various orbits of the Weyl group, and extract information on the theory without fixing a level decomposition. It seems there are no further general indices of the type of i 0 ( f ), but a survey among specific G +++ algebras will show that some of them admit other indices. It also turns out that though they do distinguish between the orbits of the Weyl groups, they are not “fine enough” in the sense that it is not possible to tell every orbit apart from others by comparing these “indices”. The requirement of invariance under shifts by 2Q G +++ is automatically built in if we use the following ansatz for a “mod 2 index”: i( f ) = 

r 

i=−2

pi αi , f mod 2

pi ∈ Z.

(45)

Poincaré Duality and G +++ Algebras

509

The mod 2 property implies that we can restrict pi to the values 0, 1 of the additive group Z2 . The question is now which combinations of pi make i( f ) invariant under Weyl-reflections. A trivial solution consists of setting all pi = 0, and the existence of i 0 ( f ) demonstrates that there is a non-trivial solution. An easy way to construct all non-trivial indices goes as follows. One can encode a candidate i( f ) by writing the Dynkin diagram of G +++ , and label all nodes i with the number pi . The action of a Weyl reflection in the simple root labeled by node i can be computed to be wi (

 j

pjαj) =



(pj − 2

j

α j , αk  pk )α j . α j , α j 

(46)

As we will be computing modulo 2, the only property of the Cartan matrix we are interested in is whether its entries are odd or even. We can then summarize the effect of a Weyl reflection in the following prescription: The effect of wi is to add a multiple of the value of pi to the pk of the nodes adjacent (connected by lines) to i. If the line connecting the nodes i and k is single or triple, then pi has to be added to pk . If the line is a double line, then pi has to be added to pk if |αi | > |αk |, but not otherwise. In graphical language: For double lines the value of pk can only go along double lines in the direction of the arrow. Hence under Weyl reflections, each pk receives contributions from the adjacent nodes. Invariance under Weyl reflections implies choosing the pk such that all the contributions from adjacent nodes cancel out. An immediate consequence is for example that if the diagram has a chain with an endpoint (all the G +++ Dynkin diagrams have at least one such endpoint), that the node adjacent to the endpoint must have pi = 0. For chains of single lines with endpoint this implies that every second node must have pi . After these preliminary considerations it is easy to solve for the remaining pi . We should note however that if i( f ) and j ( f ) are indices in the above sense, that then also (i + j)( f ) is an index. Hence, when enumerating all the indices, we can restrict to a set of linearly independent indices. For these indices we have not built in the invariance under outer automorphisms, and indeed, for the Dn+++ algebras there exist indices that are not invariant under these, and hence not under T-duality. We will exhibit specific indices for specific G +++ algebras later. 5. G +++ Algebras: Explicit Computations We will now illustrate the general findings of the previous section, and fill in the details of their application to specific G +++ -algebras. We will recover and extend the results of [28]. First however we derive a simple result that will be highly useful later. 5.1. An index for S L(2n, R) algebras. The following result turns out to be very useful. It is essentially formulated in terms of a “mod 2” index for S L(2n, R) algebra; it is useful because in many instances one can identify a contribution of this sort to an index of a G +++ algebra. Lemma. Consider the algebra S L(2n, R), which has as its Dynkin diagram a straight line connecting 2n − 1 nodes. We number its nodes 1, . . . , 2n − 1 from one end to the

510

A. Keurentjes

other. Consider an element f ∈ Q A2n−1 and compute n  I = ω2i−1 , f  mod 2.

(47)

i=1

The generators eα −exp(iπ α, f e−α generate S O( p, 2n − p), where p is odd if I = 1, and p is even when I = 0. This can be proven quickly when we use an observation from [15]. There it was noted, that for an A D−1 = S L(D, R) algebra, with a function f written as f = ωk1 + ωk2 + . . . + ωkn

k1 > k2 > . . . > kn ,

(48)

f defines the denominator subgroup S O(k, D − k) with k described by an alternating sum of the ki , k=

n  (−)i+1 ki .

(49)

i=1

We note that when computing modulo 2, p = 2n − p. Computing (49) modulo 2, one can immediately omit the signs (−)i+1 and then also skip all the even ki . But then it is easy to see that (I = k = p) mod 2.

(50)

This lemma is useful because among the G +++ algebras there are many instances where one can identify a contribution of the form I in i 0 ( f ). If this contribution corresponds to a gravitational subalgebra, then I distinguishes between the space-time signature (even, even) and (odd, odd). These space-time signatures admit (anti-)self dual forms with n-form field strength when n mod 2 = I . More precisely if (n + I ) mod 2 = 0, then the square of the Hodge-star ∗ 2 = 1; if (n + I )mod 2 = 1 then the square of the Hodge star ∗ 2 = −1. When there are n-forms transforming in some representation of an internal symmetry group, then these representations have to be real, respectively complex, otherwise they cannot obey a relation of the form ∗ F(n) = ±F(n) .

(51)

In relevant situations, this reality or complexity of representations is implied in the i 0 ( f ) condition. 5.2. Simply-laced split G +++ algebras. We now demonstrate a number of explicit computations. We will first discuss the simply laced G +++ algebras. As the gravity sub-algebra consists entirely of long roots, which can never mix with short roots under diagram automorphisms and under Weyl reflections, these are to some extent the most representative for the T-duality pattern. Moreover, in many aspects the non-simply laced and non-split algebras can be viewed as an extension of the simply laced algebras. Correspondingly, the theories based on non-simply-laced and non-split algebras are an extension of theories based on simply laced algebras, by including extra matter sectors.

Poincaré Duality and G +++ Algebras

511

+++

A1 -2

-1

0

1

n

+++

An -2

-1

0 1

Fig. 1. Dynkin diagrams of the triple extended A-algebras

5.2.1. A+++ algebras: General Relativity in various dimensions. The A+++ algebra is n n conjectured to describe symmetries of General Relativity, formulated in D = n + 3 dimensions. The highest root of the An algebra is given by n 

αH =

αi .

(52)

i=1

The consistency criterion for the index i 0 ( f ) hence takes the explicit form i 0 ( f ) = α−2 + α0 +

n 

αi , f  mod 2 = 1.

(53)

i=1

The A+++ n -algebras with n > 1 allow a diagram automorphism, whose action consists of exchanging αi with αn+1−i for 1 ≤ i ≤ n. The index i 0 remains invariant under this automorphism. Omitting the αn node and all links to it leads to the Dynkin diagram of S L(D), which is supposed to describe the simple part of the G L(D) relevant to the vielbein. After embedding the space-time signature in this S L(D)-algebra, the i 0 condition fixes the remaining sign, so apart from the space-time signature there are no adjustable signs. For n odd (General Relativity in even dimensions), there is an extra index: (n−1)/2 

i1 ( f ) = 

α2k+1 , f  mod 2.

(54)

k=0

Its meaning is more easily described in terms of the index i 0 ( f ) + i 1 ( f ). With the αn node omitted, this index is of the form I we have used in the lemma in Subsect. 5.1. Hence we can link the values of i 0 + i 1 to the space-time signature. However, consistency requires that i 0 ( f ) = 1 always, and hence i 0 ( f ) + i 1 ( f ) = i 1 ( f ) + 1. So we have i 1 ( f ) = 1 if the space-time signature (t, s) = (even, even), and i 1 ( f ) = 0 for (t, s) = (odd, odd). We can see immediately that 4-dimensional General Relativity, governed by A+++ 1 does not allow signature changing dualities. This is proven in two steps: one first notes that Weyl reflections in αi (i = 1, 2, 3) do not affect the space-time signature (as they only permute directions); the Weyl reflection in α4 can alter the f -function by integer

512

A. Keurentjes

multiples of 2α4 ∈ 2Q(A+++ 1 ), but this does not affect the space-time signature, which is a mod 2 computation. The impossibility to perform a time-like T-duality could also have been anticipated from our physical interpretation: For general relativity T-duality has to be generated by Kaluza-Klein-monopoles. But for 4-dimensional General Relativity these are not extended objects, so there cannot be any relation via “string-dualities”. For higher dimensional General Relativity, we can immediately check time-like dualities by embedding the space-time signature in an appropriate way. We start with considering the signature (0, D). The requirement i 0 ( f ) = 1 implies that the right real form of the K (A+++ n ) algebra must be a non-compact form given by: f = ωn .

(55)

Weyl reflecting in αn and reducing modulo 2 results in f = ω0 + ωn−1 + ωn .

(56)

This is easily seen to represent signature (D − 4, 4), and hence reproduces the naive argument of Sect. 2. It is amusing to note that this computation even applies for D = 4; the formulas are predicting that this theory does not transform under timelike T-duality. The signature ( p, D − p) can be implemented by (if 1 < p ≤ n − 1 = D − 4) f = ω0 + ω p + ωn .

(57)

f = ω p + ωn−1 + ωn .

(58)

Weyl reflecting in αn leads to

This represents signature (D − 4 − p, p + 4), which agrees with the derivation in Sect. 2. Note that this duality becomes impossible if p > D − 4. If 2 ≤ p ≤ D − 2 = n + 1 we can also embed the ( p, D − p) by using f = ω−2 + ω p−2 + ωn .

(59)

Now the Weyl reflection in αn leads to f = ω−2 + ω0 + ω p−2 + ωn−1 + ωn

(60)

which represents space-time signature (D − p, p). Collecting all the possible orbits under time-like T-duality, we obtain the following table, which is valid for D ≥ 7: D mod 4 0

1 2

3

(t mod 4, s mod 4) (0, 0) (1, 3) ↔ (3, 1) (2, 2) (0, 1) ↔ (1, 0) (2, 3) ↔ (3, 2) (0, 2) ↔ (2, 0) (1, 1) (3, 3) (0, 3) ↔ (3, 0) (1, 2) ↔ (2, 1)

(61)

Poincaré Duality and G +++ Algebras

513

+++

D4

3 -2

-1

0

1

2

4

+++

n

-2

-1

0

n-1

n-3

1

Dn n-2

Fig. 2. Dynkin diagrams of the triple extended D-algebras

We have required D ≥ 7 because for D = 4, 5, 6 some of these orbits break up due to the absence of the relevant duality operation. For D = 4 timelike T-dualities are impossible, as explained previously. For the dimensions D = 5, 6 the orbits are D 5

6

(t, s) (0, 5) ↔ (1, 4) (2, 3) ↔ (3, 2) (4, 1) ↔ (5, 0) (0, 6) ↔ (4, 2) ↔ (2, 4) ↔ (6, 0) (1, 5) (3, 3) (5, 1)

(62)

These orbits (some of which can also be found in [28]) of the gravitational algebra are of importance because any G +++ algebra has a A+++ algebra as subalgebra. For any of n these theories the signatures listed here should be connected by duality transformations. The only issue may be whether some of these orbits may get connected due to additional symmetries. 5.2.2. Dn+++ -algebras: String theories in various dimensions. The Dn+++ algebras (with n ≥ 4) are conjectured to represent bosonic string theory in D = n + 2 dimensions. The highest root of a Dn algebra reads αH =

n−3 

2αi

+ αn−2 + αn−1 + αn .

(63)

i=1

The i 0 criterion reads: i 0 ( f ) = α−2 + α0 + αn−2 + αn−1 + αn , f  mod 2 = 1.

(64)

The Dn+++ algebras always have multiple indices. There is always the second index i 1 ( f ) = αn−2 + αn−1 , f  mod 2.

(65)

514

A. Keurentjes

Moreover for n > 4 and even there is a third independent index n/2−2 

i2 ( f ) = 

α2k + αn−1 + αn , f  mod 2.

(66)

k=1

The Dn+++ algebras with n > 4 allow a diagram automorphism that is inherited from the Dn algebra. It exchanges the nodes αn−1 and αn−2 . Hence it leaves i 0 and i 1 invariant, but it exchanges i 2 with i 1 + i 2 . The D4+++ algebra has a highly symmetric diagram, and has apart from i 0 ( f ) the indices i 1 ( f ) = α2 + α3 , f 

i 2 ( f ) = α2 + α4 , f  mod 2.

(67)

The algebra D4+++ allows an even bigger set of automorphisms, inherited from the triality of D4 : Any permutation of α2 , α3 and α4 corresponds to an admissible automorphism. All of these leave i 0 invariant, but they permute i 1 , i 2 and i 1 + i 2 . In the theory with maximal dimension D = n +2, the S L(D) chain representing gravity is formed by the nodes α−2 , . . . , αn−2 . It is also possible to form a D-dimensional theory by instead using the chain α−2 , . . . , αn−3 , αn−1 . This formulation is related to the previous one by an outer automorphism of Dn+++ , or equivalently, the two are related by a T-duality. Whichever way we choose to embed the chain, there are two nodes not included. For concreteness, we choose these to be αn−1 and αn . The i 0 ( f ) condition fixes the product of their signs, but we can choose one sign independently. Physically, these nodes correspond to a 2-form in the maximal dimension (αn−1 ), and a D − 4-form (αn ), that is the Poincaré dual to the 2-form. Poincaré duality tells us their signs should be related, and indeed, so does its algebraic counterpart, the i 0 condition. The index i 1 ( f ) incorporates the sign of the 2-form, but not of the (D − 4)-form, and hence is an interesting quantity if we wish to keep track of the signs. If we specify the space-time signature and i 1 ( f ) the signs of the theory are essentially fixed. We note also that the roots corresponding to the “dual of the graviton” are at higher levels, but that essentially its sign is the product of the signs of the 2-form with the sign of the (D − 4)-form. In Euclidean signature (0,D) we have the two possibilities f = ωn−1 ↔ i 1 ( f ) = 1

or

f = ωn ↔ i 1 ( f ) = 0.

(68)

The option f = ωn−1 allows a Weyl-reflection in the αn−1 (string-)node, resulting in the signature (2, D − 2). Weyl reflections in the node αn (the (D − 5)-brane node) have no signature changing effect. Note that with these conventions the 2-form kinetic term has a minus sign, and hence allows extremal string solutions of world volume signature (0, 2). That implies however that the (D − 4)-form field term has the conventional sign, so a (D − 5) brane must have an odd number of time directions on its world volume, which is impossible. For the choice f = ωn the roles are reversed: Now the kinetic term for the 2-form has a plus-sign and the (D −4)-form a minus-sign, implying the existence of a (D −5)-brane but absence of a string. Now a signature changing duality induced by a Weyl reflection in the node αn is possible, resulting in the signature (D − 4, 4). We now turn to the signatures ( p, D − p) 1 ≤ p ≤ D − 3 = n − 4, and with i 1 ( f ) = 0. We set f = ωn−3 + ωn−2 + ωn−1 ,

(69)

Poincaré Duality and G +++ Algebras

515

for p = 1, and f = ωn−2− p + ωn−3 + ωn−2 + ωn−1 + qωn ,

(70)

for 1 < p ≤ n − 4 = D − 3, where q can be adjusted to make i 0 ( f ) = 1. Weyl reflecting in the string node αn−1 amounts to the disappearing of ωn−3 from the above expressions. This however does not have any effect on the signature. This is because the relevant theories all have a conventional sign in front of the 2-form, implying a string with world-volume of signature (1, 1). The rules we outlined in Sect. 2 then imply that the overall space-time signature cannot change under reflections in these nodes. Note also that i 1 ( f ) = 0 implies the same for the T-dual theory, the one where we interchange the roles of αn−1 and αn−2 . The theory with p = 1 has the standard Minkowski signature, and here we can generate time-like T-dualities only with the 5-brane node (and hence time-like T-duality must be invisible in CFT [36]). To study duality in the 5-brane node we set (1 ≤ p < n − 4 = D − 6) f = ω1 + ω p+1 + ωn .

(71)

These theories also have signature ( p, D − p) and i 1 ( f ) = 1, and it is not hard to show that they are connected to the previous theories by permutation of the coordinates (Weyl reflections in nodes other than αn−1 and αn ). A Weyl reflection in αn will result in f = ω p+1 + ωn

(72)

representing signature (D − p −4, p +4). This is generated by a reflection corresponding to a (D − 4)-brane with world volume signature ( p, D − p − 4): ( p, D − p) = ( p, D − p − 4) + (0, 4) → (D − p − 4, p) + (0, 4) = (D − p − 4, p + 4). (73) The (D − 4)-brane solutions with these world volume signatures ought to exist provided the sign in front of the corresponding kinetic term is (−) p+1 , which of course perfectly correlates with our claim that the 2-form terms had a plus-sign for their kinetic terms. The other subcases correspond to i 1 ( f ) = 1. To study duality in the node (n − 1) we set f = ωn−3− p + ωn−3 + ωn−1 + qωn ,

(74)

where again q should be set to 0 or 1 to fulfill the i 0 -condition. These represent spacetime signatures ( p, D − p) with 1 ≤ p ≤ n − 1 = D − 3. A Weyl reflection in the node αn−1 leads to f = ωn−3− p + ωn−1 + qωn ,

(75)

which represents the space-time signature ( p + 2, D − p + 2). This is consistent with the interpretation of the theory as one with a wrong signed 2-form term, which implies string solutions with world-volume signature (2, 0) and (0, 2). To study duality in the (D − 5)-brane node when i 1 ( f ) = 1 we set f = ω0 + ω p + ωn−1 + ωn

(76)

516

A. Keurentjes

-2

-1

0

1

5

6

3

4

E 6+++

2

E 7+++

7

-2

-1

0

1

2

3

4

5

6

E8+++

8

-2

-1

0

1

2

3

4

5

6

7

Fig. 3. Dynkin diagrams of the triple extended E-algebras

with 1 ≤ p ≤ n − 3 = D − 5. This represents signature ( p, D − p). Reflecting in αn leads to f = ω0 + ω1 + ω p + ωn−1 + ωn

(77)

which represents the signature (D − p − 2, p + 2). This implies there is a (D − 5)-brane with world volume signature ( p − 1, D − p − 3), meaning that the sign for the kinetic terms of the (D − 6)-form is (−) p , perfectly consistent with our claim that the theories with i 1 ( f ) have a wrong signed kinetic term for the 2-form. There are many other signature changes possible, that we will not exhibit explicitly. The theories conjectured to be described by Dn+++ algebras allow another interesting oxidation branch, leading to 6 dimensional theories (see [7]. The gravitational subalgebra is now the one corresponding to the nodes α−2 , α−1 , α0 , α1 and αn . The space-time signature is then governed by i 0 ( f ) + i 1 ( f ), if this is 1 then the signature is (odd, odd), if it is 0 then the signature is (even, even). With the constraint i 0 ( f ) = 1, this implies that the space-time signature fixes i 1 ( f ). The reason for this is that the theory admits 3-form field strengths that should be subjected to a (anti-)self-duality constraint. For 6 dimensional space-time signature (even, even) the eigenvalues of the Hodge star are imaginary, and i 1 ( f ) = 1. One can verify, using the techniques of [15] (see also the appendix of [15]), that this implies that the 6 dimensional coset symmetry is S O(n−3, n−3)/S O(n−3, C), such that the representation of the 3-forms are indeed complex valued. In contrast, when i 1 ( f ) = 0, and the space-time signature is (odd, odd), the coset is of the form S O(n − 3, n − 3)/S O(n − 3 − p, p) × S O(n − 3 − p, p), and the representations to which the 3-forms belong is real. 5.2.3. E 6+++ : Membranes in 8 dimensions. To the E 6+++ -algebra corresponds an 8 dimensional theory that has a doublet of 3-forms. This doublet transforms under S L(2)/S O(2) or S L(2)/S O(1, 1), depending on the choice of various signs. The highest root of the E 6 algebra is α H = 2α1 + 3α2 + 2α3 + α4 + 2α5 + α6 .

(78)

The E 6+++ -algebra only has one index i 0 ( f ) = α−2 + α0 + α2 + α4 + α6 , f  mod 2 = 1.

(79)

Poincaré Duality and G +++ Algebras

517

The i 0 -condition then tells us that if the signature of the 8-dimensional space-time is (odd, odd), that the coset must be of the form S L(2), R)/S O(2), while when the signature is (even, even) the coset has to be S L(2)/S O(1, 1). This correlates perfectly with Poincaré duality: In signature (odd, odd) the eigenvalues of the duality operation are ±i, and the 3-form and its dual combine in a complex combination, rotated into one another by S O(2) ∼ = U (1). In signature (even, even) however the eigenvalues of Hodge-star are ±1, which is consistent with the fact that the S O(1, 1) algebra is real. Note that according to our rules, a doublet of 3-forms in 8 dimensions gives rise to a doublet of membranes, which gives rise to a dual theory with a doublet of membranes. Of course the dual theory is related to the original one by the outer automorphism of E 6+++ . The analysis of the dual theory is identical to that for the original theory, due to the fact that i 0 ( f ) is invariant. Once we have chosen a space-time signature, the i 0 condition tells us whether the 8-dimensional coset is S L(2, R)/S O(2) or S L(2, R)/S O(1, 1). If it is S L(2, R)/S O(2), we still have some freedom in choosing consistent signs. The 8-dimensional theory has a doublet of 3-forms (coupling to membranes), and the kinetic term for this doublet can have either a plus or minus sign. This sign is encoded in a contribution of ω5 to f . In case the coset is S L(2, R)/S O(1, 1) the doublet has 1 plus and 1 minus sign, and the relative sign can be changed by a S L(2, R)/S O(1, 1) transformation, so there are no additional signs to be adjusted. A detailed analysis of the possible signature and how they connect under duality transformations/Weyl reflections gives the following two orbits: (1, 7, +), (2, 6), (3, 5, +), (5, 3, +), (6, 2), (7, 1, +),

(80)

(0, 8), (1, 7, −), (3, 5, −), (4, 4), (5, 3, −), (7, 1, −), (8, 0).

(81)

Because only in the case where the space-time signature is (odd, odd) there are unspecified signs, we have added to these signatures a + or − indicating whether the 3-form kinetic terms come with conventional (+) or unconventional sign. The first of these orbits is also discussed in [28], where another notation for the signs is used. 5.2.4. E 7+++ : 3-branes in 10-dimensions; or strings in 8-dimensions. The highest root of the E 7 algebra is given by α H = 2α1 + 3α2 + 4α3 + 3α4 + 2α5 + α6 + 2α7 .

(82)

The i 0 ( f ) index for E 7+++ then reads i 0 ( f ) = α−2 + α0 + α2 + α4 + α6 , f  mod 2 = 1.

(83)

A second index is given by i 1 ( f ) = α4 + α6 + α7 , f  mod 2.

(84)

The E 7+++ -algebra leads to a 10-dimensional theory with a self-dual 4-form3 . Selfduality is only consistent with signature (odd, odd), but then any sign for the kinetic 3 In many references this theory is not oxidized beyond 9 dimensions, but the existence of the 10-dimensional version is well-known [6, 7]. A simple way to construct the theory is by starting with the bosonic sector of type IIB theory, and then omitting the scalar sector and truncating further to the fields that are singlets under S O(2)/S O(1, 1).

518

A. Keurentjes

terms for the 3-form is possible. This is indeed exactly the content of the i 0 -condition for this theory. Once we fix the space-time signature, the sign of the 4-form kinetic term is encoded in i 1 . From this we can learn that once the sign is fixed to plus or minus, there does not exist a time-like T-duality to a 10 dimensional theory where the 4-form kinetic term has opposite sign. The 3-brane coupling to the 4-form in 10 dimensions dualizes to a triplet of strings coupling to 2-forms in 8 dimensions. Indeed, by choosing the S L(8)-chain running from the α−2 node to the α7 node, we find this second, dual theory. The indices for this theory are easier discussed if we first look at the index i 0 + i 1 . It is easy to see that if i 0 + i 1 = 1 (which, because i 0 = 1 corresponds to i 1 = 0), this corresponds to space-time signature (odd, odd) for the 8-dimensional theory, while i 0 + i 1 = 0 (i 1 = 1) corresponds to space-time signature (even, even). The 10-dimensional theories are connected in the following way under time-like T-dualities. We indicate a − if the value of i 1 = 1 and a + if i 1 = 0. Then the orbits are (3, 7, −) ↔ (7, 3, −),

(85)

(1, 9, −) ↔ (5, 5, −) ↔ (9, 1, −).

(86)

All of these theories have 8-dimensional duals that have space-time signature (even, even). The more conventional theories have i 1 ( f ) = 0, and can all be dualized into each other, that is the signatures (1, 9, +), (3, 7, +), (5, 5, +), (7, 3, +), (9, 1, +),

(87)

are all connected. Because of i 1 ( f ) = 0, these must dualize to 8-dimensional theories with space-time signature (odd, odd). 5.2.5. E 8+++ : M-theory and IIB-string theory. In [26] the possible signatures and orbits for the M-theory interpretation of the E 8+++ algebras were listed. These remain correct, but we are arguing in the present paper that there is an extra consistency condition. As the highest root for the E 8 algebra is α H = 2α1 + 3α2 + 4α3 + 5α4 + 6α5 + 4α6 + 2α7 + 3α8 ,

(88)

this extra condition takes the following form for E 8+++ : i 0 ( f ) = α−2 + α0 + α2 + α4 + α8 , f  mod 2 = 1.

(89)

This implies that some of the generalized signatures in [26] are inconsistent with Poincaré duality. The remaining ones organize in the orbits with signatures (note that the i 0 ( f )- condition fixes the sign of the 3-form to (−)T −1 , where T is the number of time directions!) (1, 10); (2, 9); (5, 6); (6, 5); (9, 2); (10, 1),

(90)

(0, 11); (3, 8); (4, 7); (7, 4); (8, 3); (11, 0).

(91)

and

The first orbit contains the standard signatures of the M, M  and M ∗ theories [11].

Poincaré Duality and G +++ Algebras

519 +++

Bn

n

-2

-1

0

1

n-2

n-1

+++

Cn

-2

-1

0

n-1

1

n

+++

F4 -2

-1

0

1

2

3

4

Fig. 4. Dynkin diagrams for the triple extended 2-laced algebras

The second orbit contains precisely the remaining signatures. Though it is standard lore that these 11-dimensional theories are non-supersymmetric, some of the 10 dimensional theories included here seem to have supersymmetric extensions [37]. The various signatures for the IIB theories can be derived straightforwardly. Note that the i 0 ( f ) condition states that the space-time signature for a IIB-theory or a variant must be (odd, odd). Variants of the IIB-theory cannot occur in signatures (even, even) because the 4-form whose field strength will be (anti-)self dual is a singlet under the S L(2) coset symmetries; we cannot tune its duality properties. The orbit with M-theories contains all the signatures and sign patterns of the I I B, I I B  and I I B ∗ theories [10, 11], the other orbit contains the signatures (1, 9, −, −), (3, 7, +, +), (3, 7, +, −), (5, 5, +, +), (7, 3, +, +), (7, 3, +, −), and (9, 1, −, −) (in the notation of [15]). 5.3. Non-simply laced and non-split G +++ -algebras. 5.3.1. Non-simply laced split. The non-simply laced algebras are distinguished by the appearance of short roots. Short roots can never be exchanged with the gravity sector (as these correspond to long roots). For the space-time signature these are hence of no interest. Signs for short roots are possible, but these just represent wrong signs in the matter sector. The time-like T-duality pattern follows in principle from the long roots. Of importance is therefore the largest simply laced algebra inside the non-simply laced algebras. These are given by Bn+++ ⊃ Dn+++

F4+++ ⊃ D4+++ .

(92)

The long roots of Cn+++ are in some sense given by a “triple extension of An1 ”. This algebra however does not allow a Dynkin diagram4 (curious readers may try n = 2), 4 The same is true for any non-simple group, which is probably one of the reasons why these theories have not been considered much so far. Note that it is completely straightforward to define theories with non-simple groups in 3-dimensions; it is their “triple extension” that is hard to define.

520

A. Keurentjes

and the easiest way to describe the corresponding algebra seems to be as a truncation of the Cn+++ algebra, so this does not have many advantages. As the highest root for the Bn algebra is n−1

 αH = 2αi + αn , (93) i=1

the index i 0 ( f ) for the Bn+++ algebra is given by i 0 ( f ) = α−2 + α0 + αn , f  mod 2 = 1.

(94)

A second index is given by i 1 ( f ) = αn−1 , f  mod 2.

(95)

This is invariant due to the 2-laced nature of the algebra. The theory allows two maximally oxidized variants. There is a D = n+2-dimensional theory, which contains General Relativity, a vector and its dual D −3-form, and a 2-form and its dual (D − 4)-form. The sign of the kinetic term for the 2-form cannot be chosen freely; the algebra requires the sign to be the square of the sign for the kinetic term of the vector, and hence this is always +. Because of this, the i 0 ( f )-condition fixes the sign of the (D − 3)-form once the space-time signature is embedded in the chain from α−2 to αn−2 . None of these considerations fixes the sign of the vector; this is encoded in a separate index i 1 ( f ). These theories can easily be seen, both from the algebraic and from the physical perspective, to have the same time-like T-duality pattern as the theories with Dn+++ algebra, and i 1 ( f ) = 1 (because the two-form sign is fixed!), so we refer the reader to the discussion there. A second branch for this theory gives a theory in 6 dimensions, with (2n − 5) 2-forms satisfying (anti-)self duality constraints. These transform in the vector representation of a real form of Bn−3 . The i 0 ( f )-condition implies that the signature of the 6-dimensional theory is (odd, odd). The (even, even) signatures cannot be realized; these would require complex representations for the 2-forms, but the relevant representations (the vector representations (n − 2, 1) ⊕ (1, n − 3) of some real form of S O(n −2)× S O(n −3)) are always real regardless of the choice of f . For the Cn algebras the highest root is n−1

 αH = 2αi + αn . (96) i=1

For the Cn+++ algebra the index i 0 ( f ) is given by i 0 ( f ) = α−2 + α0 + αn , f  mod 2 = 1.

(97)

For n even a second index reads n/2  i1 ( f ) =  α2i−1 , f  mod 2.

(98)

i=1

Associated to these algebras are 4 dimensional theories of General Relativity, coupled to a sigma model on Sp(n − 1, R)/H and a bunch of vectors and their duals transforming

Poincaré Duality and G +++ Algebras

521

+++

G2 -2

-1

0

1

2

Fig. 5. Dynkin diagram of the triple extended 3-laced algebras

in the fundamental 2n-dimensional representation. The denominator subgroup is a real form of G L(n −1, C). It can be G L(n, R), or U (n − p, p) for some value p (see Appendix A of [15]). We immediately note that G L(n, R) is a group over the reals, while the representations of U (n − p, p) are typically complex. The i 0 ( f ) condition relates the possible real form of G L(n, C) to the space-time signature. To be precise, for space-time signature (even, even) the denominator subgroup in 4 dimensions must be a real group (because of duality requirements), and it can be shown that the i 0 ( f ) condition implies it is G L(n, R). For space-time signature (odd, odd) however, the i 0 ( f )-condition, as well as duality requirements state that H must be complex (and hence, of the form U (n − p, p). The index i 1 is correlated to the fact that we can change the sign of the vectors in 4 dimensions; only for Cn+++ theories with n even this leads to a conserved quantity. The time-like T-duality pattern is rather boring, because it is the same as for A+++ 1 (4-dimensional general relativity) that does not allow any time-like T-dualities. For F4 the highest root is α H = 2α1 + 3α2 + 4α3 + 2α4 .

(99)

For F4+++ the index i 0 ( f ) is given by i 0 ( f ) = α−2 + α0 + α2 , f  mod 2 = 1.

(100)

This algebra gives rise to a 6-dimensional theory with S L(2, R) symmetry, with vectors (and their dual 3-forms) transforming in the 2 and 2-tensors in the 3. The index i 0 ( f ) implies that the 6-dimensional theory has signature (odd, odd). The (even, even) signatures are again impossible, because the sign of the 2-forms is again the square of the vectors, hence +. It is easy to establish that it is impossible to change the space-time signature. As we have argued the time-like T-duality pattern is the same as for D4+++ ; we however have to take into account the extra restriction that the space-time signature is fixed to (odd, odd), which renders any signature change impossible. Note that signs from the gravity and fields described by long roots can run into the matter sector described by the short roots, which does allow sign changes in the matter sector. It is the opposite direction that is impossible. There is only one 3-laced algebra among the triple extended algebras, which is G +++ 2 . The highest root of G 2 is α H = 2α1 + 3α2 .

(101)

i 0 ( f ) = α−2 + α0 + α2 , f  mod 2 = 1.

(102)

The index condition is

The oxidation endpoint of this theory is 5 dimensional Einstein-Maxwell gravity. The i 0 condition implies that the sign of the vector in 5 dimensions is determined by the space-time signature. This is because the dual 2-form is at level 2, and hence its sign

522

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cannot be adjusted. The i 0 condition states essentially that the sign of the vector terms is in accordance with the (fixed) sign of its dual. With this proviso, every space-time signature is possible. A computation shows that the orbits under duality are essentially the same as those for 5 dimensional gravity. 5.3.2. Non-split groups. One advantage of the derivation of the criterion i 0 ( f ) is that it is clear that it immediately generalizes to theories that are not based on split real forms of the algebra G (for a survey that encompasses all the simple algebras that are not split, and the corresponding theories see [8]). We can allow G to be any real form as long as it is non-compact. One starts from a coset sigma model in 3 dimensions on G/H (we will specify the possible H in a moment), coupled to 3-dimensional general relativity. The obvious conjecture is that such theories fit into the framework of G +++ algebras by extending their “Tits-Satake diagram” (see e.g. [31]) by three more nodes. A Tits-Satake diagram is nothing but a Dynkin diagram with decoration, so we extend the Tits-Satake diagram as we would for the Dynkin diagram, while leaving the extra nodes without decoration5 (in [8] we already defined (once) extended Tits-Satake diagrams). Also here we could specify the signature in a function f , in the same way as we have done in [26]. The only thing we wish to avoid is that f takes non-trivial values on the invariant subspace of the Cartan-involution. If we, as in [15] inscribe the values of f on the Tits-Satake diagram, it is clear that we should only allow f to have non-trivial values at white nodes, that are not decorated by arrows. As long as there exists a 3-dimensional theory, our derivation of Sect. 4 remains valid; we never used any information about the real form of G there. So also in this case we should have i 0 ( f ) = 1. We will not attempt any detailed analysis for this (vast) category of theories. 6. Discussion and Conclusions In this final section we describe, before reaching our conclusions, a number of issues that should provide the reader with food for thought, and possibly directions for future research. 6.1. The meaning of the i 0 ( f )-condition. We have derived and presented the i 0 ( f ) condition as a necessary condition to implement the action of Poincaré duality on G +++ algebras in the correct way. Though of course the relation to Poincaré duality is sufficient to highlight the importance of i 0 ( f ), there are further questions that can and should be posed. One observation is that the requirement of Poincaré duality is in a sense “external”; we have presented it as an argument related to the interpretation of the algebra as that of a physical theory. There it is rooted in the calculus of differential forms, which does not seem to lead to a direct relation with the G +++ algebra. Also the fact that we have derived it for a particular decomposition, related to 3-dimensional theories and then relied on physical arguments to extrapolate to all theories that can be reconstructed from 5 It is conceivable that the extra nodes could be decorated. But then an extension of the analysis of [8] combined with the usual framework of G +++ algebras (extraction of matter content via level decomposition, etc.) would lead us to conclude that these theories cannot be oxidized to theories in dimension 3 or higher. This hypothetical class of theories would hence be poorly understood both from the mathematical and the physical side.

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G +++ algebras is somewhat dissatisfying. One would like to have a more “internal” explanation, where the condition would follow as a consistency requirement from some framework to set up the theory directly from the algebra. The only such framework with manifest G +++ -symmetry that has been proposed thus far [23] does not really seem to require the i 0 ( f ) condition from the outset, and one may (and probably should) wonder whether the condition is hidden somewhere at a deeper level of the interpretation. Another interesting issue is what the precise mathematical meaning of the i 0 ( f ) condition is: What exactly is the distinguishing feature that separates K (G +++ ) defined by an f with i 0 ( f ) = 1 from those with an i 0 ( f ) = 0? In the application to theories with (anti-)self-dual forms, the i 0 ( f ) condition ensures that the eigenvalues of the Hodgestar, and in particular whether they are real or imaginary, can be properly represented on the K (G +++ ) algebra. This leads us to suspect that the i 0 ( f ) condition may be some sort of “reality condition” on K (G +++ ). Whether this is the case could be verified if we knew the representation theory for K (G +++ ), but at present too little is known about these algebras. The cosets G +++ /K (G +++ ) hypothetically describe a theory which originally had a finite number of degrees of freedom, by an infinite number of variables. It is an open question whether this should be interpreted as an indication for new degrees of freedom, or that the finite number of degrees of freedom should arise after imposing (infinitely many ?) constraints (as in proposals that suggest that the higher level fields contain the derivatives of lower level fields [33, 34, 23]). It is also possible that we may both need constraints and new degrees of freedom. Most interpretations agree at least on the fact that both forms and dual forms are independently represented by G +++ algebras. This indicates that the i 0 ( f )-condition either is one of the constraints necessary to reduce the number of degrees of freedom, or is implied by such constraints. Therefore we expect that it should play an essential role in the formulation of the theory with manifest G +++ /K (G +++ ) symmetry. Yet another interesting issue that should be related to the i 0 ( f )-condition is the ability to couple the theories to spinors. It was observed in [40] that these should transform in representation of K (G +++ ). This has not been made very precise thus far (but see the constructions of [39, 38] relating to infinite-dimensional sub-algebras of K (G +++ ), and the attempt of [41] to implement supersymmetry in some G +++ -algebras). Here we note the close relation of Poincaré duality to the properties of the Levi-Civita anti-symmetric tensor, and its important consequences for the representation theory of spinors, in particular their chirality properties. We therefore suspect that the i 0 ( f ) condition is relevant in this context. If the i 0 ( f ) condition is indeed some sort of reality condition on K (G +++ ), as we have conjectured, then also the existence of (pseudo-)Majorana spinors in the low-energy effective theory may be encoded by it. Needless to say, all these ingredients should be crucial for the possible supersymmetrization of some G +++ -theories, most notably for the M-theory algebra E 8+++ . 6.2. Brane solutions. There have been various attempts at relating p-brane solutions to G +++ -theories [23, 42, 43, 45, 44, 29]. Typically one postulates (extremal) brane solutions for positive (real) roots α of G +++ . Of course not every root can correspond to a genuine brane solution, for example in Minkowski signature extremal branes should have the time-like direction contained in their world-volume. In the literature also S-branes are discussed [29]. The real solutions corresponding to these cannot be extremal and hence are excluded from the category of solutions we wish to discuss. There exist so-called extremal S-branes that are essentially analytic continuations of the former real solutions, with imaginary charges. For this we would

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have to allow for complex fluxes; this would lead to a (partial) analytic continuation of the algebra, and therefore to a relaxation of the strict conditions we have required for the representation of Poincaré duality and the real form of the K (G +++ )-algebras. The restrictions on the world volume signature of extremal branes found in [12] can be neatly encoded in a simple constraint. It is easily seen that to a positive root there can be associated a genuine extremal brane solution if and only if α, f  mod 2 = 1.

(103)

In essence this equation guarantees the right number of minus-signs in the equation of motion. This simple equation covers both the minus signs coming from the space-time signature, and possible explicit signs in the kinetic terms in the Lagrangian. Because it is formulated in terms of the invariant bilinear form, it transforms appropriately under Weyl group transformations and outer automorphisms (and hence under T- and S-dualities): If the root α gives a brane solution to a theory with generalized space-time signature encoded in f , then w(α) gives a brane solution to a theory with space-time signature encoded in w( f ), where w is an element of the Weyl group or an outer automorphism. It is tempting to interpret α, f  mod 2 = 0 as a condition for “instantonic”, or S-branes. Strictly speaking however, the standard ansatz, with i 0 ( f ) = 1 and α, f  mod 2 = 0 does not lead to a solution. It was observed in [29] that S-brane solutions can be interpreted as solutions of theories in different space-time signatures. In the present language this amounts to finding, for a given root α, an element f of the coweight lattice that obeys i 0 ( f ) = 1 and α, f mod 2 = 1 simultaneously. There are in general many solutions to these equations, defining just as many “Wick rotations” of the original theory. In view of this non-uniqueness of the Wick rotation, it is not quite clear what this continuation exactly means. 6.3. The stability of Kaluza-Klein-theories. It has been argued by Witten [46] that Kaluza-Klein space-times are unstable. The original example studies the theory arising from compactifying 5 dimensional General Relativity on a circle. It is argued that this space-time decays by the nucleation of “bubbles of nothing”, with a decay time proportional to the radius of the circle. The “exotic dualities” of [19] and the suggestive analysis of our Sect. 2 seem to suggest that this theory may allow a dual description, essentially as General Relativity on a circle of inverse radius. There is clearly some tension between this T-duality, and the stability analysis: It seems impossible to reconcile a decay time proportional to the radius of the circle, with a dual description which essentially places the theory on a circle with inverse radius. Of course we should not jump to such conclusions. The analysis of [46] is only semi-classical and valid in the limit of large radius, and should not be näívely extrapolated to small radius. The arguments of Sect. 2 follow from a reckless extrapolation of semi-classical results, which also has questionable aspects. Note that in the hypothetical T-duality, in the limit of small circle size, the KaluzaKlein monopoles play a role similar to that of D0 branes in IIA-theory. Their charges organize to give the spectrum of a Kaluza-Klein theory compactified on a dual circle, that decompactifies when the radius of the original circle goes to zero. We cannot rely on supersymmetry to claim that we have actually any control in the regime of small circle size; this qualitative argument suggests however that the original space-time may no longer be the appropriate reference structure for such small circle sizes, because of the possible existence of a dual space-time which is non-geometrical in the original variables. It is clear that this would lead to novel non-perturbative phenomena that have

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played no role in the analysis of the paper [46], and may offer new perspectives on avoiding its conclusions. It is furthermore amusing to note that the paper [28] and the present paper seem to suggest that the two descriptions used in [46], the 5-dimensional Euclidean gravity on a circle and the 5-dimensional Minkowski gravity on a circle, are not only analytic continuations of each other, but actually dual space-times related by a T-duality. 6.4. Ghosts in Euclidean theories. One of the main motivations for turning to theories in Euclidean signature has always been that the path-integral, and various quantities are better behaved in such signatures. In terms of non-linear realizations this can be traced back to the compactness of the Lorentz-group in these cases. For G +++ algebras, we have however argued that compact denominator sub-algebras are inconsistent with Poincaré duality. We therefore either have to sacrifice the possibility of compact K (G +++ ), or the standard implementation of Poincaré duality. There seems to be a sense in which minus-signs just cannot be avoided in a hypothetical formulation of the theory with G +++ /K (G +++ ) symmetries. This raises interesting questions on the existence of no-ghost theorems and the elimination of non-physical degrees of freedom. Lacking a detailed formalism, it seems far too premature to discuss such issues, but it is interesting to note that, in the context of G +++ algebras we eventually have to face these sign-problems, as dealing with them by analytic continuation leads at best to ambiguities. 6.5. Conclusions. Theories that exhibit enhanced, hidden symmetries upon reduction, allow oxidations to different higher dimensional theories. The least one can say about G +++ algebras is that they form a bookkeeping device that correctly records all the possible reductions and oxidations. To correctly reproduce the space-time signatures and explicit signs for these theories requires that Poincaré duality is implemented correctly on these algebras. We have argued in the present paper that an element f of the coweight lattice Q(G +++ ) correctly encodes a space-time signature and sign-pattern consistent with Poincaré duality, if and only if it satisfies the condition i 0 ( f ) = 1.

(104)

Perhaps we should stress that this is not a conjecture: In spite of the fact that G +++ algebras are far from well understood, the analysis we presented in Sect. 5 reproduces all the results of Sect. 2 which involved only established techniques. Of course the far more ambitious conjecture, that there should exist a formulation of theories with (non-linearly realized, or linearized) G +++ symmetries, would imply that the ambiguities in reduction and oxidation correspond to physically equivalent formulations of the theories. All these formulations would then be related by “exotic T- and S-dualities”. Even theories of General Relativity have the appropriate structure to accommodate such exotic symmetries, and they are suggestive of interesting non-perturbative dynamics. Better still, a hypothetical non-perturbative extension of such a theory has desirable properties, namely the ability to avoid certain classes of singularities (vanishing circles/cycles). This property is usually advertised as a “good” property of string theories; trying to elevate it to a physical principle would place the theories with enhanced symmetries that inspired triple-extended algebras in a privileged position. It would be most interesting to check these hypothetical dualities in more general situations, but it seems that the technology to do so has not yet been developed. Studying infinite-dimensional G +++ -algebras and the corresponding theories by level expansions gives a direct and intuitive link to physics, but is necessarily limited to a

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finite number of levels, even with powerful computer algorithms. Information that can be encoded in an abstract fashion, such as (104) that restricts the signs to the ones consistent with Poincaré dualities, and the condition for the existence of extremal branes (103), does not refer to a specific theory or decomposition. We hope that more such abstract, duality invariant tools will be developed; these should provide us with insights that reach beyond any expansion to finite level. Acknowledgements. I would like to thank Sophie de Buyl, Laurent Houart and Nassiba Tabti, for sharing and discussing the results of [28] with me before publication, Axel Kleinschmidt for discussions, and Paul Cook and Peter West for correspondence. The author is a post-doctoral researcher for the “FWO-Vlaanderen”. This work was supported in part by the “FWO-Vlaanderen” through project G.0034.02, in part by the Belgian Federal Science Policy Office through the Interuniversity Attraction Pole P5/27 and in part by the European Commission FP6 RTN programme MRTN-CT-2004-005104.

References 1. Polchinski, J.: String theory. Vol. 1: An introduction to the bosonic string; String theory. Vol. 2: Superstring theory and beyond, Cambridge: Cambridge University Press, 1998; A. Giveon, M. Porrati, E. Rabinovici: Target space duality in string theory. Phys. Rept. 244, 77 (1994) 2. Cremmer, E., Julia, B.: The N=8 supergravity theory. 1. The Lagrangian. Phys. Lett. B80, 48, (1978); E. Cremmer, B. Julia,: The SO(8) supergravity. Nucl. Phys. B159, 141 (1979) 3. Julia, B.: Group disintegrations. In: C80-06-22.1.2 LPTENS 80/16 Invited paper presented at Nuffield Gravity Workshop, Cambridge, Eng., Jun 22 - Jul 12, 1980 4. Julia, B.: Kac-Moody symmetry of gravitation and supergravity theories. In: LPTENS 82/22 Invited talk given at AMS-SIAM Summer Seminar on Applications of Group Theory in Physics and Mathematical Physics, Chicago, Ill., Jul 6-16, 1982 5. Breitenlohner, P., Maison, D., Gibbons, G.W.: Four-dimensional black holes from Kaluza-Klein theories. Commun. Math. Phys. 120, 295 (1988) 6. Cremmer, E., Julia, B., Lu, H., Pope, C.N.: Higher-dimensional origin of D = 3 coset symmetries. http://arxiv.org/list/hep-th/9909099, 1999 7. Keurentjes, A.: The group theory of oxidation. Nucl. Phys. B658, 303 (2003) 8. Keurentjes, A.: The group theory of oxidation. II: Cosets of non-split groups. Nucl. Phys. B658, 348 (2003) 9. Ortin, T.: Gravity and strings. Cambridge: Cambridge University Press, 2004 10. Hull, C.M.: Timelike T-duality, de Sitter space, large N gauge theories and topological field theory. JHEP 9807, 021 (1998) 11. Hull, C.M.: Duality and the signature of space-time. JHEP 9811, 017 (1998) 12. Hull, C.M., Khuri, R.R.: Branes, times and dualities. Nucl. Phys. B536, 219 (1998) 13. Cremmer, E., Lavrinenko, I.V., Lu, H., Pope, C.N., Stelle, K.S., Tran, T.A.: Euclidean-signature supergravities, dualities and instantons. Nucl. Phys. B534, 40 (1998) 14. Hull, C.M., Julia, B.: Duality and moduli spaces for time-like reductions. Nucl. Phys. B534, 250 (1998) 15. Keurentjes, A.: Time-like T-duality algebra. JHEP 0411, 034 (2004) 16. West, P.C.: E(11) and M theory. Class. Quant. Grav. 18, 4443 (2001) 17. Gaberdiel, M.R., Olive, D.I., West, P.C.: A class of Lorentzian Kac-Moody algebras. Nucl. Phys. B645, 403 (2002) 18. West, P.: Very extended E(8) and A(8) at low levels, gravity and supergravity. Class. Quant. Grav. 20, 2393 (2003) 19. Englert, F., Houart, L., Taormina, A., West, P.: The symmetry of M-theories. JHEP 0309, 020 (2003) 20. Kleinschmidt, A., Schnakenburg, I., West, P.: Very-extended Kac-Moody algebras and their interpretation at low levels. Class. Quant. Grav. 21, 2493 (2004) 21. Kac, V.G.: Infinite Dimensional Lie Algebras. Cambridge: Cambridge University Press, 1990 22. West, P.: E(11), SL(32) and central charges. Phys. Lett. B575, 333–342 (2003) 23. Englert, F., Houart, L.: G+++ invariant formulation of gravity and M-theories: Exact BPS solutions. JHEP 0401, 002 (2004) 24. Schnakenburg, I., Miemiec, A.: E(11) and spheric vacuum solutions of eleven and ten dimensional supergravity theories. JHEP 0405, 003 (2004) 25. Keurentjes, A.: U-duality (sub-)groups and their topology. Class. Quant. Grav. 21, S1367 (2004) 26. Keurentjes, A.: E(11): Sign of the times. Nucl. Phys. B697, 302 (2004)

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27. Englert, F., Henneaux, M., Houart, L.: From very-extended to overextended gravity and M-theories. JHEP 0502, 070 (2005) 28. de Buyl, S., Houart, L., Tabti, N.: Dualities and signatures of G++ invariant theories. JHEP 0506, 084 (2005) 29. Cook, P.P., West, P.C.: M-theory solutions in multiple signatures from E(11). JHEP 0512, 041 (2005) 30. Barrett, J.W., Gibbons, G.W., Perry, M.J., Pope, C.N., Ruback, P.: Kleinian geometry and the N=2 superstring. Int. J. Mod. Phys. A9, 1457 (1994) 31. Helgason, S.: “Differential geometry, Lie groups and symmetric spaces.” New York: Academic Press 1978 32. de Wit, B., Nicolai, H.: D = 11 supergravity with local SU(8) invariance. Nucl. Phys. B274, 363 (1986) H. Nicolai: D = 11 supergravity with local SO(16) invariance. Phys. Lett. B187, 316 (1987) ; B. Drabant, M. Tox, H. Nicolai: Yet more versions of D = 11 supergravity. Class. Quant. Grav. 6, 255 (1989) 33. Damour, T., Henneaux, M., Nicolai, H.: E(10) and a ‘small tension expansion’ of M theory. Phys. Rev. Lett. 89, 221601 (2002) 34. Nicolai, H., Fischbacher, T.: Low level representations for E(10) and E(11). In: Proc of the Ramanujan International Symposium on Kac Moody Algebras and Applications, Contemp. Math. 393, Providence, RI: Amer. Math. Soc., 2004 arXiv:hep-th/0301017 35. Gibbons, G.W., Hawking, S.W.: Classification of gravitational instanton symmetries. Commun. Math. Phys. 66, 291 (1979) 36. Moore, G.W.: Finite in all directions. http://arxiv.org/list/hep-th/9305139, 1993 37. Vaula, S.: On the construction of variant supergravities in D = 11, D = 10. JHEP 0211, 024 (2002) 38. de Buyl, S., Henneaux, M., Paulot, L.: Hidden symmetries and Dirac fermions. Class. Quant. Grav. 22, 3595 (2005) 39. Nicolai, H., Samtleben, H.: On K(E(9)). Q. J. Pure Appl. Math. 1, 180 (2005) 40. Keurentjes, A.: The topology of U-duality (sub-)groups. Class. Quant. Grav. 21, 1695 (2004) 41. Miemiec, A., Schnakenburg, I.: Killing spinor equations from nonlinear realisations. Nucl. Phys. B 698, 517 (2004) 42. West, P.: Brane dynamics, central charges and E(11). JHEP 0503, 077 (2005) 43. West, P.: E(11) origin of brane charges and U-duality multiplets. JHEP 0408, 052 (2004) 44. Cook, P.P., West, P.C.: G+++ and brane solutions. Nucl. Phys. B705, 111 (2005) 45. Englert, F., Houart, L.: G+++ invariant formulation of gravity and M-theories: Exact intersecting brane solutions. JHEP 0405, 059 (2004) 46. Witten, E.: Instability of the Kaluza-Klein vacuum. Nucl. Phys. B195, 481 (1982) Communicated by G.W. Gibbons

Commun. Math. Phys. 275, 529–551 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0310-7

Communications in

Mathematical Physics

Inviscid Limit for Damped and Driven Incompressible Navier-Stokes Equations in R2 P. Constantin1 , F. Ramos2 1 Department of Mathematics, The University of Chicago, Chicago, IL 60637, USA.

E-mail: [email protected]

2 Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro RJ, 21945-970, Brazil

Received: 27 November 2006 / Accepted: 15 February 2007 Published online: 2 August 2007 – © Springer-Verlag 2007

Abstract: We consider the zero viscosity limit of long time averages of solutions of damped and driven Navier-Stokes equations in R2 . We prove that the rate of dissipation of enstrophy vanishes. Stationary statistical solutions of the damped and driven NavierStokes equations converge to renormalized stationary statistical solutions of the damped and driven Euler equations. These solutions obey the enstrophy balance. 1. Introduction The vanishing viscosity limit of solutions of Navier-Stokes equations is a subject that has been extensively studied. Boundary layers, which present the most important physical aspects of the problem, are difficult to study and their mathematical understanding is rather limited. More progress has been made in the study of the limit when boundaries are absent (flow in Rn or Tn ). Even in this restricted situation, there are two distinct concepts of vanishing viscosity limit. The finite time, zero viscosity limit is the limit limν→0 S ν (t)(ω0 ) of solutions S ν (t)(ω0 ) of the Navier-Stokes equations with a fixed initial datum ω0 and with time t in some finite interval [0, T ]. By contrast, in the infinite time zero viscosity limit, long time averages of functionals of the solutions

1 t lim (S ν (s)ω0 )ds = (ω)dµν (ω) t→∞ t 0 are considered first, at fixed ν. These are represented by measures µν in function space. The long time, zero viscosity limit is then limν→0 µν ,  
1 t (S ν (s)ω0 )ds . lim lim ν→0 t→∞ t 0 The two kinds of limits are not the same. This is most clearly seen in the situation of two dimensional, unforced Navier-Stokes equations. In this case, any smooth solution

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of the Euler equations is a finite time inviscid limit but the infinite time inviscid limit is unique: it is the function identically equal to zero. This simple example points out the fact that the infinite time zero viscosity limit is more selective. In less simple situations, when the Navier-Stokes equations are forced, the long time inviscid limit is not well understood. The finite time zero viscosity limit is the limit that has been most studied. For smooth solutions in R3 , the zero viscosity limit is given by solutions of the Euler equations, for short time, in classical ([23]), and Sobolev ([18]) spaces; the limit holds for as long as the Euler solution is smooth ([6]). The convergence occurs in the Sobolev space H s as long as the solution remains in the same space ([20]). The rates of convergence are optimal in the smooth regime, O(ν). In some nonsmooth regimes (smooth vortex patches), the finite time inviscid limit exists and optimal rates of convergence can be obtained ([1, 20]) but the rates deteriorate when the smoothness of the initial data deteriorates – for nonsmooth vortex patches ([7]). One of the most fundamental questions concerning the inviscid limit is: what happens to ideally conserved quantities? For instance, in three dimensions, the kinetic energy is conserved by smooth Euler flow, and dissipated by viscous Navier-Stokes flow. Does the rate of dissipation of kinetic energy vanish with viscosity, or is there a non-zero limit? This is the problem of anomalous dissipation. The term was coined relatively recently by field theorists but the anomaly was suggested by Onsager and Kolmogorov independently in the nineteen forties. The problem is open. In two dimensions there exist infinitely many integrals that are conserved by smooth Euler flows. One of them is the enstrophy
|ω(x, t)|2 d x, R2

where ω is the vorticity of the flow. The existence of anomalous dissipation of enstrophy is postulated in Kraichnan’s theory for two dimensional turbulence ([15]). Bounds on the dissipation of enstrophy in physical terms were given in ([2]). Anomalous dissipation of enstrophy was studied in the framework of finite time inviscid limits with rough initial data ([11, 19]). It was established that, if the initial vorticity belongs to L 2 (R2 ) then weak solutions of the Euler equations conserve enstrophy, and that implies absence of anomalous dissipation of enstrophy for finite time. In this paper we study the long time, zero viscosity limit for damped and driven two dimensional Navier-Stokes equations. Damping terms in two dimensional turbulence studies have been considered to model the Ekman pumping due to friction with boundaries, and a variety of other physical damping of turbulence mechanisms. The Charney-Stommel model of the Gulf Stream ([3]) is a two dimensional, damped and driven Euler system with a physically significant, but mathematically harmless beta effect term. Numerical studies of two dimensional turbulence employ devices to remove the energy that piles up at the large scales, and damping is the most common such device. Damping was proved to reduce the number of degrees of freedom of slightly viscous Navier-Stokes equations ([16, 17]). The fact there is no anomalous dissipation of enstrophy in damped and driven Navier-Stokes equations was suggested by D. Bernard ([5]). The paper is organized as follows. In the second section we describe the equations and a few of the properties of individual solutions of the viscous equations S N S,γ (t)(ω0 ). One of the facts that plays a significant role in the paper is that the positive semiorbit O + (t0 ) = {S N S,γ (t)(ω0 ) | t ≥ t0 > 0 } is relatively compact in L 2 (R2 ) and included in

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a bounded set in L 1 (R2 ) ∩ L ∞ (R2 ) that does not depend on the viscosity. The uniform bound uses essentially the fact that the damping factor γ > 0 is bounded away from zero independently of the vanishing viscosity. In order to prove compactness, because we work in the whole space, we need to prove also that the solution does not travel. Our results apply to the spatially periodic boundary conditions as well. The absence of anomalous dissipation of energy follows immediately from the bounds in the second section. The third section is devoted to the study of the vanishing viscosity limit of sequences of time independent individual solutions. The sequences have enough compactness to pass to convergent subsequences. The resulting solution is a weak solution of the damped and driven Euler equations. The existence of weak solutions of such equations in the case of the Charney-Stommel model was first obtained in ([3]). The weak solution of the damped and driven Euler equation is a renormalized solution in the sense of ([9]). This implies that the weak solution obeys an enstrophy balance and that is used to show that there is no anomalous dissipation. The fourth section introduces the notion of stationary statistical solution of the damped and driven Navier-Stokes equations in the spirit of ([12, 13]). In the case dω of  finite dimensional dynamical systems dt = N (ω), invariant measures µ obey ∇ω (ω)N (ω)dµ(ω) = 0 for any test function . In infinite dimensions we need to restrict the test functions to a limited class of admissible functions. Among them are generalizations of the characters exp iω, w with w a test function and an additional type of test function  (ω) that uses (β(ω )) , a mollification of a function of a mollification of ω. Such technical precautions aside, the notion of stationary statistical solution of the damped and driven Navier-Stokes equation is a natural extension of the notion of invariant measure for deterministic finite dimensional dynamical systems. We show that weak limits of stationary statistical solutions of the damped and driven Navier-Stokes equations are renormalized stationary statistical solutions of the damped and driven Euler equations, a concept that we introduce in the spirit of ([9]). We also show that if the supports of the stationary statistical solutions of the damped and driven Navier-Stokes equations are included in sets that are bounded uniformly in L p (R2 ) ∩ L ∞ (R2 ) (with p < 2 for technical reasons having to do with the slow decay at infinity of velocity in the Biot-Savart law) then the weak limits are renormalized stationary statistical solutions of the damped and driven Euler equations that obey the enstrophy balance. In the fifth section we prove our main results. We construct stationary statistical solutions µν of the damped and driven Navier-Stokes equations by the Krylov-Bogoliubov procedure of taking long time averages. We show that these solutions have good enough properties so that their weak limits are renormalized stationary statistical solutions µ0 of the damped and driven Euler equations that obey the enstrophy balance. We use this fact to prove that zero viscosity limit of the long time average enstrophy dissipation rate vanishes:  
1 t lim ν lim sup ∇ω(s + t0 )2L 2 (R2 ) ds = 0 ν→0 t→∞ t 0 holds for all solutions ω(t) = S N S,γ (t)(ω0 ), all t0 > 0, and all ω0 ∈ L p (R2 )∩ L ∞ (R2 ). We also prove that convergence in this class of statistical solutions is such that

2 ν ω L 2 (R2 ) dµ (ω) = ω2L 2 (R2 ) dµ0 (ω). lim ν→0 L 2 (R2 )

L 2 (R 2 )

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2. The Setup We consider damped and driven Navier-Stokes equations in R2  ∂t u + u · ∇u − ν u + γ u + ∇ p = f, ∇ ·u =0

(1)

with γ > 0 a fixed damping coefficient, ν > 0, f time independent with zero mean and f ∈ W 1,∞ (R2 ) ∩ H 1 (R2 ). The initial velocity is divergence-free and belongs to (L 2 (R2 ))2 . We start by stating some of the properties of the individual solutions. Theorem 2.1. Let u 0 be divergence-free, u 0 ∈ H 1 (R2 )2 . Then the solution of (1) with initial datum u 0 exists for all time, is unique, smooth, and obeys the energy equality



d |u|2 d x + γ |u|2 d x + ν |∇u|2 d x = f · ud x. (2) 2dt R2 R2 R2 R2 The kinetic energy is bounded uniformly in time, with bounds independent of viscosity:   1 1 u(·, t) L 2 (R2 ) ≤ e−γ t u(·, 0) L 2 (R2 ) −  f  L 2 (R2 ) +  f  L 2 (R2 ) . γ γ The vorticity ω (the curl of the incompressible two dimensional velocity) ω = ∂1 u 2 − ∂2 u 1 = ∇ ⊥ · u

(3)

∂t ω + u · ∇ω − ν ω + γ ω = g,

(4)

obeys

with g ∈ L 2 (R2 ), the vorticity source, g = ∇ ⊥ · f . The map t → ω(t) is continuous [0, ∞) → L 2 (R2 ). If the initial vorticity is in L p (R2 ), p ≥ 1, and g ∈ L p (R2 ), then the p-enstrophy is bounded uniformly in time, with bounds independent of viscosity   1 1 ω(·, t) L p (R2 ) ≤ e−γ t ω(·, 0) L p (R2 ) − g L p (R2 ) + g L p (R2 ) γ γ for p ≥ 1. Moreover, the solution does not travel: For every  > 0 there exists R > 0 such that,
|ω(x, t)|2 d x ≤  |x|≥R

holds for all t ≥ 0. The proof of this theorem uses well-known methods, and will not be presented here. We only sketch the proof of the last statement. We take a smooth nonnegative function φ supported in {x ∈ R2 |x| ≥ 21 } and identically equal to 1 for |x| ≥ 1, multiply the   vorticity equation (4) by φ Rx ω(x, t) and integrate in space. Denoting


x |ω(x, t)|2 d x, Y R (t) = φ R

Inviscid Limit for Damped and Driven Incompressible Navier-Stokes Equations in R2

533

we obtain:

C



Y R (t)

 |x|≥ R2

|g(x)|2 d x

d 2dt Y R (t) + γ Y R (t) ≤  + Rν2 |ω(x, t)|2 d x + R1



 |u(x, t)||ω(x, t)|2 d x .

We deduce that d dt Y R (t) + γ Y R (t) ≤

 C1 γ −1 |x|≥ R |g(x)|2 d x + νRE2 + UR ω(·, s)2L 4 (R2 ) , 2

where U is a time independent bound on u L 2 (R2 ) , depending only on γ , u 0  L 2 (R2 ) and on  f  L 2 (R2 ) , and E is a time independent bound on the enstrophy, depending only on γ , ω0  L 2 (R2 ) and g L 2 (R2 ) . We observe that
∞ ν eγ (t−s) ∇ω(·, s)2L 2 (R2 ) ds 0

is bounded in terms of γ , initial enstrophy and the norm of g in L 2 (R2 ), a fact that follows immediately from the enstrophy balance. From the uniform bound on enstrophy and a Sobolev embedding theorem we deduce that
∞ eγ (t−s) ω(·, s)2L 4 (R2 ) ds ≤ F, 0

where F is bounded in terms of γ , the viscosity, initial enstrophy and norm of g in L 2 (R2 ). It then follows that  
Eν U F Y R (t) ≤ e−γ t Y R (0) + C2 γ −1 |g(x)|2 d x + + . γ R2 R |x|≥ R2 Choosing R large enough proves the claim. We note that R can be chosen uniformly for all initial vorticities ω0 ∈ L 2 (R2 ) that are uniformly bounded in L 2 (R2 ) and satisfy a uniform centering property (see below).  We are going to use the notation  f, g = R2 f (x)g(x)d x, and sometimes write S N S,γ (t)(ω0 ) for the vorticity ω(x, t) solution of (4). Theorem 2.2. Let ω0 ∈ X , where X ⊂ L 2 (R2 ) is a bounded set that satisfies the uniform centering property ∀ > 0, ∃R > 0, ∀ω0 ∈ X ,
|ω0 (x)|2 d x ≤ . |x|≥R

Then, for any t0 > 0, the set

O + (t0 , X ) = cl S N S,γ (t)ω0 | ω0 ∈ X, t ≥ t0

(where cl(O) is the L 2 (R2 ) closure of the set O) is compact in L 2 (R2 ). The proof of this theorem follows from an uniform bound in H 1 (R2 ) for ω(t) for t ≥ t0 and the uniform “no-travel”property of the previous theorem.

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3. Stationary Deterministic Solutions Let (u (ν) , ω(ν) ) be a sequence of solutions of  −ν u + γ u + ∇ p + u · ∇u = f, ∇ · u = 0,

(5)

and the corresponding vorticity equation  γ ω + u · ∇ω − ν ω = g, ω = ∇ ⊥ · u.

(6)

We let ν → 0 but keep f, g, γ fixed. The solutions u (ν) exist, are smooth and decay rapidly at infinity. Moreover, the energy balance
f · u (ν) d x γ u (ν) 2L 2 (R2 ) + ν∇u (ν) 2L 2 (R2 ) = R2

implies that the sequence u (ν) is bounded in L 2 (R2 ). The enstrophy balance
γ ω2L 2 (R2 ) + ν∇ω2L 2 (R2 ) =

R2

gωd x

(7)

implies that the sequence ω(ν) is bounded in L 2 (R2 ). Passing to a subsequence, we consider the weak limit ω(0) = w − lim ω(ν)

(8)

ν→0

in L 2 (R2 ). Because of the compact restriction H 1 (R2 )2 → L q ()2 , for any relatively compact open set  ⊂ R2 and any 1 ≤ q < ∞, we may assume, by passing to a subse1 x⊥ quence, that u (ν) = K ω(ν) (where K = 2π is the Biot-Savart kernel) converge to |x|2 u (0) strongly in L q ()2 .

Theorem 3.1. The function ω(0) is a renormalized solution of the inviscid equation  (0) γ ω + u (0) · ∇ω(0) = g, (9) ω(0) = ∇ ⊥ · u (0) . −1,q

In addition, ω(0) ∈ L 2 (R2 ), u (0) ∈ H 1 (R2 ), the equation holds in Wloc (R2 ) for any 1 < q < 2, and
γ ω(0) 2L 2 (R2 ) = gω(0) d x (10) R2

holds. Remark. Renormalized solutions have been introduced in ([9]). The existence of weak solutions for damped and driven Euler equations using a vanishing viscosity method was obtained in ([3]).

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Proof. The facts that ω(0) ∈ L 2 (R2 ), and u (0) ∈ L 2 (R2 )2 follow from the construction and uniform bounds on the solutions u (ν) , ω(ν) . Furthermore, the solutions u (ν) 2 (R2 )2 to u (0) . The vorticities are are bounded in H 1 (R2 )2 and converge strongly in L loc 2 2 bounded in L (R ) and converge weakly. If φ is a test function then (u (ν) ·∇)φ converge strongly to (u (0) · ∇)φ in L 2 (R2 )2 and, because the scalar product of weak and strong convergent sequences is convergent, we have:

lim u (ν) · ∇φω(ν) d x = u (0) · ∇φω(0) d x. ν→0 R2

R2

This means that u (0) , ω(0) is a weak solution of the inviscid equation. Because u (0) ∈ H 1 (R2 )2 , ω(0) ∈ L 2 (R2 ) and g ∈ L 2 (R2 ), we are under the conditions of consistency in ([9]), Thm II. 3, and the same proof applied to our case shows that u 0 , w 0 is a renormalized solution of the inviscid equation, that is, γ ω(0) β  (ω(0) ) + u (0) · ∇β(ω(0) ) = gβ  (ω(0) )

(11)

holds in the sense of distributions for any β ∈ C 1 that is bounded, has bounded derivative and vanishes near the origin. We present the proof here, for the sake of completeness. It 1,2 2 (R2 ), then is easy to prove (see Lemma II.1 in [9]), that if u 0 ∈ (Wloc (R2 ))2 , ω0 ∈ L loc



1 u 0 · ∇ω0 j − u 0 · ∇ ω0 j → 0 in L loc (R2 ) (12) when  tends to zero. Here (and hereafter) j is a standard mollifier – j (z) =  −2 j ( −1 z) with j (z) a fixed smooth, even, compactly supported nonnegative function with  j (z)dz = 1 – and a b denotes convolution. Then, considering the mollified functions ω0 = ω j , u 0 = u 0 j and g = g j , it follows immediately from (12) that u 0 · ∇ω0 + γ ω0 − g = q ,

(13)

1 (R2 ) as  tends to holds in the sense of distributions, and q converges to zero in L loc 1 zero. From this equation, we obtain that if β ∈ C (R), and β is bounded with bounded first derivative, then

u 0 · ∇β(ω0 ) + γ ω0 β  (ω0 ) − g β  (ω0 ) = q β  (ω0 )

(14)

also holds in the sense of distributions. Letting  tend to zero, we prove (11). In order to prove (10), we mollify b = β(ω(0) ), where β is a C 1 function with compact support b = b j . We use the identity ([8]) (u ⊗ b) − u  ⊗ b = ρ (u, b),

(15)

ρ (u, b) = r (u, b) − (u − u  ) ⊗ (b − b ),

(16)

with

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P. Constantin, F. Ramos

and with


r (u, b) =

Because

R2

j (z)(u(x − z) − u(x)) ⊗ (b(x − z) − b(x))dz.
R2

it follows that

(bu) · ∇b d x = R2

Tr ((u  ⊗ b ) ∇b ) d x = 0,


R2

r (u, b) · ∇b −

R2

(u − u  )(b − b )∇b d x.

(17)

Now b = β(ω) ∈ L 1 ∩ L ∞ (R2 ) and we can pass to the limit in (17) using the fact that u − u  is O() in L 2 (R2 ) (because of the uniform bound in H 1 (R2 )), and working in L 4 (R2 ) with b : ∇b is O()−1 in L 4 (R2 ), and b − b converges to zero in L 4 (R2 ). We deduce that

(0)  (0) (0) γ ω β (ω )β(ω )d x = gβ  (ω(0) )β(ω(0) )d x R2

R2

holds for any β ∈ C 1 with compact support. Taking a sequence of functions that approximate β(ω) = ω, with β  uniformly bounded, we deduce (10). Theorem 3.2. Let u (ν) , ω(ν) be a sequence of solutions of (5, 6). Then the enstrophy dissipation vanishes in the limit ν → 0:
|∇ω(ν) |2 d x = 0 lim ν ν→0

R2

holds. Proof. Taking the limit superior in the enstrophy balance equation (7), using Fatou’s lemma and the fact that ω(ν) converge to ω(0) weakly in L 2 (R2 ), we have:
lim sup ν∇ω(ν) 2L 2 (R2 ) ≤ lim sup gω(ν) d x − lim inf γ ω(ν) 2L 2 (R2 ) 2 ν→0 ν→0 ν→0 R (18)
≤ gω(0) d x − γ ω(0) 2L 2 (R2 ) = 0. R2

4. Stationary Statistical Solutions In this section we follow the methods of Foias, see [12–14], and define a notion of a stationary statistical solution of the damped and driven incompressible Navier-Stokes equations in the vorticity phase space. The solution is a Borel probability measure in L 2 (R2 ). We note that L 2 (R2 ) is a separable Hilbert space and therefore the Borel σ -algebra associated to the strong (norm) topology is the same as the Borel σ -algebra associated to the weak topology. (Indeed, any open set is a countable union of open balls, any open ball is a countable union of closed balls and closed balls are convex, hence weakly closed, hence weakly Borel.)

Inviscid Limit for Damped and Driven Incompressible Navier-Stokes Equations in R2

537

Definition 4.1. A stationary statistical solution of the damped and driven Navier-Stokes equation (SSSNS) in vorticity phase space is a Borel probability measure µν in L 2 (R2 ) such that
ω2H 1 (R2 ) dµν (ω) < ∞, (1) 2 (R 2 ) L
(2) u · ∇ω + γ ω − g,   (ω) + ν∇x ω, ∇x   (ω)dµν (ω) = 0 L 2 (R 2 )

1 x⊥ for any test functional  ∈ T , with u = ω, and 2π |x|2


(3) γ ω2L 2 (R2 ) + ν ω2H 1 (R2 ) − g, ω dµν (ω) ≤ 0, E 1 ≤ω L 2 (R2 ) ≤E 2

E 1 , E 2 ≥ 0.

The class of cylindrical test functions T is given by: Definition 4.2. The class of test functions T is the set of functions  : L 2 (R2 ) → R of the form (ω) :=  I (ω) = ψ (ω, w1 , . . . , ω, wm ) ,

(19)

(ω) :=  (ω) = ψ (α (ω), w1 , . . . , α (ω), wm ) ,

(20)

or

where ψ is a C 1 scalar valued function defined on Rm , m ∈ N; w1 , . . . , wm belong to C02 (R2 ) and α (ω) = J β(J ω), where β ∈ C 3 is a compactly supported function of one real variable, and J is the convolution operator J (ω) = j ω, with j ≥ 0 a fixed  smooth, nonnegative, even ( j (−z) = j (z)) function supported in |z| ≤ 1 and with R2 j (z)dz = 1. The test functions  used in the definition are all locally bounded and weakly sequentially continuous in L 2 (R2 ). We note the trivial but very important distinction between weakly continuous and strongly continuous functions defined on L 2 (R2 ): any weakly continuous function is strongly continuous, but there exist strongly continuous functions – for instance, the norm - that are not weakly continuous. Because the SSSNS is a Borel probability, bounded continuous functions are integrable. In the sequel we will pass to weak limits of SSSNS, µν → µ E , and then the distinction between weakly continuous and continuous  functions is important: although for strongly continuous functions  the integrals dµν are defined  and finite,  it is only for weakly continuous functions  that the weak limit limν→0 dµν = dµ E holds by definition. We will obtain stronger information as well, but that needs to be proved carefully. We discuss now the definition of SSSNS and comment on its mathematical soundness. We will also verify the fact that for each test function, the integrand in (2) is a weakly continuous function on L 2 . We start by making sense of (1): the integrand can

538

P. Constantin, F. Ramos

be viewed as a Borel measurable function defined for all ω ∈ L 2 (R2 ), equal to infinity for ω ∈ H 1 (R2 ). The fact that this function is Borel measurable follows from the fact that ω2H 1 is everywhere the limit of the sequence of continuous (hence measurable) functions J ω2H 1 obtained by taking a fixed a sequence  → 0 and convolving with a mollifier. The requirement (3) is a local enstrophy balance; it implies ([14]) that the SSSNS has bounded support. We define the set   g L 2 (R2 ) B := ω ∈ L 2 (R2 ); ω L 2 (R2 ) ≤ . (21) γ Proposition 4.3. The support of any stationary statistical solution of the damped and driven Navier-Stokes equations in vorticity phase space is included in the bounded set in B ⊂ L 2 (R2 ): supp µν ⊂ B.

(22)

Proof. It follows from Definition 4.1 item (3), that if

E = ω ∈ L 2 (R2 ); E 12 ≤ ω2L 2 (R2 ) ≤ E 22 , then


γ E

ω2L 2 (R2 ) dµν (ω) ≤ g L 2 (R2 )


E

ω L 2 (R2 ) dµν (ω)




≤ g L 2 (R2 )

E

ω2L 2 (R2 ) dµν (ω)

1/2 .

Hence,
E

ω2L 2 (R2 ) dµν (ω) ≤

g2L 2 (R2 ) γ2

.

Thus, 
 g2L 2 (R2 ) 2 ω L 2 (R2 ) − dµν (ω) ≤ 0. γ2 E

(23)

If E 12 = g2L 2 (R2 ) /γ 2 and E 2 → ∞, then by (23), we have µ(E) = 0, and the result follows immediately.   We compute now   for the test functions  ∈ T . Clearly α : L 2 (R2 ) → L 2 (R2 ) continuously differentiable and bounded uniformly on bounded sets of L 2 (R2 ); moreover 1 (R2 ). α (ω) · φ = ((β  (ω )) φ ) , ∀φ ∈ L loc

(24)

For  ∈ T we have thus ∇ω  (ω) · φ =

m  j=1

∂ j ψ(α (ω), w1 , . . . , α(ω) , wm )(β  (ω )φ ) , w j , (25)

Inviscid Limit for Damped and Driven Incompressible Navier-Stokes Equations in R2

539

and for ψ I ∈ T we have ∇ω  I (ω) · φ =

m 

∂ j ψ(ω, w1 , . . . , ω, wm ))φ, w j ,

(26)

j=1

where ∂ j ψ denotes the derivative of ψ with respect to its j th variable. Clearly, in both cases, φ → ∇ω (ω) · φ is a bounded linear continuous functional on L 2 (R2 ) and thus, by the Riesz representation theorem, there exists an element   (ω) ∈ L 2 (R2 ) such that ∇ω (ω) · v =   (ω), v, ∀v ∈ L 2 (R2 ). This is the identification implied in the shorthand notation   (ω) used in Definition 4.1. For instance  I (ω) =

m 

∂ j ψ(ω, w1 , . . . , ω, wm )w j .

(27)

∂ j ψ(ω, w1 , . . . , ω, wm )∂x(k) w j

(28)

j=1

Consequently ∂x(k)  I (ω) =

m  j=1

for any multi-index k with |k| ≤ 2. For  , a similar computation yields m 

∂k(k)  (ω) =

  ∂ j ψ(α (ω), w1 , . . . , α (ω), wm )∂x(k) β  (ω )w j   .

(29)

j=1

Lemma 4.1. Let  ∈ T and ω ∈ B ⊂ L 2 (R2 ) with B a bounded set in L 2 (R2 ). Then   (ω) ∈ C02 (R2 ), and there exists a constant depending only on  and B such that   (ω)W 2,2 (R2 ) +   (ω)W 2,∞ (R2 ) ≤ C

(30)

holds for all ω ∈ B. Consider, for any Fi : L 2 (R2 ) → R, i = 1, 2, 3 given by F1 (ω) =   (ω), γ ω − g, F2 (ω) = ∇x   (ω), ∇x ω and F3 (ω) =   (ω), u · ∇ω,

u=

1 x⊥ ω. 2π |x|2

These three maps are well defined for ω ∈ L 2 (R2 ), weakly continuous and bounded uniformly on bounded sets B ⊂ L 2 (R2 ).

540

P. Constantin, F. Ramos

Remarks. If β ∈ C0k+1 (R) and w j ∈ C0k (R2 ) then   (ω) ∈ C0k (R2 ). The expressions ∇x   (ω), ∇x ω = − x   (ω), ω and   (ω), u · ∇ω = −u · ∇x   (ω), ω make 2 (R2 ), ω ∈ B. sense for k ≥ 2, u ∈ L loc (k)

(k)

Proof. It is easy to see that ∂x ( ) (ω) and ∂x  I (ω) are uniformly bounded in L ∞ (R2 ) ∩ L 2 (R2 ) for all ω ∈ B, |k| ≤ 2. This is verified for ∂x(k)  I (ω) directly by (k) inspection of (28) and for ∂x  by inspection of (29). We check the bounds for  :  As α (ω) is bounded on bounded subsets of L 2 (R2 ), and ψ is of class C 1 , we have that   ∂ j ψ(α  (ω), w1 , . . . , α  (ω), wm ) ≤ C, ∀ω ∈ B. (31) The fact that β  (ω ) ∈ L ∞ (R2 ) is bounded uniformly for ω ∈ B implies that      (k)   ∂x (β (ω )w j   

L p (R 2 )

≤

 C  w j  p 2 L (R ) |k| 

(32)

holds uniformly, for all p, 1 ≤ p ≤ ∞. By (31) and (32), we have from (29) that    (k)   ∂x  (ω)

L p (R 2 )

≤

Cp  |k|

(33)

holds for 1 ≤ p ≤ ∞ with C p uniform all ω ∈ B. Thus, ∂x(k)   (ω) are bounded in L ∞ (R2 ) ∩ L 2 (R2 ). Concerning the statements about the maps Fi , we start with F1 (ω) =   (ω), γ ω − g = ∇ω (ω) · (γ ω − g); ω ∈ L 2 (R2 ).

(34)

This function is weakly continuous. Indeed, for  I we have by (27),  I (ω), γ ω − g m  ∂ j ψ(ω, w1 , . . . , ω, wm )w j , γ ω − g, = j=1

and it is clear that this is a weakly continuous function of ω ∈ L 2 (R2 ). It is also quite obvious that it is uniformly bounded for ω ∈ B. In the case of  , by (25) we have ∇ω  (ω) · (γ ω − g) m  ∂ j ψ(α  (ω), w1 , . . . , α  (ω), wm )((β  (ω )(γ ω − g) ) , w j . = j=1

The weak continuity here follows from the fact that if ω j converges weakly to ω then j j ω → ω converge pointwise, and it is bounded. Consequently, (β  (ω )(γ ω j − g) ) converges pointwise and is uniformly bounded. Therefore we can use the Lebesgue dominated convergence theorem in the integral against a fixed w from the finite list w1 , . . . wm appearing in   . It is also clear that    α (ω) · (γ ω − g) 2 2 ≤ C(ω L 2 (R2 ) + g L 2 (R2 ) ), ∀ω ∈ L 2 (R2 ). (35)  L (R )

Inviscid Limit for Damped and Driven Incompressible Navier-Stokes Equations in R2

541

Thus, we have F1 (ω) ≤ c3 (ω L 2 (R2 ) + g L 2 (R2 ) ) ≤ C; ∀ω ∈ B.

(36)

Therefore, F1 (ω) is weakly continuous and bounded uniformly for ω ∈ B. The fact that F2 is well defined follows from the fact that   (ω) ∈ L 2 (R2 ) and F2 (ω) = −   (ω), ω. The weak continuity for F2 follows as for F1 : in the case of  I it is straightforward, and in the case of  it follows because weak convergence becomes pointwise convergence and we can apply the Lebesgue dominated convergence theorem. 1 x⊥ For F3 , we note first that, if u = 2π ω, then, by classical singular integral |x|2

theory ([22]) u ∈ L rloc (R2 ), r < ∞, and ∇ · u = 0. Because ∇x   (ω) is bounded and compactly supported, u · ∇x   (ω) ∈ L 2 (R2 ) and F3 (ω) = −u · ∇x   (ω), ω is well defined. If ωk converge weakly in L 2 (R2 ) to ω, then the corresponding velocities u k converge strongly to u in L 2 on compact sets K , by the compact embedding H 1 (K ) ⊂⊂ L 2 (K ). The case of  I follows then because the functions w j in the list w1 , . . . , wm have compact supports, and therefore the functions u k · ∇w j converge strongly as k → ∞ to u · ∇w j in L 2 (R2 ). The scalar products u k · ∇ωk , w j  = −ωk , u k · ∇w j  converge, as k → ∞ to −ω, u · ∇w j , because the scalar products of weakly convergent and strongly convergent sequences converge. Therefore the function F3 is weakly continuous for this class of test functions. It is easy to see that the function is uniformly bounded locally in L 2 (R2 ). In the case of  , a similar argument shows that, if ωk converges weakly in L 2 (R2 ) to ω, then (u k · ∇ωk ) (x) → (u · ∇ω) (x) R2 ,

holds for each x ∈ and these functions are uniformly bounded as x ∈ R2 . Also, the  functions β ((ωk ) ) converge pointwise and are bounded. This implies that F3 is weakly continuous; the uniform boundedness is easily verified.   We define the notion of a renormalized stationary statistical solution of the Euler equation. Definition 4.4. A Borel probability measure µ0 on L 2 (R2 ) is a renormalized stationary statistical solution of the damped and driven Euler equation if
u · ∇ω + γ ω − g,   (ω)dµ0 (ω) = 0 (37) L 2 (R 2 )

(with u =

1 2π

x⊥ |x|2

ω) holds for any test functional  ∈ T .

We say that a renormalized stationary statistical solution µ0 of the Euler equation satisfies the enstrophy balance if


γ ω2L 2 (R2 ) − g, ω dµ0 (ω) = 0 (38) L 2 (R 2 )

holds.

542

P. Constantin, F. Ramos

We recall Prokhorov’s theorem (see for instance [21]): Theorem 4.5. Let X be a complete separable metrizable topological space, and let M be a set of Borel probability measures on X . For each sequence in M to contain a weakly convergent subsequence it is sufficient that for each  > 0, there is a compact subset K of X such that µ(X \ K ) <  for each µ ∈ M. We recall that a sequence of Borel probability measures πn on a topological space X converges weakly to a Borel probability measure π on X if for every continuous bounded real-valued function  on X ,

lim (s)dπn (s) = (s)dπ(s). (39) n→∞ X

X

Theorem 4.6. Given a sequence of stationary statistical solutions of the damped and driven NSE in vorticity phase space, {µν }, with ν → 0, there exists a subsequence, denoted also {µν }, and a Borel probability measure µ0 on L 2 (R2 ), such that

lim (ω)dµν (ω) = (ω)dµ0 (ω), (40) ν→0 L 2 (R2 )

L 2 (R 2 )

holds for all weakly continuous, locally bounded real-valued functions . Furthermore, the weak limit measure µ0 is a renormalized stationary statistical solution of the damped and driven Euler equation. Proof. The ball B defined in (22) endowed with the weak topology is a complete separable metrizable compact space ([10]). By (22), we have supp µν ⊂ B, and thus µν satisfy the sufficient condition of Theorem 4.5. Therefore there exists a subsequence µν that converges weakly in B to a Borel probability measure µ0 on B. Because B is weakly closed in L 2 (R2 ), we can extend the measure µ0 to L 2 (R2 ) by setting µ0 (X ) = µ0 (X ∩ B) for any Borelian set X . We claim that µ0 is a renormalized statistical solution of the damped and driven Euler equation. Indeed, for any  ∈ T , for each i = 1, 2, 3,

Fi (ω)dµν (ω) = Fi (ω)dµ0 (ω) lim ν→0

holds in view of Lemma 4.1 because each Fi is bounded and weakly continuous. In particular, the sequence F2 (ω)dµν (ω) is bounded, and so
lim ν F2 (ω)dµν (ω) = 0 ν→0

holds. The fact that µν are SSSNS implies by Definition (4.1), (2),

ν (F1 (ω) + F3 (ω)) dµ (ω) = −ν F2 (ω)dµν (ω). Passing to the limit ν → 0 we deduce
(F1 (ω) + F3 (ω)) dµ0 (ω) = 0, which is the condition (37). Hence µ0 is a renormalized stationary statistical solution of the damped and driven Euler equation.  

Inviscid Limit for Damped and Driven Incompressible Navier-Stokes Equations in R2

543

We consider the sets

  B∞ p (r ) = ω ∈ B; ω L p (R2 ) ≤ r, ω L ∞ (R2 ) ≤ r

defined for r > 0, 1 ≤ p < 2. Theorem 4.7. Let {µν } be a sequence of stationary statistical solutions of the damped and driven NSE in vorticity phase space, with ν → 0. Assume that there exists 1 < p < 2 and r > 0 such that supp µν ⊂ B ∞ p (r ). Then, the limit µ0 of any weakly convergent subsequence is a renormalized stationary statistical solution of the damped and driven Euler equation (37) that is supported in B∞ p (r ) and satisfies the enstrophy balance (38). Proof. The set B ∞ closed in B. Indeed, if ω j ∈ B ∞ (r ) converges weakly p (r ) is weakly      to ω and if φ ∈ C0∞ (R2 ) then  ωφd x  = lim j→∞ ω j φd x  ≤ r φ L 1 (R2 ) implies  that ω L ∞ (R2 ) ≤ r . Similarly, we obtain  ωφd x  ≤ r φ L p (R2 ) where p  > 2 is the dual exponent, and deduce that ω L p (R2 ) ≤ r . By Theorem 4.6, the limit µ0 of a weakly convergent subsequence is a Borel probability measure supported in B and a renormalized statistical solution of the damped and driven Euler equation (37). The set 0 ν U = L 2 (R2 ) \ B ∞ p (r ) is weakly open and µ (U ) ≤ lim inf ν→0 µ (U ) = 0 follows by 0 general properties of weak convergence. Thus, the support of µ is included in B ∞ p (r ). In order to prove the enstrophy balance we consider the function ψ (m) (a1 , . . . , am ) = 

1 |ak |2 . 2 m

k=1



Let w j be a complete orthonormal basis in L 2 (R2 ), formed with functions w j ∈ C02 (R2 ). Then, for each fixed m,  (m,) (ω) = ψ (m) ((β(ω )) , w1 , . . . , (β(ω )) , wm ) is a function in T , and ( (m,) ) (ω), (γ ω − g) =

m  (β(ω )) , w j ((β  (ω ))(γ ω − g) ) , w j . (41) j=1





Because w j is an orthonormal basis in L 2 (R2 ), it follows by Parseval’s theorem that lim ( m, ) (ω), (γ ω − g) = (β(ω )) , (β  (ω )(γ ω − g) )

m→∞

holds for each ω in B. Moreover, because the functions (β(ω )) and (β  (ω )(γ ω−g) ) are bounded in L 2 (R2 ) as ω ∈ B, it follows that the sequence ( m, ) (ω), (γ ω − g) is bounded uniformly for ω ∈ B. Thus, we may apply the Lebesgue dominated convergence theorem to deduce

lim ( m, ) (ω), (γ ω − g)dµ0 (ω) = (β(ω )) , (β  (ω )(γ ω − g) ) dµ0 (ω). m→∞

(42)

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Because (u · ∇ω) = ∂k (u k ω) we have ( (m,) ) (ω), u · ∇ω =

m  (β(ω )) , w j (β  (ω )∂k (u k ω) ) , w j .

(43)

j=1

In order to establish the pointwise limit lim ( (m,) ) (ω), u · ∇ω = (β(ω )) , (β  (ω )∂k (u k ω) ) 

m→∞

and the uniform bounds on (β(ω )) and (β  (ω )∇x (uω) ) in L 2 (R2 ) for ω ∈ B we need to split the Biot-Savart expression u=

1 x⊥ ω 2π |x|2

in two pieces, corresponding to 1 x⊥ = K 1 (x) + K 2 (x), 2π |x|2 1 x⊥ K 1 (x) = 1|x|≤1 , 2π |x|2 1 x⊥ K 2 (x) = 1|x|>1 . 2π |x|2 Clearly, because each component of K 1 ∈ L 1 (R2 ), it follows that u 1 = K 1 ω is in L 2 (R2 ) by the Hausdorff-Young inequality, and its norm in L 2 is bounded by a constant  uniformly for ω ∈ B. On the other hand, because each component of K 2 ∈ L p (R2 ), with p  > 2 the dual exponent of p < 2, we have that u 2 = K 2 ω ∈ L ∞ (R2 ) with 1 2 2 2 norm bounded by r , as long as ω ∈ B ∞ p (r ). Therefore u ⊗ ω ∈ L (R ) + L (R ), ∞ 2 with norm bounded uniformly for ω ∈ B ∞ p (r ). Consequently, (u ⊗ ω) ∈ L (R ) with ∞  norm uniformly bounded for ω ∈ B p (r ) and, because β (ω ) is uniformly bounded in L 2 (R2 ), we may use the Lebesgue dominated convergence theorem to deduce

(m,)  0 lim ) (ω), u · ∇ωdµ (ω) = (β(ω )) , (β  (ω )∂k (u k ω) ) dµ0 (ω). ( m→∞

(44) Because of (37, 42, 44) we have then
(β(ω )) , (β  (ω )(γ ω − g) ) dµ0 (ω) L 2 (R 2 )
+ (β(ω )) , (β  (ω )∂k (u k ω) ) dµ0 (ω) = 0. L 2 (R 2 )

Now we are going to investigate the term
  Iβ, = (β(ω )) β  (ω )∂k (u k ω)  d x. R2

(45)

Inviscid Limit for Damped and Driven Incompressible Navier-Stokes Equations in R2

545

Integrating by parts we write Iβ, = Jβ, + K β, , with


Jβ, = −

and

R2


K β, = −

R2

  ∂k (β(ω )) β  (ω )(u k ω)  d x,

  (β(ω ) ) β  (ω )(∂k ω )(u k ω)  d x.

We split Jβ, further, using (15, 16): Jβ, = L β, + Mβ, , with


L β, = −

and

R2


Mβ, = −

R2

  ∂k (β(ω )) β  (ω )(u k ) (ω)  d x,

  ∂k (β(ω )) β  (ω )ρ (u k , ω)  d x.

We estimate 1 |Mβ, | ≤ C sup |β| sup |β  | ρ (u, ω) L 1 (R2 ) .  We used the fact that 1 ∂k (β)  L ∞ (R2 ) ≤ C β L ∞ (R2 ) .  We claim that |Mβ, | ≤ C sup |β| sup |β  |ω L 2


R2

j (z)(1 + |z|)δz ω L 2 dz,

where (δh ω)(x) = ω(x − h) − ω(x). Indeed this follows from a bound on ρ (u, ω) and the uniform bound δz u L 2 (R2 ) ≤ |z|ω L 2 (R2 ) . We fix  > 0 and we consider a sequence of compactly supported functions β(y) that converge uniformly on the compact R∞ = [−2 gγL ∞ , 2 gγL ∞ ] together with two derivatives to the function y , (i.e. β → y, β  → 1, β  → 0) and such that |β(y)| + |β  (y)| + |β  (y)|) ≤ C. It is easy to see that for fixed  > 0,
(L β, + K β, )dµ0 (ω) = 0. lim β→y

L 2 (R 2 )

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Indeed, K β, (ω) is a continuous function of ω ∈ L 2 (R2 ), uniformly bounded on K ∞ p and converging pointwise to zero. As for L β, , it is also continuous, bounded and converges to the stated limit because 0 = ∂k (ω ) ((u k ) ) (ω ) d x. R2

On the other hand, from

|Mβ, |dµ0 (ω) ≤ C L 2 (R 2 )


L 2 (R 2 )

R2

j (z)(1 + |z|)δz ω L 2 (R2 ) dzdµ(0) (ω),

with C uniform for all β in the sequence, it follows from the Lebesgue dominated convergence theorem that
lim lim sup |Mβ, |dµ(0) = 0. →0

β→y

L 2 (R 2 )

By (45) and the estimates above it follows that
(β(ω )) , (β  (ω )(γ ω − g) ) dµ0 (ω) = 0. lim lim sup →0

β→y

L 2 (R 2 )

On the other hand, by the Lebesgue dominated convergence theorem again,  lim→0 lim supβ→y L 2 (R2 ) (β(ω )) , (β  (ω )(γ ω − g) ) dµ0 (ω)

 = L 2 (R2 ) γ ω2L 2 (R2 ) − g, ω dµ0 (ω), which proves (38). 5. Long Time Averages and the Inviscid Limit In this section we consider SSSNSs obtained as generalized (Banach) limits of long time averages of functionals of deterministic solutions of the damped and driven Navier-Stokes equations. These SSSNS have good enough properties to pass to the inviscid limit and are used to prove that the time averaged enstrophy dissipation vanishes in the zero viscosity limit. Definition 5.1. A generalized limit (Banach limit) is a linear continuous functional Lim t→∞ : BC([0, ∞)) → R such that 1. Lim t→∞ (g) ≥ 0; ∀g ∈ BC([0, ∞)) with g(s) ≥ 0 ∀s ≥ 0, 2. Lim t→∞ (g) = limt→∞ g(t), whenever the usual limit exists. The space BC[0, ∞) is the Banach space of all bounded continuous functions defined on [0, ∞), and the functional Lim t→∞ is constructed as an easy application of the Hahn-Banach theorem. It can be shown that any generalized limit satisfies lim inf g(T ) ≤ Lim t→∞ (g) ≤ lim sup g(T ), ∀g ∈ BC([0, ∞)). T →∞

T →∞

(46)

Furthermore, given a particular g0 ∈ BC([0, ∞)), and a sequence t j → ∞ for which g0 (t j ) converges to a number l, we can construct a generalized limit Lim t→∞ satisfying Lim t→∞ (g0 ) = l, see [4, 14]. This implies that one can choose a functional Lim t→∞ so that Lim t→∞ g0 = lim supt→∞ g0 (t).

Inviscid Limit for Damped and Driven Incompressible Navier-Stokes Equations in R2

547

Theorem 5.1. Let u 0 ∈ L 2 (R2 ) and ∇ ⊥ u 0 = ω0 ∈ L 1 (R2 ) ∩ L ∞ (R2 ). Let f ∈ W 1,1 (R2 ) ∩ W 1,∞ (R2 ). Let t0 > 0. Let Lim t→∞ be a Banach limit. Then

1 t (ω)dµν (ω) = Lim t→∞ (S N S,γ (s + t0 ))(ω0 ))ds (47) t 0 L 2 (R 2 ) is a statistical stationary solution of the damped and driven Navier-Stokes equations. For any p > 1 there exists r depending only on γ , f, ω0 but not ν nor t0 such that suppµν ⊂ B ∞ p (r ). The inequality
ν

L 2 (R 2 )

∇ω2L 2 (R2 ) dµν (ω)


≤

L 2 (R 2 )

  g, ω − γ ω2L 2 dµν (ω)

(48)

(49)

holds. Proof. By Theorem 2.2, the set O + (t0 , {ω0 }) = cl{S N S,γ (s + t0 )(ω0 ), |s ≥ 0 } is compact in L 2 (R2 ). By Theorem 2.1, (S N S,γ (s + t0 )(ω0 )) is a continuous bounded function on [0, ∞) and so is its time average on [0, t]. Thus, the generalized limit
1 t (S N S,γ (s + t0 ))(ω0 ))ds Lim t→∞ t 0   exists. Moreover, it is a positive functional on C O + (t0 , {ω0 }) . Because of the Riesz representation theorem on compact spaces, there exists a Borel measure µν on the compact O + (t0 , {ω0 }) that represents the limit. The measure µν is supported in O + (t0 , {ω0 }), µν (X ) = µν (X ∩ O + (t0 , {ω0 })), for any X Borelian in L 2 (R2 ). The inclusions (48) follow from Theorem 2.1. We take a test function  ∈ T . Then
  (ω), u · ∇ω + γ ω − ν ωdµν (ω) L 2 (R 2 )

= Lim t→∞

1 t




t 0

d (S N S,γ (s + t0 )(ω0 ))ds = 0 ds

holds. This verifies definition 2 (2). In order to verify conditions (1) and (3) we take the solution ω(t) = S N S,γ (t)(ω0 ) mollify it, ω (t) = J (ω(t)) and take the enstrophy balance. We obtain from (4) d 2 2dt ω (t) L 2 (R2 )

+ ν∇ω (t)2L 2 (R2 ) + γ ω (t)2L 2 (R2 ) − g , ω (t) = ρ (u(t), ω(t)), ∇ω (t).

(50)

Integrating in time we deduce    t 1 t 2 γ ω (s + t ) − g , ω (s + t ) ds + νt 0 ∇ω (s + t0 )2L 2 (R2 ) ds  0   0 2 2 t 0 L (R )   = 2t1 ω (t0 )2L 2 (R2 ) − ω (t + t0 )2L 2 (R2 ) t + 1t 0 ρ (u(s + t0 ), ω(s + t0 )), ∇ω (s + t0 )ds. (51)

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P. Constantin, F. Ramos

Fixing  > 0, we may apply Lim t→∞ :  t  Lim t→∞ 1t 0 γ ω (s + t0 )2L 2 (R2 ) − g , ω (s + t0 ) ds    = L 2 (R2 ) γ ω 2L 2 (R2 ) − g , ω  dµν (ω) and 1 Lim t→∞ t




t 0


∇ω (s

+ t0 )2L 2 (R2 ) ds

=

L 2 (R 2 )

∇ω 2L 2 (R2 ) dµν (ω)

hold because the functionals are continuous. From (51) we have     2 2 ν ν L 2 (R2 ) γ ω  L 2 (R2 ) − g , ω  dµ (ω) + ν L 2 (R2 ) ∇ω  L 2 (R2 ) dµ (ω) t = Lim t→∞ 1t 0 ρ (u(s + t0 ), ω(s + t0 )), ∇ω (s + t0 )ds. (52) We estimate the right-hand side taking ∇ω in L ∞ (R2 ), where it costs  −1 , where  is a time independent bound on S N S,γ (s + t0 )ω0 ) L ∞ (R2 ) (from Theorem 2.1). Then we are left with   t   Lim t→∞ 1t 0 ρ (u(s + t0 ), ω(s + t0 )), ∇ω (s + t0 )ds    t ≤ Lim t→∞ 1t 0 R2 j (z)δz ω(s + t0 ) L 2 (R2 ) ds, where  is a bound on sups≥0 ω(s + t0 ) L ∞ (R2 ) ω(s + t0 ) L 2 (R2 ) . We use crucially now the fact that O + (t0 , {ω0 }) is compact in L 2 (R2 ). Then for every small number h > 0 there exists  > 0 so that δz ω(s + t0 ) L 2 (R2 ) ≤ h holds for all s ≥ 0, and all z in the compact support of j. Therefore we have from (52),         L 2 (R2 ) γ ω 2L 2 (R2 ) − g , ω  dµν (ω) + ν L 2 (R2 ) ∇ω 2L 2 (R2 ) dµν (ω) ≤ h()

(53)

with 0 ≤ h(), a function satisfying lim→0 h() = 0. We remove now the mollifier, carefully. First we note that

    2 ν γ ω L 2 (R2 ) −g, ω dµ (ω) = lim γ ω 2L 2 (R2 ) −g , ω  dµν (ω) →0 L 2 (R2 )

L 2 (R 2 )

holds trivially because µν is a Borel measure. This, together with (53) implies that

  γ ω2L 2 (R2 ) − g, ω dµν (ω) ν lim sup ∇ω 2L 2 (R2 ) dµν (ω) ≤ − →0 L 2 (R2 )

which implies, by Fatou’s lemma

∇ω2L 2 (R2 ) dµν (ω) ≤ − ν L 2 (R 2 )

L 2 (R 2 )

L 2 (R 2 )

  γ ω2L 2 (R2 ) − g, ω dµν (ω). (54)

Because the right-hand side is finite, this proves (1) and (49). The proof of (3) for arbitrary E 1 , E 2 follows from a very similar computation as the one above. We take

Inviscid Limit for Damped and Driven Incompressible Navier-Stokes Equations in R2

549

χ  (y), a smooth, nonnegative, compactly supported function defined for y ≥ 0. Then y χ (y) = 0 χ  (e)de is bounded on R+ and d d χ (ω (t)2L 2 (R2 ) ) = χ  (ω (t)2L 2 (R2 ) ) ω (t)2L 2 (R) . dt dt We multiply (50) by 2χ  (ω (t)2L 2 (R2 ) ) and we proceed as above by taking time average, long time limit and removing the mollifier. We obtain


χ  (ω2L 2 (R2 ) ) ν∇ω2L 2 (R2 ) + γ ω2L 2 (R2 ) − g, ω dµν (ω) ≤ 0 L 2 (R 2 )

and letting χ  (y) → 1[E 2 ,E 2 ] pointwise, with 0 ≤ χ  (y) ≤ 2, concludes the proof. 1

 

2

Theorem 5.2. Let f ∈ W 1,1 (R2 ) ∩ W 1,∞ (R2 ). Let u 0 ∈ L 2 (R2 ) be divergence-free and let ∇ ⊥ u 0 = ω0 ∈ L 1 (R2 ) ∩ L ∞ (R2 ). Let ων (t) = S N S,γ (t)(ω0 ) be the vorticity of the solution of the damped and driven Navier-Stokes equation. Then,  
2 1 t ν   ∇ω (s + t0 ) L 2 (R2 ) ds = 0, lim ν lim sup (55) ν→0 t→∞ t 0 holds for any t0 > 0. Proof. We argue by contradiction and assume that the statement is false. Then, there exists a sequence νk → 0 and δ > 0, such that, for each fixed νk , there exists a sequence of times t j → ∞ (that may depend on k) such that νk tj




tj

  ν ∇ω k (s + t0 )2 2

L (R 2 )

0

ds ≥ δ

(56)

holds for all t j → ∞. Because of the enstrophy balance δ≤

νk tj




tj 0

∇ωνk (s + t0 )2L 2 (R2 ) =

1 tj




tj  0

 −γ ωνk (s +t0 )2L 2 (R2 ) + g, ωνk (s + t0 ) ds

 1  νk ω (t0 )2L 2 (R2 ) − ωνk (t0 + t j )2L 2 (R2 ) + 2t j

It follows that 1 lim sup t→∞ t


t  −γ ωνk (s + t0 )2L 2 (R2 ) + g, ωνk (s + t0 ) ds ≥ δ.

(57)

0

Because the function −γ ω2L 2 (R2 ) + g, ω is continuous on cl O + (t0 , {ω0 }), by the remark after Definition 5.1, we can choose a generalized limit such that
  2 1 t −γ ωνk (s + t0 ) L 2 (R2 ) + g, ωνk (s + t0 ) ds Lim t→∞ t 0 (58)
  2 1 t = lim sup −γ ωνk (s + t0 ) L 2 (R2 ) + g, ωνk (s + t0 ) ds. t→∞ t 0

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P. Constantin, F. Ramos

Now, by Theorem 5.1, this means that we have a SSSNS µνk that satisfies (48) and that also satisfies, in view of (57) and (58),


−γ ω2L 2 (R2 ) + g, ω dµνk (ω) ≥ δ > 0. (59) L 2 (R 2 )

Passing to a weakly convergent subsequence we find with Theorem 4.7 that there exists a renormalized statistical solution of the damped and driven Euler equations µ0 that satisfies the enstrophy balance (38). Because the function ω → g, ω is weakly continuous, we have

νk lim g, ωdµ (ω) = g, ωdµ0 (ω). (60) k→∞ L 2 (R2 )

L 2 (R 2 )

On the other hand, by Fatou’s lemma

ω2L 2 (R2 ) dµ0 (ω) ≤ γ lim inf γ

ω2L 2 (R2 ) dµνk (ω).

(61)

From (59) and (60) we have

2 νk γ lim inf ω L 2 (R2 ) dµ (ω) ≤

g, ωdµ0 (ω) − δ,

(62)

L 2 (R 2 )

k→∞ L 2 (R2 )

k→∞ L 2 (R2 )

L 2 (R 2 )

and from (61) and (62) we obtain


γ ω2L 2 (R2 ) − g, ω dµ0 (ω) ≤ −δ < 0. L 2 (R 2 )

This is a contradiction because (38) holds. Thus (55) holds.

(63)

 

Theorem 5.3. Let f ∈ W 1,1 (R2 ) ∩ W 1,∞ (R2 ). Let u 0 ∈ L 2 (R2 ) be divergence-free and let ∇ ⊥ u 0 = ω0 ∈ L 1 (R2 ) ∩ L ∞ (R2 ). Let µν be SSSNS associated to long time averages given by (47) that converge weakly as ν → 0 to a renormalized statistical solution µ0 of the damped and driven Euler equation. Then

2 ν ω L 2 (R2 ) dµ (ω) = ω2L 2 (R2 ) dµ0 (ω) (64) lim ν→0 L 2 (R2 )

L 2 (R 2 )

holds. Proof. Indeed, by Theorem 4.7 we know that µ0 satisfies (38). From (49) and (60) we have

lim sup γ ω2L 2 (R2 ) dµν (ω) ≤ g, ωdµ0 (ω). (65) ν→0 L 2 (R2 )

Using (38) we obtain
lim sup

ν→0 L 2 (R2 )

L 2 (R 2 )

γ ω2L 2 (R2 ) dµν (ω)

From (61) we obtain (64).


≤

L 2 (R 2 )

γ ω2L 2 (R2 ) dµ0 (ω).

(66)

 

Acknowledgement. The work of P.C. is partially supported by NSF-DMS grant 0504213. The work of F.R is partially supported by the Pronex in Turbulence, CNPq and FAPERJ. Brazil. grant number E-26/171.198/2003, and by CAPES Foundation. Brazil. grant number BEX4427/05-0. F.R. also wants to thank the Department of Mathematics of The University of Chicago for its hospitality.

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References 1. Abidi, H., Danchin, R.: Optimal bounds for the inviscid limit of Navier-Stokes equations. Asymptot. Anal. 38, 35–46 (2004) 2. Alexakis, A., Doering, C.: Energy and enstrophy dissipation in steady state 2D turbulence. Phys. Lett. A 359, 652–657 (2006) 3. Barcilon, V., Constantin, P., Titi, E.: Existence of solutions to the Stommel-Charney model of the Gulf Stream. SIAM J. Math. Anal. 19, 1355–1364 (1988) 4. Bercovici, H., Constantin, P., Foias, C., Manley, O.P.: Exponential decay of the power spectrum of turbulence. J. Stat. Phys. 80(3-4), 579–602 (1995) 5. Bernard, D.: Influence of friction on the direct cascade of 2D forced turbulence. Europhys. Lett. 50, 333–339 (2000) 6. Constantin, P.: Note on loss of regularity for solutions of the 3-D incompressible Euler and related equations. Commun. Math. Phys. 104, 311–326 (1986) 7. Constantin, P., Wu, J.: Inviscid limit for vortex patches. Nonlinearity 8, 735–742 (1995) 8. Constantin, P., E, W., Titi, E.: Onsager conjecture on the energy conservation for solutions of Euler’s equation, Commun. Math. Phys. 165, 207–209 (1994) 9. DiPerna, R., Lions, P-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989) 10. Dunford, N., Schwartz, J.: Linear operators. Part I. General theory. A Wiley-Interscience Publication. New York: John Wiley and Sons, Inc., 1988 11. Eyink, G.: Dissipation in turbulent solutions of 2D Euler equations. Nonlinearity 14, 787–802 (2001) 12. Foias, C.: Statistical study of the Navier-Stokes equations I. Rend. Sem. Mat. Univ. Padova 48, 219–348 (1972) 13. Foias, C.: Statistical study of the Navier-Stokes equations II. Rend. Sem. Mat. Univ. Padova 49, 9–123 (1973) 14. Foias, C., Manley, O.P., Rosa, R., Temam, R.: Navier-Stokes Equations and Turbulence. In: Encyclopedia of Mathematics and its Applications, Vol. 83, Cambridge: Cambridge University Press, 2001 15. Frisch, U.: Turbulence. Cambridge: Cambridge University Press, 1995 16. Ilyin, A.A., Miranville, A., Titi, E.S.: Small viscosity sharp estimates for the global attractor of the 2D damped-driven Navier-Stokes equations. Comm. Math. Sci 2, 403–425 (2004) 17. Ilyin, A.A., Titi, E.S.: Sharp estimates of the number of degrees of freedom for the damped-driven 2D Navier-Stokes equations. J. Nonl. Sci. 16, 233–253 (2006) 18. Kato, T.: Nonstationary flows of viscous and ideal fluids in R3 . J. Funct. Anal. 9, 296–305 (1972) 19. Lopes Filho, M., Mazzucato, A., Nussenzveig Lopes, H.: Weak solutions, renormalized solutions and enstrophy defects in 2D turbulence. ARMA 179, 353–387 (2006) 20. Masmoudi, N.: Remarks about the inviscid limit of the Navier-Stokes system. Commun. Math. Phys., to appear (2007) 21. Smolyanov, O.G., Fomin, S.V.: Measures on linear topological spaces. Russ. Math. Surv. 31(4), 1–53 (1976) 22. Stein, E.: Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press Princeton, NJ, 1970 23. Swann, H.S.G.: The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in R3 . Trans. Amer. Math. Soc. 157, 698–726 (1971) Communicated by A. Kupiainen

Commun. Math. Phys. 275, 553–580 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0317-0

Communications in

Mathematical Physics

The Periodic Oscillation of an Adiabatic Piston in Two or Three Dimensions Paul Wright Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012, USA. E-mail: [email protected] Received: 14 December 2006 / Accepted: 15 February 2007 Published online: 10 August 2007 – © Springer-Verlag 2007

Abstract: We study a heavy piston of mass M that separates finitely many ideal, unit mass gas particles moving in two or three dimensions. Neishtadt and Sinai previously determined a method for finding this system’s averaged equation and showed that its solutions oscillate periodically. Using averaging techniques, we prove that the actual motions of the piston converge in probability to the predicted averaged behavior on the time scale M 1/2 when M tends to infinity while the total energy of the system is bounded and the number of gas particles is fixed. 1. Introduction Consider the following simple model of an adiabatic piston separating two gas containers: A massive piston of mass M
1 divides a container in Rd into two halves. The piston has no internal degrees of freedom and can only move along one axis of the container. On either side of the piston there are a finite number of ideal, unit mass, point gas particles that interact with the walls of the container and with the piston via elastic collisions. When M = ∞, the piston remains fixed in place, and each gas particle performs billiard motion at a constant energy in its sub-container. We make an ergodicity assumption on the behavior of the gas particles when the piston is fixed. Then we study the motions of the piston when the number of gas particles is fixed, the total energy of the system is bounded, but M is very large. Heuristically, after some time, one expects the system to approach a steady state, where the energy of the system is equidistributed amongst the particles and the piston. However, even if we could show that the full system is ergodic, an abstract ergodic theorem says nothing about the time scale required to reach such a steady state. Because the piston will move much slower than a typical gas particle, it is natural to try to determine the intermediate behavior of the piston by averaging techniques. By averaging over the motion of the gas particles on a time scale chosen short enough that the piston is nearly fixed, but long enough that the ergodic behavior of individual gas particles is

554

P. Wright

observable, we will show that the system does not approach the expected steady state on the time scale M 1/2 . Instead, the piston oscillates periodically, and there is no net energy transfer between the gas particles. This paper follows earlier work by Neishtadt and Sinai [NS04, Sin99]. They determined that for a wide variety of Hamiltonians for the gas particles, the averaged behavior of the piston is periodic oscillation, with the piston moving inside an effective potential well whose shape depends on the initial position of the piston and the gas particles’ Hamiltonians. They pointed out that an averaging theorem due to Anosov [Ano60, LM88], proved for smooth systems, should extend to this case. This paper proves that Anosov’s theorem extends to the particular gas particle Hamiltonian described above. Thus, if we examine the actual motions of the piston with respect to the slow time τ = t/M 1/2 , then, as M → ∞, in probability (with respect to Liouville measure) most initial conditions give rise to orbits whose actual motion is accurately described by the averaged behavior for 0 ≤ τ ≤ 1, i.e. for 0 ≤ t ≤ M 1/2 . Gorelyshev and Neishtadt [GN06] and we [Wri06] have already proved that when d = 1, i.e. when the gas particles move on a line, the convergence of the actual motions to the averaged behavior is uniform over all initial conditions, with the size of the deviations being no larger than O(M −1/2 ) on the time scale M −1/2 . The system under consideration in this paper is a simple model of an adiabatic piston. The general adiabatic piston problem [Cal63], well-known from physics, consists of the following: An insulating piston separates two gas containers, and initially the piston is fixed in place, and the gas in each container is in a separate thermal equilibrium. At some time, the piston is no longer externally constrained and is free to move. One hopes to show that eventually the system will come to a full thermal equilibrium, where each gas has the same pressure and temperature. Whether the system will evolve to thermal equilibrium and the interim behavior of the piston are mechanical problems, not adequately described by thermodynamics [Gru99], that have recently generated much interest within the physics and mathematics communities. One expects that the system will evolve in at least two stages. First, the system relaxes toward a mechanical equilibrium, where the pressures on either side of the piston are equal. In the second, much longer, stage, the piston drifts stochastically in the direction of the hotter gas, and the temperatures of the gases equilibrate. See for example [GPL03, CL02] and the references therein. So far, rigorous results have been limited mainly to models where the effects of gas particles recolliding with the piston can be neglected, either by restricting to extremely short time scales [LSC02, CLS02] or to infinite gas containers [Che04]. A recent study involving some similar ideas by Chernov and Dolgopyat [CD06a] considered the motion inside a two-dimensional domain of a single heavy, large gas particle (a disk) of mass M
1 and a single unit mass point particle. They assumed that for each fixed location of the heavy particle, the light particle moves inside a dispersing (Sinai) billiard domain. By averaging over the strongly hyperbolic motions of the light particle, they showed that under an appropriate scaling of space and time the limiting process of the heavy particle’s velocity is a (time-inhomogeneous) Brownian motion on a time scale O(M 1/2 ). It is not clear whether a similar result holds for the piston problem, even for gas containers with good hyperbolic properties, such as the Bunimovich stadium. In such a container the motion of a gas particle when the piston is fixed is only nonuniformly hyperbolic because it can experience many collisions with the flat walls of the container immediately preceding and following a collision with the piston. The present work provides a weak law of large numbers, and it is an open problem to describe the sizes of the deviations for the piston problem [CD06b]. Although our result

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does not yield concrete information on the sizes of the deviations, it is general in that it imposes very few conditions on the shape of the gas container. Most studies of billiard systems impose strict conditions on the shape of the boundary, generally involving the sign of the curvature and how the corners are put together. The proofs in this work require no such restrictions. In particular, the gas container can have cusps as corners and need satisfy no hyperbolicity conditions. We begin in Sect. 2 by giving a physical description of our results. Precise assumptions and our main result, Theorem 1, are stated in Sect. 3, and a proof is presented in the following sections. 2. Physical Motivation for the Results Before precisely stating our assumptions and results, we briefly review the physical motivations for our results and introduce some notation. Consider a massive, insulating piston of mass M that separates a gas container D in Rd , d = 2 or 3. See Fig. 1. Denote the location of the piston by Q, its velocity by d Q/dt = V , and its cross-sectional length (when d = 2, or area, when d = 3) by . If Q is fixed, then the piston divides D into two subdomains, D1 (Q) = D1 on the left and D2 (Q) = D2 on the right. By E i we denote the total energy of the gas inside Di , and by |Di | we denote the area (when d = 2, or volume, when d = 3) of Di . We are interested in the dynamics of the piston when the system’s total energy is bounded and M → ∞. When M = ∞, the piston remains fixed in place, and each energy E i remains constant. When M is large but finite, M V 2 /2 is bounded, and so V = O(M −1/2 ). It is natural to define ε = M −1/2 , W =

V , ε

so that W is of order 1 as ε → 0. This is equivalent to scaling time by ε. If we let Pi denote the pressure of the gas inside Di , then heuristically the dynamics of the piston should be governed by the following differential equation: dQ = V, dt

M

dV = P1  − P2 , dt

i.e. dQ = εW, dt

dW = ε P1  − ε P2 . dt

Fig. 1. A gas container D ⊂ R2 separated by a piston

(1)

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To find differential equations for the energies of the gases, note that in a short amount of time dt, the change in energy should come entirely from the work done on a gas, i.e. the force applied to the gas times the distance the piston has moved, because the piston is adiabatic. Thus, one expects that d E1 = −εW P1 , dt

d E2 = +εW P2 . dt

(2)

To obtain a closed system of differential equations, it is necessary to insert an expression for the pressures. Because the pressure of an ideal gas in d dimensions is proportional to the energy density, with the constant of proportionality 2/d, we choose to insert Pi =

2E i . d |Di |

Later, we will make assumptions to justify this substitution. However, if we accept this definition of the pressure, and define the slow time τ = εt, we obtain the following ordinary differential equations for the four macroscopic variables of the system: ⎡ ⎤ ⎡ ⎤ W Q 2E 1  2E 2  ⎥ d ⎢W ⎥ ⎢ ⎢ d|D1 (Q)| − d|D2 (Q)| ⎥ (3) ⎥. ⎣E ⎦ = ⎢ 2W E 1  − d|D1 (Q)| 1 ⎣ ⎦ dτ E2  E2 + d|2W D2 (Q)| Neishtadt and Sinai [Sin99, NS04] pointed out that the solutions of Eq. (3) have the piston moving according to an effective Hamiltonian. This can be seen as follows. Since ∂ |D1 (Q)| ∂ |D2 (Q)| ==− , ∂Q ∂Q d ln(E i )/dτ = −(2/d)d ln(|Di (Q)|)/dτ , and so   |Di (Q(0))| 2/d E i (τ ) = E i (0) . |Di (Q(τ ))| Hence d 2 Q(τ ) 2 E 1 (0) |D1 (Q(0))|2/d 2 E 2 (0) |D2 (Q(0))|2/d = − , 1+2/d 2 dτ d |D1 (Q(τ ))| d |D2 (Q(τ ))|1+2/d and so (Q, W ) behave as if they were the coordinates of a Hamiltonian system describing a particle undergoing motion inside a potential well. The effective Hamiltonian may be expressed as 1 2 E 1 (0) |D1 (Q(0))|2/d E 2 (0) |D2 (Q(0))|2/d W + + . 2 |D1 (Q)|2/d |D2 (Q)|2/d

(4)

The question is, do the solutions of Eq. (3) give an accurate description of the actual motions of the macroscopic variables when M tends to infinity? The main result of this paper is that, for an appropriately defined system, the answer to this question is

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affirmative for 0 ≤ t ≤ M 1/2 , at least for most initial conditions of the microscopic variables. Observe that one should not expect the description to be accurate on time scales much longer than O(M 1/2 ) = O(ε−1 ). The reason for this is that, presumably, there are corrections of size O(ε2 ) in Eqs. (1) and (2) that we are neglecting. On the time scale ε−1 , these errors roughly add up to no more than size O(ε−1 · ε2 = ε), but on a longer time scale they should become significant. Such higher order corrections for the adiabatic piston were studied by Crosignani et al. [CDPS96]. 3. Statement of the Main Result 3.1. Description of the model. We begin by describing the gas container. It is a compact, connected billiard domain D ⊂ Rd with a piecewise C 3 boundary, i.e. ∂D consists of a finite number of C 3 embedded hypersurfaces, possibly with boundary and a finite number of corner points. The container consists of a “tube,” whose perpendicular crosssection P is the shape of the piston, connecting two disjoint regions. P ⊂ Rd−1 is a compact, connected domain whose boundary is piecewise C 3 . Then the “tube” is the region [0, 1] × P ⊂ D swept out by the piston for 0 ≤ Q ≤ 1, and [0, 1] × ∂P ⊂ ∂D. If d = 2, P is just a closed line segment, and the “tube” is a rectangle. If d = 3, P could be a circle, a square, a pentagon, etc. Our fundamental assumption is as follows: Main Assumption. For almost every Q ∈ [0, 1] the billiard flow of a single particle on an energy surface in either of the two subdomains Di (Q) is ergodic (with respect to the invariant Liouville measure). If d = 2, the domain could be the Bunimovich stadium [Bun79]. Another possible domain is indicated in Fig. 1. Polygonal domains satisfying our assumptions can also be constructed [Vor97]. Suitable domains in d = 3 dimensions can be constructed using a rectangular box with shallow spherical caps adjoined [BR98]. Note that we make no assumptions regarding the hyperbolicity of the billiard flow in the domain. The Hamiltonian system we consider consists of the massive piston of mass M located at position Q, as well as n 1 + n 2 gas particles, n 1 in D1 and n 2 in D2 . Here n 1 and n 2 are fixed positive integers. For convenience, the gas particles all have unit mass, though all that is important is that each gas particle has a fixed mass. We denote the positions of the gas particles in Di by qi, j , 1 ≤ j ≤ n i . The gas particles are ideal point particles that interact with ∂D and the piston by hard core, elastic collisions. Although it has no effect on the dynamics we consider, for convenience we complete our description of the Hamiltonian dynamics by specifying that the piston makes elastic collisions with walls located at Q = 0, 1 that are only visible to the piston. We denote velocities by d Q/dt = V = εW and dqi, j /dt = vi, j , and we set E i, j = vi,2 j /2,

Ei =

ni

E i, j .

j=1

Our system has d(n 1 + n 2 ) + 1 degrees of freedom, and so its phase space is (2d(n 1 + n 2 ) + 2)-dimensional. We let h(z) = h = (Q, W, E 1,1 , E 1,2 , . . . , E 1,n 1 , E 2,1 , E 2,2 , . . . , E 2,n 2 ),

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P. Wright

so that h is a function from our phase space to Rn 1 +n 2 +2 . We often abbreviate h = (Q, W, E 1, j , E 2, j ), and we refer to h as consisting of the slow variables because these quantities are conserved when ε = 0. We let h ε (t, z) = h ε (t) denote the actual motions of these variables in time for a fixed value of ε. Here z represents the initial condition in phase space, which we usually suppress in our notation. One should think of h ε (·) as being a random variable that takes initial conditions in phase space to paths (depending on the parameter t) in Rn 1 +n 2 +2 . 3.2. The averaged equation. From the work of Neishtadt and Sinai [NS04], one can derive ⎡ ⎤ W ⎡ ⎤ Q 2E 1  2E 2  ⎥ ⎢ d|D d ⎢ W ⎥ ⎢ 1 (Q)| − d|D2 (Q)| ⎥ ¯ 2W E 1, j  (5) ⎥ ⎣ ⎦ = H (h) := ⎢ − d|D1 (Q)| ⎣ ⎦ dτ E 1, j 2W E  E 2, j 2, j + d|D2 (Q)|

as the averaged equation (with respect to the slow time τ = εt) for the slow variables. Later, in Sect. 4.3, we will give another heuristic derivation of the averaged equation that is more suggestive of our proof. As in Sect. 2, the solutions of Eq. (5) have (Q, W ) behaving as if they were the coordinates of a Hamiltonian system describing a particle undergoing motion inside a potential well. The effective Hamiltonian is given by Eq. (4). ¯ z) = h(τ ¯ ) be the solution of Let h(τ, d h¯ ¯ = H¯ (h), dτ

¯ h(0) = h ε (0).

¯ as being a random variable. Again, think of h(·) 3.3. The main result. The solutions of the averaged equation approximate the motions of the slow variables, h ε (t), on a time scale O(1/ε) as ε → 0. Precisely, fix a compact set V ⊂ Rn 1 +n 2 +2 such that h ∈ V ⇒ Q ⊂⊂ (0, 1), W ⊂⊂ R, and E i, j ⊂⊂ (0, ∞) for each i and j.1 We will be mostly concerned with the dynamics when h ∈ V. Define Q min = inf Q, h∈V

E min

Q max = sup Q, h∈V

1 = inf W 2 + E 1 + E 2 , h∈V 2

1 E max = sup W 2 + E 1 + E 2 . h∈V 2

For a fixed value of ε > 0, we only consider the dynamics on the invariant subset of phase space defined by Mε = {(Q, V, qi, j , vi, j ) ∈ R2d(n 1 +n 2 )+2 : Q ∈ [0, 1], qi, j ∈ Di (Q), M 2 E min ≤ V + E 1 + E 2 ≤ E max }. 2 Let Pε denote the probability measure obtained by restricting the invariant Liouville measure to Mε . Define the stopping time ¯ )∈ Tε (z) = Tε = inf{τ ≥ 0 : h(τ / V or h ε (τ/ε) ∈ / V}. 1 We have introduced this notation for convenience. For example, h ∈ V ⇒ Q ⊂⊂ (0, 1) means that there exists a compact set A ⊂ (0, 1) such that h ∈ V ⇒ Q ∈ A, and similarly for the other variables.

Periodic Oscillation of an Adiabatic Piston in 2 or 3 Dimensions

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Theorem 1. If D is a gas container in d = 2 or 3 dimensions satisfying the assumptions in Subsect. 3.1 above, then for each T > 0,



¯ ) → 0 in probability as ε = M −1/2 → 0, sup h ε (τ/ε) − h(τ 0≤τ ≤T ∧Tε

i.e. for each fixed δ > 0,  Pε

sup

0≤τ ≤T ∧Tε



h ε (τ/ε) − h(τ ¯ ) ≥ δ

→ 0 as ε = M −1/2 → 0.

It should be noted that the stopping time in the above result is not unduly restrictive. If the initial pressures of the two gasses are not too mismatched, then the solution to the averaged equation is a periodic orbit, with the effective potential well keeping the piston away from the walls. Thus, if the actual motions follow the averaged solution closely for 0 ≤ τ ≤ T ∧ Tε , and the averaged solution stays in V, it follows that Tε > T . The techniques of this paper should immediately generalize to prove the analogue of Theorem 1 above in the nonphysical dimensions d > 3, although we do not pursue this here. 4. Preparatory Material Concerning a Two-Dimensional Gas Container with only One Gas Particle on Each Side Our results and techniques of proof are essentially independent of the dimension and the fixed number of gas particles on either side of the piston. Thus, we focus on the case when d = 2 and there is only one gas particle on either side. Later, in Sect. 6, we will indicate the simple modifications that generalize our proof to the general situation. For clarity, in this section and next, we denote q1,1 by q1 , v2,1 by v2 , etc. We decompose the gas particle coordinates according to whether they are perpendicular to or parallel to the  piston’s face, for example q1 = (q1⊥ , q1 ). See Fig. 2. The Hamiltonian dynamics define a flow on our phase space. We denote this flow by z ε (t, z) = z ε (t), where z = z ε (0, z). One should think of z ε (·) as being a random variable that takes initial conditions in phase space to paths in phase space. Then h ε (t) = h(z ε (t)). By the change of coordinates W = V /ε, we may identify all of the Mε defined in Sect. 3 with the space M = {(Q, W, q1 , v1 , q2 , v2 ) ∈ R10 : Q ∈ [0, 1], q1 ∈ D1 (Q), q2 ∈ D2 (Q), 1 E min ≤ W 2 + E 1 + E 2 ≤ E max }, 2

Fig. 2. A choice of coordinates on phase space

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P. Wright

and all of the Pε with the probability measure P on M, which has the density 







d P = const d QdW dq1⊥ dq1 dv1⊥ dv1 dq2⊥ dq2 dv2⊥ dv2 . (Throughout this work we will use const to represent generic constants that are independent of ε.) We will assume that these identifications have been made, so that we may consider z ε (·) as a family of measure preserving flows on the same space that all preserve the same probability measure. We denote the components of z ε (t) by Q ε (t), ⊥ (t), etc. q1,ε

The set {z ∈ M : q1 = Q = q2 } has co-dimension two, and so t z ε (t){q1 = Q = q2 } has co-dimension one, which shows that only a measure zero set of initial conditions will give rise to three particle collisions. We ignore this and other measure zero events, such as gas particles hitting singularities of the billiard flow, in what follows. Now we present some background material, as well as some lemmas that will assist us in our proof of Theorem 1. We begin by studying the billiard flow of a gas particle when the piston is infinitely massive. Next we examine collisions between the gas particles and the piston when the piston has a large, but finite, mass. Then we present a heuristic derivation of the averaged equation that is suggestive of our proof. Finally we prove a lemma that allows us to disregard the possibility that a gas particle will move nearly parallel to the piston’s face – a situation that is clearly bad for having the motions of the piston follow the solutions of the averaged equation. 4.1. Billiard flows and maps in two dimensions. In this section, we study the billiard flows of the gas particles when M = ∞ and the slow variables are held fixed at a specific value h ∈ V. We will only study the motions of the left gas particle, as similar definitions and results hold for the motions of the right gas particle. Thus we wish to study √ the billiard flow of a point particle moving inside the domain D1 at a constant speed 2E 1 . The results of this section that are stated without proof can be found in [CM06a]. Let T D1 denote the tangent bundle to D1 . The billiard flow takes place √ in the threedimensional space M1h = M1 = {(q1 , v1 ) ∈ T D1 : q1 ∈ D1 , |v1 | = 2E 1 }/ ∼. Here the quotient means that when q1 ∈ ∂D1 , we identify velocity vectors pointing outside of D1 with those pointing inside D1 by reflecting through the tangent line to ∂D1 at q1 , so that the angle of incidence with the unit normal vector to ∂D1 equals the angle of reflection. Note that most of the quantities defined in this subsection depend on the fixed value of h. We will usually suppress this dependence, although, when necessary, we will indicate it by a subscript h. We denote the resulting flow by y(t, y) = y(t), where y(0, y) = y. As the billiard flow comes from a Hamiltonian system, it preserves Liouville measure restricted to the energy surface. We denote the √ resulting probability measure by µ. This measure has the density dµ = dq1 dv1 /(2π 2E 1 |D1 |). Here  √ dq1 1 2 : |v | = 2E . represents area on R2 , and dv1 represents length on S√ = v ∈ R 1 1 1 2E 1 There is a standard cross-section to√the billiard flow, the collision cross-section  = {(q1 , v1 ) ∈ T D1 : q1 ∈ ∂D1 , |v1 | = 2E 1 }/ ∼. It is customary to parameterize  by {x = (r, ϕ) : r ∈ ∂D1 , ϕ ∈ [−π/2, +π/2]}, where r is arc length and ϕ represents the angle between the outgoing velocity vector and the inward pointing normal vector to ∂D1 . It follows that  may be realized as the disjoint union of a finite number of rectangles and cylinders. The cylinders correspond to fixed scatterers with smooth boundary placed inside the gas container. If F : 
is the collision map, i.e. the return map to the collision cross-section, then F preserves the projected probability measure ν, which has the density dν = cos ϕ dϕ dr/(2 |∂D1 |). Here |∂D1 | is the length of ∂D1 .

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We suppose that the flow is ergodic, and so F is an invertible, ergodic measure preserving transformation. Because ∂D1 is piecewise C 3 , F is piecewise C 2 , although it does have discontinuities and unbounded derivatives near discontinuities corresponding to grazing collisions. Because of our assumptions on D1 , the free flight times and the curvature of ∂D1 are uniformly bounded. It follows that if x ∈ / ∂ ∪ F −1 (∂), then F is differentiable at x, and const D F(x) ≤ , (6) cos ϕ(F x) where ϕ(F x) is the value of the ϕ coordinate at the image of x. ˆ of  correspondFollowing the ideas in Appendix A, we induce F on the subspace  ing to collisions with the (immobile) piston. We denote the induced map by Fˆ and the ˆ by {(r, ϕ) : 0 ≤ r ≤ , ϕ ∈ [−π/2, +π/2]}. induced measure by νˆ . We parameterize  ˆ As ν  = / |∂D1 |, it follows that νˆ has the density d νˆ = cos ϕ dϕ dr/(2). For x ∈ , define the billiard particle √ ζ x to be the free flight time, i.e. the time it takes traveling at speed 2E 1 to travel from x to F x. If x ∈ / ∂ ∪ F −1 (∂), Dζ (x) ≤

const . cos ϕ(F x)

(7)

Santaló’s formula [San76, Che97] tells us that π |D1 | . |v1 | |∂D1 |

Eν ζ =

(8)

ˆ → R is the free flight time between collisions with the piston, then it follows If ζˆ :  from Proposition 10 that π |D1 | . (9) E νˆ ζˆ = |v1 | 



The expected value of v1⊥ when the left gas particle collides with the (immobile) piston is given by √ 



  2E 1 +π/2 π

⊥

E νˆ v1 = E νˆ 2E 1 cos ϕ = cos2 ϕ dϕ = 2E 1 . 2 4 −π/2

(10)

t

We wish to compute limt→∞ t −1 0 2v1⊥ (s) δq ⊥ (s)=Q ds, the time average of the 1 change in momentum of the left gas particle when it collides with the piston. If this limit exists and is equal for almost every initial condition of the left gas particle, then it makes sense to define the pressure inside D1 to be this quantity divided by . Because the collisions are hard-core, we cannot directly apply Birkhoff’s Ergodic Theorem to ˆ compute this limit. However, we can compute this limit by using the map F. Lemma 2. If the billiard flow y(t) is ergodic, then for µ − a.e. y ∈ M1 , 1 t→∞ t lim

 t



⊥

v1 (s) δq ⊥ (s)=Q ds = 0

1

E1 . 2 |D1 (Q)|

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Proof. Because the billiard flow may be viewed as a suspension flow over the collision cross-section with ζ as the height function, it suffices to show that the convergence ˆ For an initial condition x ∈ , ˆ define Nˆ t (x) = Nˆ t = takes ∈ .  place for νˆ − a.e.x  ˆ # s ∈ (0, t] : y(s, x) ∈  . By the Poincaré Recurrence Theorem, Nˆ t → ∞ as t → ∞, νˆ − a.e. But  Nˆ t

1 t

⊥

1

⊥ ˆ n ( F x) ≤

v1

v (s) δq ⊥ (s)=Q ds  Nˆ t 1 t 0 1 ˆ ( Fˆ n x) Nˆ t n=1 ζ n=0 Nˆ t

Nˆ t

Nˆ t 1

⊥ ˆ n ≤ ˆ

v1 ( F x), Nt −1 ˆ ˆ n ζ ( F x) Nˆ t n=0 n=0

and so the result follows from Birkhoff’s Ergodic Theorem and Eqs. (9) and (10).

 

Corollary 3. If the billiard flow y(t) is ergodic, then for each δ > 0,

 t

 



E 1 

⊥

1 1 ≥ δ → 0 as t → ∞. µ y∈M :

v1 (s) δq ⊥ (s)=Q ds − 1 t 0 2 |D1 (Q)|

4.2. Analysis of collisions. In this section, we return to studying our piston system when ε > 0. We will examine what happens when a particle collides with the piston. For convenience, we will only examine in detail collisions between the piston and the left gas particle. Collisions with the right gas particle can be handled similarly. When the left gas particle collides with the piston, v1⊥ and V instantaneously change according to the laws of elastic collisions:  ⊥+     ⊥−  1 1− M 2M v1 v1 = . 2 M − 1 V− V+ 1+M In our coordinates, this becomes  ⊥+   2 1 ε −1 v1 = W+ 2ε 1 + ε2

2ε 1 − ε2

  ⊥−  v1 . W−

(11)

Recalling that v1 , W = O(1), we find that to first order in ε, v1⊥+ = −v1⊥− + O(ε),

W + = W − + O(ε).

(12)

Observe that a collision can only take place if v1⊥− > εW − . In particular, v1⊥− > √ −ε 2E max . Thus, either v1⊥− > 0 or v1⊥− = O(ε). By expanding Eq. (11) to second order in ε, it follows that





E 1+ − E 1− = −2εW v1⊥ + O(ε2 ),



(13)



W + − W − = +2ε v1⊥ + O(ε2 ).

Periodic Oscillation of an Adiabatic Piston in 2 or 3 Dimensions

563

Note that

it is immaterial whether we use the pre-collision or post-collision values of W and v1⊥ on the right-hand side of Eq. (13), because any ambiguity can be absorbed into the O(ε2 ) term. It is convenient for us to define a “clean collision” between the piston and the left gas particle: Definition 1. The left gas particle √ experiences a clean collision with the piston if and only if v1⊥− > 0 and v1⊥+ < −ε 2E max . In particular, after a clean collision, the left gas particle will escape from the piston, i.e. the left gas particle will have to move into the region q1⊥ ≤ 0 before it can experience another collision with the piston. It follows that there exists a constant C1 > 0, which on the set V, such that for all ε sufficiently small, so long as Q ≥ Q min

depends

and v1⊥ > εC1 when q1⊥ ∈ [Q min , Q], then the left gas particle will experience only clean collisions √ with the piston, and the time between these collisions will be greater than 2Q min /( 2E max ). (Note that when we write expressions such as q1⊥ ∈ [Q min , Q], we implicitly mean that q1 is positioned √ inside the “tube” discussed at the beginning of Sect. 3.) One can verify that C1 = 5 2E max would work. Similarly, we can define clean collisions between the right gas particle and the piston. We assume that C1 was chosen

sufficiently large such that for all ε sufficiently small, so long as Q ≤ Q max and v2⊥ > εC1 when q2⊥ ∈ [Q, Q max ], then the right gas particle will experience only clean collisions with the piston. Now we define three more stopping times, which are functions of the initial conditions in phase space:



⊥

⊥ Tε = inf{τ ≥ 0 : Q min ≤ q1,ε (τ/ε) ≤ Q ε (τ/ε) ≤ Q max and v1,ε (τ/ε) ≤ C1 ε},



⊥

⊥ Tε = inf{τ ≥ 0 : Q min ≤ Q ε (τ/ε) ≤ q2,ε (τ/ε) ≤ Q max and v2,ε (τ/ε) ≤ C1 ε}, T˜ε =T ∧ Tε ∧ Tε ∧ Tε . Define H (z) by ⎡

⎤ W

⊥

⎢+2 v1 δq ⊥ =Q − 2 v2⊥ δq ⊥ =Q ⎥ ⎢ ⎥ 1

2

H (z) = ⎢ ⎥. −2W v1⊥ δq ⊥ =Q ⎣ ⎦

⊥ 1 +2W v2 δq ⊥ =Q 2

Here we make use of Dirac delta functions. All integrals involving these delta functions may be replaced by sums. The following lemma is an immediate consequence of Eq. (13) and the above discussion: Lemma 4. If 0 ≤ t1 ≤ t2 ≤ T˜ε /ε, the piston experiences O((t2 − t1 ) ∨ 1) collisions with gas particles in the time interval [t1 , t2 ], all of which are clean collisions. Furthermore,  t2 h ε (t2 ) − h ε (t1 ) = O(ε) + ε H (z ε (s))ds. t1

Here any ambiguities arising from collisions occurring at the limits of integration can be absorbed into the O(ε) term.

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4.3. Another heuristic derivation of the averaged equation. The following heuristic derivation of Eq. (5) when d = 2 was suggested in [Dol05]. Let t be a length of time long enough such that the piston experiences many collisions with the gas particles, but short enough such that the slow variables change very little, in this time interval. From each collision with the left gas particle, Eq. (13) states that W changes by an amount

+2ε v1⊥ + O(ε2 ), and from Eq. (10) the average change in W at these collisions should √ 2 be approximately √ επ 2E 1 /2 + O(ε ). From Eq. (9) the frequency of these collisions is approximately 2E 1 /(π |D1 |). Arguing similarly for collisions with the other particle, we guess that W E1 E2  =ε −ε + O(ε2 ). |D1 (Q)| |D2 (Q)| t With τ = εt as the slow time, a reasonable guess for the averaged equation for W is dW E1 E2  = − . |D1 (Q)| |D2 (Q)| dτ Similar arguments for the other slow variables lead to the averaged equation (5), and this explains why we used Pi = E i / |Di | for the pressure of a 2-dimensional gas in Sect. 2. There is a similar heuristic derivation of the averaged equation in d > 2 dimensions. Compare the analogues of Eqs. (9) and (10) in Subsect. 6.2.

4.4. A priori estimate on the size of a set of bad initial conditions. In this section, we give an a priori estimate on the size

conditions that should not give rise

of a set of initial ¯ ) is small. In particular, when proving to orbits for which sup0≤τ ≤T ∧Tε h ε (τ/ε) − h(τ Theorem 1, it is convenient to focus on orbits that only contain clean collisions with the piston. Thus, we show that P{T˜ε < T ∧ Tε } vanishes as ε → 0. At first, this result may seem surprising, since P{Tε ∧ Tε = 0} = O(ε), and one would expect T /ε ∪t=0 z ε (−t){Tε ∧ Tε = 0} to have a size of order 1. However, the rate at which orbits escape from {Tε ∧ Tε = 0} is very small, and so we can prove the following: Lemma 5. P{T˜ε < T ∧ Tε } = O(ε). In some sense, this lemma states that the probability of having a gas particle move nearly parallel to the piston’s face within the time interval [0, T /ε], when one would expect the other gas particle to force the piston to move on a macroscopic scale, vanishes as ε → 0. Thus, one can hope to control the occurrence of the “nondiffusive fluctuations” of the piston described in [CD06a] on a time scale O(ε−1 ). Proof. As the left and the right gas particles can be handled similarly, it suffices to show that P{Tε < T } = O(ε). Define





Bε = {z ∈ M : Q min ≤ q1⊥ ≤ Q ≤ Q max and v1⊥ ≤ C1 ε}.

Periodic Oscillation of an Adiabatic Piston in 2 or 3 Dimensions

565

√ T /ε Then {Tε < T } ⊂ ∪t=0 z ε (−t)Bε , and if γ = Q min / 8E max , ⎞ ⎞ ⎛ ⎞ ⎛ ⎛ T /ε T /ε T /ε    P ⎝ z ε (−t)Bε ⎠ = P ⎝ z ε (t)Bε ⎠ = P ⎝Bε ∪ ((z ε (t)Bε )\Bε )⎠ t=0

t=0

⎛

≤ PBε + P ⎝  ≤ PBε +

T /(εγ )

 z ε (kγ )

t=0 γ 

⎞

(z ε (t)Bε )\Bε ⎠

t=0

k=0

  γ T +1 P (z ε (t)Bε )\Bε . εγ t=0

" 2 Now PBε = O(ε), so if we can show that P t=0 (z ε (t)Bε )\Bε = O(ε ), then it  will follow that P{Tε < T } = O(ε).

⊥



γ



If z ∈ t=0 (z ε (t)Bε )\Bε , it is still true that v1⊥ = O(ε). This

⊥ is because v1



changes by at most O(ε) at the collisions, and if a collision forces v1 > C1 ε, then the gas particle must escape to the region q1⊥ ≤ 0 before v1⊥ can change again, and this will

γ take time greater than γ . Furthermore, if z ∈ t=0 (z ε (t)Bε )\Bε , then at least one of the following four possibilities must hold:



• q1⊥ − Q min ≤ O(ε), • |Q − Q min | ≤ O(ε), • |Q − Q max

| ≤ O(ε), • Q − q1⊥ ≤ O(ε). ! γ " 2 It follows that P t=0 (z ε (t)Bε )\Bε = O(ε ). For example,  1{

v ⊥

≤O(ε),

q ⊥ −Q min

≤O(ε)} d P 1 1 M    ⊥ ⊥



= const   1{ v ⊥ ≤O (ε)} dW dv1 dv1 dv2 dv2 ! γ

 ×

E min ≤W 2 /2+v12 /2+v22 /2≤E max

{Q∈[0,1], q1 ∈D1 , q2 ∈D2 }

1





1{

q ⊥ −Q min

≤O(ε)} d Qdq1⊥ dq1 dq2⊥ dq2 1

= O(ε2 ).   5. Proof of the Main Result for Two-Dimensional Gas Containers with only One Gas Particle on Each Side As in Sect. 4, we continue with the case when d = 2 and there is only one gas particle on either side of the piston. 5.1. Main steps in the proof of convergence in probability. By Lemma 5, it suffices to

¯ ) → 0 in probability as ε = M −1/2 → 0. Several show that sup0≤τ ≤T˜ε h ε (τ/ε) − h(τ of the ideas in the steps below were inspired by a recent proof of Anosov’s averaging theorem for smooth systems that is due to Dolgopyat [Dol05].

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P. Wright

¯ ) satisfies the integral Step 1: Reduction using Gronwall’s Inequality. Observe that h(τ equation  τ ¯h(τ ) − h(0) ¯ ¯ ))dσ, = H¯ (h(σ 0

while from Lemma 4,



h ε (τ/ε) − h ε (0) = O(ε) + ε

τ/ε

0

 = O(ε) + ε

τ/ε

0

for 0 ≤ τ ≤ T˜ε . Define eε (τ ) = ε



H (z ε (s))ds H (z ε (s)) − H¯ (h ε (s))ds +

τ/ε 0

 0

τ

H¯ (h ε (σ/ε))dσ

H (z ε (s)) − H¯ (h ε (s))ds.

It follows from Gronwall’s Inequality that 



¯ ) ≤ O(ε) + sup |eε (τ )| eLip( H¯ |V )T . sup h ε (τ/ε) − h(τ 0≤τ ≤T˜ε

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0≤τ ≤T˜ε

Gronwall’s Inequality is usually stated for continuous paths, but the

standard proof

¯ )

(found in [SV85]) still works for paths that are merely integrable, and h ε (τ/ε) − h(τ is piecewise smooth. Step 2: Introduction of a time scale for ergodization. Let L(ε) be a real valued function such that L(ε) → ∞, but L(ε)  log ε−1 , as ε → 0. In Sect. 5.2 we will place precise restrictions on the growth rate of L(ε). Think of L(ε) as being a time scale that grows as ε → 0 so that ergodization, i.e. the convergence along an orbit of a function’s time average to a space average, can take place. However, L(ε) doesn’t grow too fast, so that on this time scale z ε (t) essentially stays on the submanifold {h = h ε (0)}, where we have our ergodicity assumption. Set tk,ε = k L(ε), so that





 tk+1,ε

H (z ε (s)) − H¯ (h ε (s))ds . sup |eε (τ )| ≤ O(εL(ε)) + ε



t ˜ k,ε 0≤τ ≤T T˜ε εL(ε) −1

ε

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k=0

Step 3: A splitting according to particles. Now H (z)− H¯ (h(z)) divides into two pieces, each of which depends on only one gas particle when the piston is held fixed: ⎡ ⎤ ⎤ ⎡ 0 0

⊥

E1  E2  ⊥





⎢ 2 v1 δq ⊥ =Q − |D (Q)| ⎥ ⎢ |D (Q)| − 2 v2 δq ⊥ =Q ⎥ 1 2 2 ⎥. ⎥ ⎢

1 H (z) − H¯ (h(z)) = ⎢ W E1  ⎦ + ⎣ ⎣−2W v ⊥ δ ⊥ ⎦ 0 +



q1 =Q 1 |D1 (Q)| W E2  ⊥



− |D2 (Q)| + 2W v2 δq ⊥ =Q 0 2

We will only deal with the piece depending on the left gas particle, as the right particle can be handled similarly. Define



E1



¯ G(z) = v1⊥ δq ⊥ =Q , G(h) = . (16) 1 2 |D1 (Q)|

Periodic Oscillation of an Adiabatic Piston in 2 or 3 Dimensions

567

Returning to Eq. (15), we see that in order to prove Theorem 1, it suffices to show that both





 tk+1,ε ¯ ε (s))ds

and ε G(z ε (s)) − G(h

tk,ε

T˜ε εL(ε) −1

k=0





 tk+1,ε ! "

¯ ε (s)) ds

ε Wε (s) G(z ε (s)) − G(h

tk,ε

T˜ε εL(ε) −1

k=0

converge to 0 in probability as ε → 0. Step 4: A splitting for using the triangle inequality. Now we let z k,ε (s) be the orbit of the ε = 0 Hamiltonian vector field satisfying z k,ε (tk,ε ) = z ε (tk,ε ). Set h k,ε (t) = h(z k,ε (t)). Observe that h k,ε (t) is independent of t. We emphasize that so long as 0 ≤ t ≤ T˜ε /ε, the times between collisions of a specific gas particle and piston are uniformly bounded greater than 0, as explained before Lemma 4. It follows that, so long as tk+1,ε ≤ T˜ε /ε, sup

tk,ε ≤t≤tk+1,ε



h k,ε (t) − h ε (t) = O(εL(ε)).

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This is because the slow variables change by at most O(ε) at collisions, and d Q ε /dt = O(ε). Also, 

tk+1,ε

tk,ε

! " ¯ ε (s)) ds Wε (s) G(z ε (s)) − G(h  = O(εL(ε)2 ) + Wk,ε (tk,ε )

tk+1,ε tk,ε

¯ ε (s))ds, G(z ε (s)) − G(h

and so





 tk+1,ε ! "

¯ ε (s)) ds

ε Wε (s) G(z ε (s)) − G(h

tk,ε

T˜ε εL(ε) −1

k=0







 tk+1,ε ¯ ε (s))ds

. ≤ O(εL(ε)) + ε const G(z ε (s)) − G(h

tk,ε

T˜ε εL(ε) −1

k=0

Thus, in order to prove Theorem 1, it suffices to show that T˜ε



εL(ε) −1



 tk+1,ε





Ik,ε + I Ik,ε + I I Ik,ε

¯ ε (s))ds

≤ ε ε G(z ε (s)) − G(h

tk,ε

T˜ε εL(ε) −1

k=0

k=0

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P. Wright

converges to 0 in probability as ε → 0, where  Ik,ε = I Ik,ε = I I Ik,ε =

tk+1,ε

tk,ε  tk+1,ε tk,ε  tk+1,ε tk,ε

G(z ε (s)) − G(z k,ε (s))ds, ¯ k,ε (s))ds, G(z k,ε (s)) − G(h ¯ ε (s))ds. ¯ k,ε (s)) − G(h G(h

The term I Ik,ε represents an “ergodicity term” that can be controlled by our assumptions on the ergodicity of the flow z 0 (t), while the terms Ik,ε and I I Ik,ε represent “continuity terms” that can be controlled by controlling the drift of z ε (t) from z k,ε (t) for tk,ε ≤ t ≤ tk+1,ε . Step 5: Control of drift from the ε = 0 orbits. Now G¯ is uniformly Lipschitz on the compact set V, and so it follows from Eq. (17) that I I Ik,ε = O(εL(ε)2 ). Thus, T˜ε

 εL(ε) −1

I I Ik,ε = O(εL(ε)) → 0 as ε → 0. ε k=0  T˜ε

 εL(ε) −1



Next, we show that for fixed δ > 0, P ε k=0 Ik,ε ≥ δ → 0 as ε → 0. For initial conditions z ∈ M and for integers k ∈ [0, T /(εL(ε)) − 1] define $ # δ T˜ε 1



Ak,ε = z : Ik,ε > and k ≤ −1 , L(ε) 2T εL(ε)   Az,ε = k : z ∈ Ak,ε . Think of these sets as describing “poor continuity” between solutions of the ε = 0 and the ε > 0 Hamiltonian vector fields. For example, roughly speaking, z ∈ Ak,ε if the orbit z ε (t) starting at z does not follow z k,ε (t) for tk,ε ≤ t ≤ tk+1,ε .

closely

One can easily check that Ik,ε ≤ O(L(ε)) for k ≤ T˜ε /(εL(ε))−1, and so it follows that T˜ε εL(ε) −1

ε

δ

Ik,ε ≤ + O(εL(ε)#(Az,ε )). 2 k=0

! " Therefore it suffices to show that P #(Az,ε ) ≥ δ(const εL(ε))−1 → 0 as ε → 0. By Chebyshev’s Inequality, we need only show that T εL(ε) −1

E P (εL(ε)#(Az,ε )) = εL(ε)



P(Ak,ε )

k=0

tends to 0 with ε. Observe that z ε (tk,ε )Ak,ε ⊂ A0,ε . In words, the initial conditions giving rise to orbits that are “bad” on the time interval [tk,ε , tk+1,ε ], moved forward by time tk,ε , are initial

Periodic Oscillation of an Adiabatic Piston in 2 or 3 Dimensions

569

conditions giving rise to orbits which are “bad” on the time interval [t0,ε , t1,ε ]. Because the flow z ε (·) preserves the measure, we find that T εL(ε) −1

εL(ε)



P(Ak,ε ) ≤ const P(A0,ε ).

k=0

To estimate P(A0,ε ), it is convenient to use a different probability measure, which is uniformly equivalent to P on the set {z ∈ M : h(z) ∈ V} ⊃ {T˜ε ≥ εL(ε)}. We denote this new probability measure by P f , where the f stands for “factor.” If we choose coordinates on M by using h and the billiard coordinates on the two gas particles, then P f is defined on M by d P f = dh dµ1h dµ2h , where dh represents the uniform measure on V ⊂ R4 , and the factor measure dµih represents the invariant billiard measure of the i th gas particle coordinates for a fixed value of the slow variables. One can verify that 1{h(z)∈V } d P ≤ const d P f , but that P f is not invariant under the flow z ε (·) when ε > 0. We abuse notation, and consider µ1h to be a measure on the left particle’s initial billiard coordinates once h and the initial coordinates of the right gas particle are fixed. In this context, µ1h is simply the measure µ from Subsect. 4.1. Then P f (A0,ε ) #  2 1 ≤ dh dµh · µh z



$

δ

1  L(ε)



˜ and εL(ε) ≤ Tε , G(z ε (s)) − G(z 0 (s))ds ≥ :

2T

L(ε) 0

and we must show that the last term tends to 0 with ε. By the Bounded Convergence Theorem, it suffices to show that for almost every h ∈ V and initial condition for the right gas particle,

#

$

1  L(ε) δ



1 ˜ and εL(ε) ≤ Tε → 0 as ε → 0. G(z ε (s)) − G(z 0 (s))ds ≥ µh z :

2T

L(ε) 0 (18) Note that if G were a smooth function and z ε (·) were the flow of a smooth family of vector fields Z (z, ε) that depended smoothly on ε, then from Gronwall’s Inequality, it would follow that sup0≤t≤L(ε) |z ε (t) − z 0 (t)| ≤ O(εL(ε)eLip(Z )L(ε) ). If this were the

 L(ε)



case, then L(ε)−1 0 G(z ε (s)) − G(z 0 (s))ds = O(εL(ε)eLip(Z )L(ε) ), which would tend to 0 with ε. Thus, we need a Gronwall-type inequality for billiard flows. We obtain the appropriate estimates in Sect. 5.2. Step 6: Use of ergodicity  along˜ fibers to control I Ik,ε . All that remains to be shown is Tε

 εL(ε) −1

I Ik,ε ≥ δ → 0 as ε → 0. that for fixed δ > 0, P ε k=0 For initial conditions z ∈ M and for integers k ∈ [0, T /(εL(ε)) − 1] define # $ ˜ε

δ 1

T Bk,ε = z : I Ik,ε > and k ≤ −1 , L(ε) 2T εL(ε)   Bz,ε = k : z ∈ Bk,ε .

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P. Wright

Think of these sets as describing “bad ergodization.” For example, roughly speaking, z ∈ Bk,ε if the orbit z ε (t) starting at z spends the time between tk,ε and tk+1,ε in a region of phase space where the function G(·) is “poorly ergodized” on the time

scale L(ε) by the flow z 0 (t) (as measured by the parameter δ/2T ). Note that G(z) = v1⊥ δq ⊥ =Q is not 1 t really a function, but that we may still speak of the convergence of t −1 0 G(z 0 (s))ds as ¯ 0 ) for almost every initial condition. t → ∞. As we showed in Lemma 2, the limit is G(h Proceeding as in Step 5 above, we find that it suffices to show that for almost every h ∈ V,

  t 

1 δ 1



¯ µh z :

G(z 0 (s))ds − G(h 0 (0)) ≥ → 0 as t → ∞. t 0 2T But this is simply a question of examining billiard flows, and it follows immediately from Corollary 3 and our Main Assumption.

5.2. A Gronwall-type inequality for billiards. We begin by presenting a general version of Gronwall’s Inequality for billiard maps. Then we will show how these results imply the convergence required in Eq. (18). 5.2.1. Some inequalities for the collision map. In this section, we consider the value of the slow variables to be fixed at h 0 ∈ V. We will use the notation and results presented in Sect. 4.1, but because the value of the slow variables is fixed, we will omit it in our notation. Let ρ, γ , and λ satisfy 0 < ρ  γ  1  λ < ∞. Eventually, these quantities will be chosen to depend explicitly on ε, but for now they are fixed. Recall that the phase space  for the collision map F is a finite union of disjoint rectangles and cylinders. Let d(·, ·) be the Euclidean metric on connected components of . If x and x  belong to different components, then we set d(x, x  ) = ∞. The invariant measure ν satisfies ν < const · (Lebesgue measure). For A ⊂  and a > 0, let Na (A) = {x ∈  : d(x, A) < a} be the a-neighborhood of A. For x ∈  let xk (x) = xk = F k x, k ≥ 0, be its forward orbit. Suppose x ∈ / Cγ ,λ , where ! " ! λ " ∪k=0 F −k Nγ (F −1 Nγ (∂)) . Cγ ,λ = ∪λk=0 F −k Nγ (∂) Thus for 0 ≤ k ≤ λ, xk is well defined, and from Eq. (6) it satisfies d(x  , xk ) ≤ γ ⇒ d(F x  , xk+1 ) ≤

const d(x  , xk ). γ

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Next, we consider any ρ-pseudo-orbit xk obtained from x by adding on an error of size  ≤ ρ at each application of the map, i.e. d(x0 , x0 ) ≤ ρ, and for k ≥ 1, d(xk , F xk−1 ) ≤ ρ.  Provided d(x j , x j ) < γ for each j < k, it follows that d(xk , xk )

≤ρ

 k  const j j=0

γ

 ≤ const ρ

const γ

k .

(20)

Periodic Oscillation of an Adiabatic Piston in 2 or 3 Dimensions

In particular, if ρ, γ , and λ were chosen such that   const λ const ρ < γ, γ

571

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then Eq. (20) will hold for each k ≤ λ. We assume that Eq. (21) is true. Then we can also control the differences in elapsed flight times using Eq. (7):  k



ζ xk − ζ x  ≤ const ρ const . (22) k γ γ It remains to estimate the size νCγ ,λ of the set of x for which the above estimates do not hold. Using Lemma 6 below, ! " νCγ ,λ ≤ (λ+1) νNγ (∂)+νNγ (F −1 Nγ (∂)) ≤ O(λ(γ +γ 1/3 )) = O(λγ 1/3 ). (23) Lemma 6. As γ → 0, νNγ (F −1 Nγ (∂)) = O(γ 1/3 ). This estimate is not necessarily the best possible. For example, for dispersing billiard tables, where the curvature of the boundary is positive, one can show that νNγ (F −1 Nγ (∂)) = O(γ ). However, the estimate in Lemma 6 is general and sufficient for our needs. Proof. First, we note that it is equivalent to estimate νNγ (FNγ (∂)), as F has the measure-preserving involution I(r, ϕ) = (r, −ϕ), i.e. F −1 = I ◦ F ◦ I [CM06b]. Fix α ∈ (0, 1/2), and cover Nγ (∂) with O(γ −1 ) starlike sets, each of diameter no greater than O(γ ). For example, these sets could be squares of side length γ . Enumerate   the sets as {Ai }. Set G = i : F Ai ∩ Nγ α (∂) = ∅% . % If i ∈ G, F| Ai is a diffeomorphism satisfying % D F| Ai % ≤ O(γ −α ). See Eq. (6). ! " Thus diameter (F Ai ) ≤ O(γ 1−α ), and so diameter Nγ (F Ai ) ≤ O(γ 1−α ). Hence νNγ (F Ai ) ≤ O(γ 2(1−α) ), and νNγ (∪i∈G F Ai ) ≤ O(γ 1−2α ). If i ∈ / G, Ai ∩ F −1 (Nγ α (∂)) = ∅. Thus Ai might be cut into many pieces −1 by F (∂), but each of these pieces must be mapped near ∂. In fact, F Ai ⊂ NO(γ α ) (∂). This is because outside F −1 (Nγ α (∂)), D F ≤ O(γ −α ), and so points in F Ai are no more than a distance O(γ /γ α ) away from Nγ α (∂), and γ < γ 1−α < γ α . It follows that Nγ (F Ai ) ⊂ NO(γ α ) (∂), and νNO(γ α ) (∂) = O(γ α ). Thus νNγ (F −1 Nγ (∂)) = O(γ 1−2α + γ α ), and we obtain the lemma by taking α = 1/3.   5.2.2. Application to a perturbed billiard flow. Returning to the end of Step 5 in Sect. 5.1, let the initial conditions of the slow variables be fixed at h 0 = (Q 0 , W0 , E 1,0 , E 2,0 ) ∈ V throughout the remainder of this section. We can assume that the billiard dynamics of the left gas particle in D1 (Q 0 ) are ergodic. Also, fix a particular value of the initial conditions for the right gas particle for the remainder of this section. Then z ε (t) and T˜ε may be thought of as random variables depending on the left gas particle’s initial conditions y ∈ M1 . Now if h ε (t) = (Q ε (t), Wε (t), E 1,ε (t), E 2,ε (t)) denotes the actual motions of the slow variables when ε > 0, it follows from Eq. (17) that, provided εL(ε) ≤ T˜ε , sup 0≤t≤L(ε)

|h 0 − h ε (t)| = O(εL(ε)).

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P. Wright

Furthermore, we only need to show that



# $  L(ε)



δ

1 1 and εL(ε) ≤ T˜ε → 0 µ y∈M :

G(z ε (s)) − G(z 0 (s))ds ≥

L(ε) 0

2T (25) as ε → 0, where G is defined in Eq. (16). For definiteness, we take the following quantities from Subsect. 5.2.1 to depend on ε as follows: 1 L(ε) = L = log log , ε γ (ε) = γ = e−L , (26) 2 λ(ε) = λ = L, Eν ζ εL ρ(ε) = ρ = const . γ The constant in the choice of ρ and ρ’s dependence on ε will be explained in the proof of Lemma 8, which is at the end of this subsection. The other choices may be explained as follows. We wish to use continuity estimates for the billiard map to produce continuity estimates for the flow on the time scale L. As the divergence of orbits should be exponentially fast, we choose L to grow sublogarithmically in ε−1 . Since from Eq. (8) the expected  flight time between collisions with ∂D1 (Q 0 ) when ε = 0 is E ν ζ = π |D1 (Q 0 )| /( 2E 1,0 |∂D1 (Q 0 )|), we expect to see roughly λ/2 collisions on this time scale. Considering λ collisions gives us some margin for error. Furthermore, we will want orbits to keep a certain distance, γ , away from the billiard discontinuities. γ → 0 as ε → 0, but γ is very large compared to the possible drift O(εL) of the slow variables on the time scale L. In fact, for each C, m, n > 0,   εL m C λ 2 = O(ε econst L ) → 0 as ε → 0. (27) γn γ Let X : M1 →  be the map taking y ∈ M1 to x = X (y) ∈ , the location of the billiard orbit of y in the collision cross-section that corresponds to the most recent time in the past that the orbit was in the collision cross-section. We consider the set of initial conditions $ # λ & −1 −1 k Eε = X (\Cγ ,λ ) X ζ (F x) > L . x ∈: k=0

Now from Eqs. (23) and (26), νCγ,λ → 0as ε → 0. Furthermore, by the ergodicity of  λ λ k −1 k F, ν x ∈  : k=0 ζ (F x) ≤ L = ν x ∈  : λ k=0 ζ (F x) ≤ E ν ζ /2 → 0 as ε → 0. But because the free flight time is bounded above, µX −1 ≤ const · ν, and so µEε → 1 as ε → 0. Hence, the convergence in Eq. (25) and the conclusion of the proof in Sect. 5.1 follow from the lemma below and Eq. (27). Lemma 7 (Analysis of deviations along good orbits). As ε → 0,  

 L

λ

1

const

sup G(z ε (s)) − G(z 0 (s))ds

= O ρ + O(L −1 ) → 0.   L γ 0 ˜ y∈Eε ∩ εL≤Tε

Periodic Oscillation of an Adiabatic Piston in 2 or 3 Dimensions

573

  Proof. Fix a particular value of y ∈ Eε ∩ εL ≤ T˜ε . For convenience, suppose that y = X (y) = x ∈ . Let y0 (t) denote the time evolution of the billiard coordinates for the left gas particle when ε = 0. Then there is some N ≤ λ such that the orbit xk = F k x = (rk , ϕk ) for 0 ≤ k ≤ N corresponds to all of the instances (in order) when y0 (t) enters the collision cross-section  = h 0 corresponding to collisions with ∂D1 (Q 0 ) for 0 ≤ t ≤ L. We write h 0 to emphasize that in this subsection we are only considering the collision cross-section corresponding to the billiard dynamics in the domain D1 (Q 0 ) at the energy level E 1,0 . In particular, F will always refer to the return map on h 0 . Also, define an increasing sequence of times tk corresponding to the actual times y0 (t) enters the collision cross-section, i.e. t0 = 0, tk = tk−1 + ζ xk−1 for k > 0. Then xk = y0 (tk ). Furthermore, define inductively N1 = inf {k > 0 : tk corresponds to a collision with the piston},   N j = inf k > N j−1 : tk corresponds to a collision with the piston . Next, let yε (t) denote the time evolution of the billiard coordinates for the left gas  = (r  , ϕ  ) of points in  particle when ε > 0. We will construct a pseudo-orbit xk,ε h0 k,ε k,ε that essentially track the collisions (in order) of the left gas particle with the boundary under the dynamics of yε (t) for 0 ≤ t ≤ L.  corresponding to the actual times First, define an increasing sequence of times tk,ε yε (t) experiences a collision with the boundary of the gas container or the moving piston. Define    ≤L , Nε = sup k ≥ 0 : tk,ε     N1,ε = inf k > 0 : tk,ε corresponds to a collision with the piston ,    N j,ε = inf k > N j−1,ε : tk,ε corresponds to a collision with the piston . Because L ≤ T˜ε (y)/ε, we know that as long as N j+1,ε ≤ Nε , then N j+1,ε − N j,ε ≥ 2.  ∈  by See the discussion in Subsect. 4.2. Then we define xk,ε h0  xk,ε

⎧    ⎨ yε (t  ) if k ∈ , / N k,ε j,ε  =  ⎩ F −1 x  k+1,ε if k ∈ N j,ε .

Lemma 8. Provided ε is sufficiently small, the following hold for each k ∈ [0, N ∧ Nε ). Furthermore, the requisite smallness of ε and the sizes of the constants inthese estimates  may be chosen independent of the initial condition y ∈ Eε ∩ εL ≤ T˜ε and of k:    is well defined. In particular, if k ∈  ) corresponds to a collision (a) xk,ε / N j,ε , yε (tk,ε point on ∂D1 (Q 0 ),  andnot to a collision point on a piece of ∂D to the right of Q 0 .  = F x (b) If k > 0 and k ∈ / N j,ε , then xk,ε k−1,ε .

574

P. Wright

(c) If k > 0 and k ∈



  , F x N j,ε , then d(xk,ε k−1,ε ) ≤ ρ and the ϕ coordinate of

 ) satisfies ϕ(y (t  )) = ϕ  + O(ε). yε (tk,ε ε k,ε k,ε  ) ≤ const ρ(const/γ )k . (d) d(xk , xk,ε (e) k = N j,ε if and only if k = N j . k  − t (f) If k > 0, tk,ε k−1,ε = tk − tk−1 + O(ρ (const/γ ) ).

We defer the proof of Lemma 8 until the end of this subsection. Assuming that ε is sufficiently small for the conclusions of Lemma 8 to be valid, we continue with the proof of Lemma 7. Set M = N ∧ Nε − 1. Note that M ≤ λ ∼ L. From (f) in Lemma 8 and Eqs. (26) and (27), we see that   M





 constλ

t M − t  ≤

t − t  →0 M,ε k,ε k−1,ε − (tk − tk−1 ) = O ρ γλ

as ε → 0.

k=1

 −t  Because the flight times tk,ε k−1,ε and tk −tk−1 are uniformly bounded above, it follows  ≥ L − const. But from Subsect. 4.2, the from the definitions of N and Nε that t M , t M,ε time between the collisions of the left gas particle with the piston are uniformly bounded away from zero. Using (c) and Eq. (24), it follows that

 L

1



G(z (s)) − G(z (s))ds ε 0

L

0

 *

 ) cos(ϕ  + O(ε))

= O(L −1 ) +

2E 1,0 cos ϕk − 2E 1,ε (tk,ε k,ε k∈{ N j :N j ≤M }

 



 = O(L −1 ) + + O(εL)

2E 1,0 cos ϕk − 2E 1,0 cos ϕk,ε k∈{ N j :N j ≤M } 



cos ϕk − cos ϕ  . = O(L −1 ) + O(εL 2 ) + 2E 1,0 k,ε k∈{ N j :N j ≤M } But using (d), k∈{ N j :N j ≤M }

M



cos ϕk − cos ϕ  ≤ O(ρ(const/γ )k ) = O(ρ(const/γ )λ ). k,ε k=0

 Since εL 2 = O(ρ(const/γ )λ ), this finishes the proof of Lemma 7.  Proof of Lemma 8. The proof is by induction. We take ε to be so small that Eq. (21) is satisfied. This is possible by Eq. (27). It is trivial to verify (a)–(f) for k = 0. So let 0 < l < N ∧ Nε , and suppose that (a)–(f) have been verified for all k < l. We have three cases to consider:

Periodic Oscillation of an Adiabatic Piston in 2 or 3 Dimensions

575

  Case 1. l − 1 and l ∈ / N j,ε : In this case, verifying (a)–(f) for k = l is a relatively straightforward application of the machinery developed in Subsect. 5.2.1, because for   , y (t) traces out the billiard orbit between x   tl−1,ε ≤ t ≤ tl,ε ε l−1,ε and xl,ε corresponding to free flight in the domain D1 (Q 0 ). We make only two remarks.  = y (t  ) corresponds First, as long as ε is sufficiently small, it really is true that xl,ε ε l,ε to a true collision point on ∂D1 (Q 0 ). Indeed, if this were not the case, then it must be that  ) > Q , and y (t  ) would have to correspond to a collision with the side of the Q ε (tl,ε 0 ε l,ε  = F x  “tube” to the right of Q 0 . But then xl,ε l−1,ε ∈ h 0 would correspond to a collision  ) ≤ const ρ(const/γ )k ≤ with an immobile piston at Q 0 and would satisfy d(xk , xk,ε λ const ρ(const/γ ) = o(γ ), using Eqs. (20) and (27). But xk ∈ / Nγ (∂h 0 ), and so it follows that when the trajectory of yε (t) crosses the plane {Q = Q 0 }, it is at least a distance ∼γ away from the boundary of the face of the piston, and its velocity vector is pointed no  )− Q = O(εL) = o(γ ), closer than ∼γ to being parallel to the piston’s face. As Q ε (tl,ε 0 and it is geometrically impossible (for small ε) to construct a right triangle whose sides s1 , s2 satisfy |s1 | ≥∼ γ , |s2 | ≤ O(εL), with the measure of the acute angle adjacent to s1 being greater than ∼γ , we have a contradiction. After crossing the plane {Q = Q 0 }, yε (t) must experience  its next collision with the face of the piston, which violates the   fact that l ∈ / N j,ε .  − t  Second, tl,ε = ζ xl−1,ε + O(εL), because v1,ε = v1,0 + O(εL). See Eq. (24).

l−1,ε



 From Eq. 22, ζ xl−1 − ζ xl−1,ε

≤ O((ρ/γ ) (const/γ )l−1 ). As tl − tl−1 = ζ xl−1 and

εL = O((ρ/γ ) (const/γ )l−1 ), we obtain (f).  : For definiteness, we suppose that Q (t  ) ≥ Case 2. There exists i such that l = Ni,ε ε l,ε Q 0 , so that the left gas particle collides with the piston to the right of Q 0 . The case when  ) ≤ Q can be handled similarly. Q ε (tl,ε 0 We know that xl−1 , xl , xl+1 ∈ / Nγ (∂h 0 ) ∪ Nγ (F −1 Nγ (∂h 0 )). Using the inductive hypothesis and Eq. (20), we can define   xl,ε = F xl−1,ε ,

  xl+1,ε = F 2 xl−1,ε ,

 ) ≤ const ρ(const/γ )l , d(x , x  l+1 and d(xl , xl,ε l+1 l+1,ε ) ≤ const ρ(const/γ ) . In particu and x  lar, xl,ε l+1,ε are both a distance ∼γ away from ∂h 0 . Furthermore, when the left gas particle collides with the moving piston, it follows from Eq. (12) that the difference between its angle of incidence and its angle of reflection is O(ε). Referring to Fig. 3,  = ϕ  + O(ε). Geometric arguments similar to the one given in this means that ϕl,ε l,ε Case 1 above show that the yε -trajectory of the left gas particle has precisely one collision with the piston and no other collisions with the sides of the gas container when  ). Note that x  was defined the gas particle traverses the region Q 0 ≤ Q ≤ Q ε (tl,ε l,ε to be the point in the collision cross-section h 0 corresponding to the return of the yε trajectory into the region Q ≤ Q 0 . See Fig. 3. From this figure, it is also evident that  , r  ) ≤ O(εL/γ ). Thus d(x  , x  ) = O(εL/γ ), and this explains the choice of d(rl,ε l,ε l,ε l,ε ρ(ε) in Eq. (26). From the above discussion and the machinery of Subsect. 5.2.1, (a)–(e) now follow readily for both k = l and k = l + 1. Furthermore, property (f) follows in much the  − t same manner as it did in Case 1 above. However, one should note that tl,ε l−1,ε =     ζ xl−1,ε + O(εL) + O(εL/γ ) and tl+1,ε − tl,ε = ζ xl,ε + O(εL) + O(εL/γ ), because

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P. Wright

Fig. 3. An analysis of the divergences of orbits when ε > 0 and the left gas particle collides with the moving piston to the right of Q 0 . Note that the dimensions are distorted for visual clarity, but that εL and εL/γ  ∈ (−π/2 + γ /2, π/2 − γ /2) and ϕ  = ϕ  + O(ε), and so are both o(γ ) as ε → 0. Furthermore, ϕl,ε l,ε l,ε  = r  + O(εL/γ ). In particular, the y -trajectory of the left gas particle has precisely one collision with rl,ε ε l,ε the piston and no other collisions with the sides of the gas container when the gas particle traverses the region  ) Q 0 ≤ Q ≤ Q ε (tl,ε

of the extra distance O(εL/γ ) that the gas particle travels to the right of Q 0 . But εL/γ = O((ρ/γ ) (const/γ )l−1 ), and so property (f) follows.  . As mentioned above, the inductive step Case 3. There exists i such that l − 1 = Ni,ε in this case follows immediately from our analysis in Case 2.  

6. Generalization to a Full Proof of Theorem 1 It remains to generalize the proof in Sects. 4 and 5 to the cases when n 1 , n 2 ≥ 1 and d = 3. 6.1. Multiple gas particles on each side of the piston. When d = 2, but n 1 , n 2 ≥ 1, only minor modifications are necessary to generalize the proof  above. As in Subsect. 4.2, one  defines a stopping time T˜ε satisfying P T˜ε < T ∧ Tε = O(ε) such that for 0 ≤ t ≤ T˜ε /ε, gas particles will only experience clean collisions with the piston. Next, define H (z) by ⎡ ⎤ W





  ⎢+2 n 1

v ⊥

δ ⊥ 2 ⊥

− 2 nj=1

v2, j δq ⊥ =Q ⎥ ⎢ ⎥ j=1 1, j q1, j =Q 2, j



⎢ ⎥

⊥

H (z) = ⎢ ⎥. −2W v1, j δq ⊥ =Q ⎢ ⎥

1, j ⎣ ⎦

⊥

+2W v2, j δq ⊥ =Q 2, j

t It follows that for 0 ≤ t ≤ T˜ε /ε, h ε (t) − h ε (0) = O(ε) + ε 0 H (z ε (s))ds. From here, the rest of the proof follows the same steps made in Subsect. 5.1. We note that at Step

Periodic Oscillation of an Adiabatic Piston in 2 or 3 Dimensions

577

3, we find that H (z) − H¯ (h(z)) divides into n 1 + n 2 pieces, each of which depends on only one gas particle when the piston is held fixed. 6.2. Three dimensions. The proof of Theorem 1 in d = 3 dimensions is essentially the same as the proof in two dimensions given above. The principal differences are due to differences in the geometry of billiards. We indicate the necessary modifications. In analogy with Sect. 4.1, we briefly summarize the necessary facts for the billiard flows of the gas particles when M = ∞ and the slow variables are held fixed at a specific value h ∈ V. As before, we will only consider the motions of one gas particle moving in D1 . Thus we consider √ the billiard flow of a point particle moving inside the domain D1 at a constant speed 2E 1 . Unless otherwise noted, we use the notation from Sect. 4.1. 1 The billiard flow √takes place in the five-dimensional space M = {(q1 , v1 ) ∈ T D1 : q1 ∈ D1 , |v1 | = 2E 1 }/∼. Here the quotient means that when q1 ∈ ∂D1 , we identify velocity vectors pointing outside of D1 with those pointing inside D1 by reflecting orthogonally through the tangent plane to ∂D1 at q1 . The billiard flow preserves Liouville measure restricted to the energy surface. This measure has the density dµ = dq1 represents dq1 dv1 /(8π volume on R3 , and dv1 represents area on   E 1 |D31 |). Here √ 2 S√2E = v1 ∈ R : |v1 | = 2E 1 . 1 √ The collision cross-section  = {(q1 , v1 ) ∈ T D1 : q1 ∈ ∂D1 , |v1 | = 2E 1 }/∼ is properly thought of as a fiber bundle, whose base consists of the smooth pieces of ∂D1 and whose fibers are the set of outgoing velocity vectors at q1 ∈ ∂D1 . This and other facts about higher-dimensional billiards, with emphasis on the dispersing case, can be found in [BCST03]. For our purposes,  can be parameterized as follows. We decompose ∂D1 into a finite union ∪ j  j of pieces, each of which is diffeomorphic via coordinates r to a compact, connected subset of R2 with a piecewise C 3 boundary. The  j are nonoverlapping, except possibly on their boundaries. Next, if (q1 , v1 ) ∈  and v1 is the outward ˆ It going velocity vector, let vˆ = v1 / |v1 |. Then  can be parameterized by {x = (r, v)}. follows that  is diffeomorphic to ∪ j  j × S 2+ , where S 2+ is the upper unit hemisphere,

and by ∂ we mean the subset diffeomorphic to (∪ j ∂ j × S 2+ ) (∪ j  j × ∂ S 2+ ). If x ∈ , we let ϕ ∈ [0, π/2] represent the angle between the outgoing velocity vector and the inward pointing normal vector n to ∂D1 , i.e. cos ϕ = v, ˆ n. Note that we no longer allow ϕ to take on negative values. The return map F : 
preserves the projected probability measure ν, which has the density dν = cos ϕ d vˆ dr/(π |∂D1 |). Here |∂D1 | is the area of ∂D1 . F is an invertible, measure preserving transformation that is piecewise C 2 . Because of our assumptions on D1 , the free flight times and the curvature of ∂D1 are uniformly bounded. The bound on D F(x) given in Eq. (6) is still true. A proof of this fact for general three-dimensional billiard tables with finite horizon does not seem to have made it into the literature, although see [BCST03] for the case of dispersing billiards. For completeness, we provide a sketch of a proof for general billiard tables in Appendix B. We suppose that the billiard flow is ergodic, so that F is ergodic. Again, we induce F ˆ of  corresponding to collisions with the (immobile) piston to obtain on the subspace  ˆ
that preserves the induced measure νˆ . the induced map Fˆ :  The free flight time ζ :  → R again satisfies the derivative bound given in Eq. (7). The generalized Santaló’s formula [Che97] yields Eν ζ =

4 |D1 | . |v1 | |∂D1 |

578

P. Wright

ˆ → R is the free flight time between collisions with the piston, then it follows If ζˆ :  from Proposition 10 that 4 |D1 | . E νˆ ζˆ = |v1 | 



The expected value of v1⊥ when the left gas particle collides with the (immobile) piston is given by √ 



  2E 1 2

⊥

E νˆ v1 = E νˆ 2E 1 cos ϕ = cos2 ϕ d vˆ1 = 2E 1 . π 3 S 2+ As a consequence, we obtain Lemma 9. For µ − a.e. y ∈ M1 ,  1 t

⊥

E1 lim .

v1 (s) δq ⊥ (s)=Q ds = 1 t→∞ t 0 3 |D1 (Q)| Compare the proof of Lemma 2. With these differences in mind, the rest of the proof of Theorem 1 when d = 3 proceeds in the same manner as indicated in Sects. 4, 5 and 6.1 above. The only notable difference occurs in the proof of the Gronwall-type inequality for billiards. Due to dimensional considerations, if one follows the proof of Lemma 6 for a three-dimensional billiard table, one finds that νNγ (F −1 Nγ (∂)) = O(γ 1−4α +γ α ). The optimal value of α is 1/5, and so νNγ (F −1 Nγ (∂)) = O(γ 1/5 ) as γ → 0. Hence νCγ ,λ = O(λγ 1/5 ), which is a slightly worse estimate than the one in Eq. (23). However, it is still sufficient for all of the arguments in Sect. 5.2.2, and this finishes the proof. A. Inducing Maps on Subspaces Here we present some well-known facts on inducing measure preserving transformations on subspaces. Let F : (, B, ν)
be an invertible, ergodic, measure preserving transformation of the probability space  endowed with the σ -algebra B and the probability ˆ ∈ B satisfy 0 < ν  ˆ < 1. Define R :  ˆ → N to be the first return time measure ν. Let  ˆ := {B ∩  ˆ i.e. Rω = inf{n ∈ N : F n ω ∈ }. ˆ Then if νˆ := ν(· ∩ )/ν ˆ ˆ and B ˆ : to ,  ˆ ν) ˆ = F Rω ω is also an invertible, ergodic, ˆ B, B ∈ B}, Fˆ : (, ˆ
defined by Fω  ˆ −1 . measure preserving transformation [Pet83]. Furthermore E νˆ R = ˆ R d νˆ = (ν ) This last fact is a consequence of the following proposition:  R−1 Proposition 10. If ζ :  → R≥0 is in L 1 (ν), then ζˆ = n=0 ζ ◦ F n is in L 1 (ˆν ), and E νˆ ζˆ =

1 E ν ζ. ˆ ν

Proof. ˆ ν

 R−1 ˆ  n=0

ζ ◦ F n d νˆ = =

 R−1 ˆ  n=0

∞ k−1  k=1 n=0

because

ˆ ∩ {R {F n (

ζ ◦ F n dν =

∞ 

k−1

ˆ k=1 ∩{R=k} n=0

ζ ◦ F n dν



ˆ F n (∩{R=k})

ζ dν =



ζ dν,

= k}) : 0 ≤ n < k < ∞} is a partition of .

 

Periodic Oscillation of an Adiabatic Piston in 2 or 3 Dimensions

579

B. Derivative Bounds for the Billiard Map in Three Dimensions Returning to Sect. 6.2, we need to show that for a billiard table D1 ⊂ R3 with a piecewise C 3 boundary and the free flight time uniformly bounded above, the billiard map F satisfies the following: If x0 ∈ / ∂ ∪ F −1 (∂), then D F(x0 ) ≤

const . cos ϕ(F x0 )

Fix x0 = (r0 , vˆ0 ) ∈ , and let x1 = (r1 , vˆ1 ) = F x0 . Let  be the plane that perpendicularly bisects the straight line between r0 and r1 , and let r1/2 denote the point of intersection. We consider  as a “transparent” wall, so that in a neighborhood of x0 , we can write F = F2 ◦ F1 . Here, F1 is like a billiard map in that it takes points (i.e. directed velocity vectors with a base) near x0 to points with a base on  and a direction pointing near r1 . (F1 would be a billiard map if we reflected the image velocity vectors orthogonally through .) F2 is a billiard map that takes points in the image of%F1 and maps% them near x1 . Let x1/2 = F1 x0 = F2−1 x1 . Then D F(x0 ) ≤ D F1 (x0 ) % D F2 (x1/2 )%. It is easy to verify that D F1 (x0 ) ≤ const, with the constant depending only on the curvature of % ∂D1 at r0 .%In other words, the constant may be chosen independent of x0 . % % Similarly, % D F2−1 (x1 )% ≤ const. Because billiard maps preserve a probability measure with a density proportional to cos ϕ, detD F2−1 (x1 ) = cos ϕ1 / cos ϕ1/2 = cos ϕ1 . As  is 4-dimensional, it follows from Cramer’s Rule for the inversion of linear transformations that % %3 % % −1 F const (x ) %D 1 % % % 2 const % D F2 (x1/2 )% ≤ ≤ , −1 cos ϕ1 detD F2 (x1 ) and we are done. Acknowledgements. The author is grateful to D. Dolgopyat, who first introduced him to this problem, and who generously shared his unpublished notes on averaging [Dol05]. The author also thanks L.-S. Young for useful discussions regarding this project and P. Balint for many helpful comments on the manuscript. This research was partially supported by the National Science Foundation Graduate Research Fellowship Program.

References [Ano60]

Anosov, D.V.: Averaging in systems of ordinary differential equations with rapidly oscillating solutions. Izv. Akad. Nauk SSSR Ser. Mat. 24, 721–742 (1960) [BCST03] Bálint, P., Chernov, N., Szász, D., Tóth, I.P.: Geometry of multi-dimensional dispersing billiards. Astérisque 286, xviii, 119–150 (2003) [BR98] Bunimovich, L.A., Rehacek, J.: On the ergodicity of many-dimensional focusing billiards. Ann. Inst. H. Poincaré Phys. Théor. 68(4), 421–448 (1998) [Bun79] Bunimovich, L.A.: On the ergodic properties of nowhere dispersing billiards. Commun. Math. Phys. 65(3), 295–312 (1979) [Cal63] Callen, H.B.: Thermodynamics. New York: Wiley, 1963 [CD06a] Chernov, N., Dolgopyat, D.: Brownian brownian motion - I. Memoirs of the American Mathematical Society, to appear, 2006 [CD06b] Chernov, N., Dolgopyat, D.: Hyperbolic billiards and statistical physics. In: Proceedings of the International Congress of Mathematicians, (Madrid, Spain, 2006), Vol. II, Berlin: European Math. Soc., 2007, pp. 1679–1704 [CDPS96] Crosignani, B., Di Porto, P., Segev, M.: Approach to thermal equilibrium in a system with adiabatic constraints. Am. J. Phys. 64(5), 610–613 (1996)

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P. Wright

Chernov, N.: Entropy, lyapunov exponents, and mean free path for billiards. J. Stat. Phys. 88(1-2), 1–29 (1997) Chernov, N.: On a slow drift of a massive piston in an ideal gas that remains at mechanical equilibrium. Math. Phys. Electron. J. 10, Paper 2, 18 pp. (electronic), (2004) Chernov, N., Lebowitz, J.L.: Dynamics of a massive piston in an ideal gas: oscillatory motion and approach to equilibrium. J. Stat. Phys. 109(3–4), 507–527 (2002) Chernov, N., Lebowitz, J.L., Sinai, Ya.: Scaling dynamics of a massive piston in a cube filled with ideal gas: exact results. J. Stat. Phys. 109(3–4), 529–548 (2002) Chernov, N., Markarian, R.: Chaotic Billiards. Number 127 in Mathematical Surveys and Monographs. Providence, RI: Amer. Math. Soc. 2006 Chernov, N., Markarian, R.: Dispersing billiards with cusps: slow decay of correlations. Commun. Math. Phys. 270(3), 727–758 (2006) Dolgopyat, D.: Introduction to averaging. Available online at http://www.math.umd.edu/∼ dmitry/IANotes.pdf, 2005 Gorelyshev, I.V., Neishtadt, A.I.: On the adiabatic perturbation theory for systems with impacts. Prikl. Mat. Mekh. 70(1), 6–19 (2006); English trans. in J. Appl. Math. and Mech. 70, 4–17 (2006) Gruber, Ch., Pache, S., Lesne, A.: Two-time-scale relaxation towards thermal equilibrium of the enigmatic piston. J. Stat. Phys. 112(5–6), 1177–1206 (2003) Gruber, Ch.: Thermodynamics of systems with internal adiabatic constraints: time evolution of the adiabatic piston. Eur. J. Phys. 20, 259–266 (1999) Lochak, P., Meunier, C.: Multiphase Averaging for Classical Systems, New York: SpringerVerlag, 1988 Lebowitz, J., Sinai, Ya.G., Chernov, N.: Dynamics of a massive piston immersed in an ideal gas. Usp. Mat. Nauk., 57(6(348)), 3–86 (2002); English trans. in Russ. Math. Surv. 57(6) 1045–1125 (2002) Neishtadt, A.I., Sinai, Ya.G.: Adiabatic piston as a dynamical system. J. Stat. Phys. 116(1–4), 815–820 (2004) Petersen, K.: Ergodic Theory. Cambridge: Cambridge University Press, 1983 Santaló, L.A.: Integral Geometry and Geometric Probability. Reading, MA: Addison Wesley, 1976 Sinai, Ya.G.: Dynamics of a massive particle surrounded by a finite number of light particles. Teoret. Mat. Fiz. 121(1), 110–116 (1999); English trans. in Theoret. and Math. Phys. 121(1) 1351–1357 (1999) Sanders, J.A., Verhulst, F.: Averaging Methods in Nonlinear Dynamical Systems. New York: Springer-Verlag, 1985 Vorobets, Ya.B.: Ergodicity of billiards in polygons. Mat. Sb. 188(3), 65–112 (1997) Wright, P.: A simple piston problem in one dimension. Nonlinearity 19, 2365–2389 (2006)

Communicated by G. Gallavotti

Commun. Math. Phys. 275, 581–585 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0321-4

Communications in

Mathematical Physics

Erratum

A Strong Szeg˝o Theorem for Jacobi Matrices E. Ryckman University of California, Los Angeles, CA 90095, USA. E-mail: [email protected] Received: 4 June 2007 / Accepted: 4 June 2007 Published online: 7 August 2007 – © Springer-Verlag 2007

Commun. Math. Phys. 271, 791–820 (2007)

1. Introduction In this erratum an error in [1] is pointed out and corrected. In that paper we were concerned with the spectral theory of Jacobi matrices, that is semi-infinite tridiagonal matrices ⎞ ⎛ b1 a1 0 0 ⎟ ⎜ ⎜a1 b2 a2 0⎟ ⎟ ⎜ J =⎜ ⎟ ⎜ 0 a2 b3 . . . ⎟ ⎠ ⎝ .. .. . . 0 0 where an > 0 and bn ∈ R. We assume that an2 − 1 and bn are conditionally summable and define λn := − κn := −

∞  k=n+1 ∞ 

bk , (1.1) (ak2

− 1)

k=n+1

for n = 0, 1, . . . . The goal of Sect. 5 of [1] was to prove the following result (which appears there as Proposition 5.1): The online version of the original article can be found under doi:10.1007/s00220-007-0195-5.

582

E. Ryckman

Proposition 1.1. There are solutions ψs and ψb to J ψ = Eψ at energy E = ±2 such that    ψs (k)     ψ (k)  → 0 b and ±

ψs (k + 1) = 1 + 21 . ψs (k)

Moreover, for either solution and for k sufficiently large, (±1)k ψ(k) is sign definite and non-zero. In [1] we attempted to use Asymptotic Integration to prove this proposition. Due to two separate multiplication errors Eqs. (5.1) and (5.2) are incorrect, and the subsequent arguments are therefore not applicable. So far our attempts to resurrect the Asymptotic Integration argument have failed, so instead in the next section we present a complete proof of Proposition 1.1 based on a contraction mapping argument. With these corrections the remainder of the argument in [1] is valid as stated. 2. Proof of Proposition 1.1. For this section all norms refer to 21 . We will use the notation x
y if x ≤ cy for some constant c > 0 (which may change from one line to the next). We will also need the following lemma, which is a special case of Lemma 2.4 in [1]: 2 Lemma 2.1. Let β, γ ∈ 21 , and define a sequence ηn := ∞ k=n βk γk . Then η ∈ 1 and η ≤ β · γ . The rest of this section is devoted to a proof of Proposition 1.1 for E = 2, the proof for E = −2 being analogous. We will begin by constructing the small solution ψs . By dividing the recurrence equation

 ak+1 ψ(k + 1) + bk+1 − E ψ(k) + ak ψ(k − 1) = 0 by ψ(k) and writing r (k) =

ψ(k) ψ(k − 1)

we deduce the discrete Riccati equation ak+1 r (k + 1) +

ak = 2 − bk+1 . r (k)

(2.1)

Lemma 2.2. There is a k0 > 0 and a solution r (k), k ≥ k0 , to the discrete Riccati equation (2.1) of the form r (k) = 1 + ε(k) with ε < 1.

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583

Proof. Write r (k) = 1 + ε(k) and ε(k)2 1 = 1 − ε(k) + r (k) 1 + ε(k) so (2.1) becomes ak+1 ε(k + 1) − ak ε(k) + ak

ε(k)2 = 2 − ak+1 − ak − bk+1 . 1 + ε(k)

Summing to infinity then yields ε(k) =



∞  ε(l)2 −1 λk + κk + κk−1 − . al ak 1 + ε(l)

(2.2)

l=k

We will use a contraction mapping argument to show (2.2) has a solution ε ∈ 21 . To this end, let [F(ε)](k) denote the right-hand side of (2.2). We must show that F maps some ball in 21 to itself and is a contraction there. Notice that because 0 < ak → 1 we have  ∞  β(l)2  
λ + κ + β2 ,  F(β)
λ + κ +  1 + β(l)  l=k

where the second inequality follows from β ∈ 21 and Lemma 2.1. So for any fixed δ < 1, if λ, κ are small enough (depending on δ) then F : {β ∈ 21 : β ≤ δ} → {β ∈ 21 : β ≤ δ}

(2.3)

as desired. Next we show that F is a contraction:  ∞     β(l)2  γ (l)2   [F(β)](k) − [F(γ )](k)
 1 + β(l) − 1 + γ (l)  l=k

= ≤

∞  |β(l)2 − γ (l)2 + β(l)2 γ (l) − γ (l)2 β(l)| l=k ∞  l=k

|(1 + β(l))(1 + γ (l))|

|β(l) + γ (l)| + |β(l)γ (l)| |β(l) − γ (l)|. |(1 + β(l))(1 + γ (l))|

So from Lemma 2.1 we have

   |β + γ | + |βγ |   F(β) − F(γ )
  |(1 + β)(1 + γ )|  · β − γ .

In particular, if β and γ  are small enough then F(β) − F(γ ) ≤ ηβ − γ  with η < 1, and F is a contraction.

(2.4)

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Now, choose δ small enough and k0 large enough that (2.3) and (2.4) hold for the ∞ 2 sequences {λk }∞ k=k0 and {κk }k=k0 (which is possible since λ, κ ∈ 1 ). Then by the Banach Fixed Point Theorem one finds a solution ε ∈ {β ∈ 21 (k0 , k0 + 1, . . . ) : β ≤ δ < 1} to the equation ε(k) = [F(ε)](k). That is, ε solves (2.2) for k ≥ k0 , as claimed.

 

Corollary 2.3. There is a solution ψs (k) to J ψ = 2ψ so that for k ≥ k0 we have ψs (k) =

k 

(1 + ε(l))

(2.5)

l=k0

with ε < 1. In particular, for k ≥ k0 , ψs (k) is sign-definite and non-zero. Proof. The first statement follows from ψs (k) = r (k) = 1 + ε(k) ψs (k − 1) and Lemma 2.2. The “in particular” part follows from ε < 1, which shows |ε(k)| < 1  for all k ≥ k0 .  Now we construct the big solution ψb : Lemma 2.4. There is a solution ψb to J ψ = 2ψ so that    ψs (k)     ψ (k)  → 0 b

as k → ∞. Proof. Let ψs be the solution from Corollary 2.3 and let ψb be the solution with

 1 = Wk (ψs , ψb ) = ak ψs (k)ψb (k + 1) − ψb (k)ψs (k + 1) and ψb (k0 ) = 1 + ε(k0 ) = ψs (k0 ). Then ψb (k + 1) ψb (k) Wk (ψs , ψb ) 1 − = = ψs (k + 1) ψs (k) ak ψs (k)ψs (k + 1) ak ψs (k)ψs (k + 1) and so  ψb (k) 1 =1+ . ψs (k) al ψs (l)ψs (l + 1) k−1

l=k0

Therefore to show

   ψs (k)     ψ (k)  → 0 b

Erratum

585

we must show k  l=k0

1 →∞ al ψs (l)ψs (l + 1)

as k → ∞. From (2.5) and the expansion log(1 + x) = x + O(|x|2 ) we have ψs (k) = exp

k 

 log 1 + ε(l)

l=k0

= exp

k 

ε(l) + O(|ε(l)|2 )



l=k0


exp

k 

 ε(l) exp(ε2 ).

l=k0

 √  √  Writing ε(l) = ε(l) l 1/ l and using Cauchy-Schwarz and ε ∈ 21 results in the bound   ψs (k)
exp log(k) which we claim is bounded by k γ for any 0 < γ < 1/2. To see this simply notice that   exp log(k)  = log(k) − γ log(k) → −∞ log γ k as k → ∞. So all together we have ψs (k)
k γ for any 0 < γ < 1/2, and so   k k   ψb (k)   1 =   l −2γ → ∞  ψ (k)  al ψs (l)ψs (l + 1) s l=k0

as k → ∞, as claimed.

l=k0

 

Finally we have: Proof of Proposition 1.1. The small solution ψs was constructed in Corollary 2.3 and the big solution ψb was constructed in Lemma 2.4. The ratio statements then follow from Lemmas 2.2 and 2.4.   Acknowledgement. The author would like to thank Rowan Killip for many helpful conversations during the preparation of this erratum.

Reference 1. Ryckman, E.: A strong Szeg˝o theorem for Jacobi matrices. Comm. Math. Phys. 271(3), 791–820 (2007) Communicated by B. Simon

Commun. Math. Phys. 275, 587–605 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0326-z

Communications in

Mathematical Physics

Twistor Spinors with Zero on Lorentzian 5-Space
Felipe Leitner Institut für Geometrie und Topologie, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany. E-mail: [email protected] Received: 19 April 2006 / Accepted: 17 April 2007 Published online: 21 August 2007 – © Springer-Verlag 2007

Abstract: We present in this paper a C 1 -metric of Lorentzian signature (1, 4) on an open neighbourhood of the origin in R5 , which admits a solution to the twistor equation for spinors with a unique isolated zero at the origin. The metric is not conformally flat in any neighbourhood of the origin and the associated conformal Killing vector to the twistor generates a one-parameter group of essential conformal transformations. The construction is based on the Eguchi-Hanson metric in dimension 4. 1. Introduction For spinor fields of suitable weight on semi-Riemannian manifolds there exist two conformally covariant linear differential equations of first order, the Dirac equation and the twistor equation. The twistor equation is overdetermined and the existence of solutions, which we call conformal Killing spinors (or simply twistor spinors), is constrained by curvature conditions on the underlying space. The twistor equation was first introduced by R. Penrose in General Relativity. In the second half of the 1980s A. Lichnerowicz started a systematic investigation of twistor spinors on Riemannian spin manifolds (cf. [Lic88, Lic89, Lic90]). Special solutions of the twistor equation are Killing spinors and parallel spinors, for which nowadays many geometric structure results are known. From the viewpoint of conformal geometry twistor spinors with zeros are of particular interest (cf. [Lic90, KR95, KR96, KR98]). This is for various reasons. In the Riemannian case the length square of a twistor gives rise to a rescaled Ricci-flat metric in the conformal class on the complement of the zero set, which consists of isolated points. Such spaces are sometimes called almost conformally Einstein manifolds (cf. [Gov04]). A result by A. Lichnerowicz states that a compact Riemannian space admitting a twistor
The research in this work was supported by a Junior Research Fellowship grant of the Erwin-SchrödingerInternational (ESI) Institute in Vienna financed by the Austrian Ministry of Education, Science and Culture (BMBWK) and by a fellowship grant of the Sonderforschungsbereich 647’ Space-Time-Matter - Geometric and Analytic Structures’ of the Deutsche Forschungsgemeinschaft (DFG) at Humboldt University Berlin.

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spinor with zero is conformally isometric to the round n-sphere S n . Any twistor on S n admits exactly one isolated zero. A construction by W. Kühnel and H.-B. Rademacher also shows that there exist twistors with zeros on complete, non-compact Riemannian spaces, which are not conformally flat. Such solutions occur typically on the conformal completion space to infinity of asymptotically Euclidean spaces with special holonomy (cf. [KR96, KR98]). In the Lorentzian setting solutions of the twistor equation always give rise to conformal Killing vector fields. The associated conformal Killing vector to a twistor with zero has the interesting property that its (local) flow consists of essential conformal transformations, i.e., transformations which are not isometries for any metric in the conformal class of the underlying space. Essential conformal transformation groups are non-compact. A conjecture by A. Lichnerowicz states that compact Lorentzian spaces with essential conformal transformation group are conformally flat. For comparison, in the Riemannian case it is well known that the only complete spaces with essential (i.e. non-compact) conformal transformation group are the n-sphere S n (compact case) and the Euclidean space (non-compact case) (cf. [Ale72, Yos75, Oba71, LF71]). In this paper we construct a family of Lorentzian metrics on (non-compact) open subsets of R5 , which admit twistor spinors with a unique isolated zero and essential conformal transformation groups. The conformal geometry of these metrics is not flat in any neighbourhood of the zero. The construction is based on the conformal completion of the Eguchi-Hanson metric defined on the complement of a closed ball in R4 (cf. [KR96]). The constructed family of metrics is of class C 1 , i.e., with respect to the standard coordinates on R5 the coefficients of the metrics are continuously differentiable exactly once. The course of the paper is very simple. In Sect. 2 we present explicitly the family of metrics in question with some chosen frame and a spinor with zero. In Sect. 3 we prove that the metrics in that family are of class C 1 and that the given spinor solves the twistor equation.

2. Metric with Frame and Spinor Let us consider the 5-dimensional real vector space R5 with canonical coordinates x = (x0 , x1 , x2 , x3 , x4 ). We set n = 5. The Minkowski metric is given by g0 = −d x02 + d x12 + d x22 + d x32 + d x42 . This metric is of Lorentzian signature (1, 4) and is flat on R5 . We aim to rewrite the Minkowski metric in cylindrical coordinates. So let E be the 4-dimensional vector subspace in R5 defined by x0 = 0 and denote by r=




x12 + x22 + x32 + x42

the radial coordinate on E. The space E \{0} (with deleted origin) is diffeomorphic to R+ × S 3 . Thereby, S 3 denotes the 3-sphere, which is given in E by the equation r = 1. As the group of elements with unit length in E ∼ = H the 3-sphere S 3 is isomorphic to the semisimple Lie group SU(2). The round metric g S 3 on S 3 is SU(2)-invariant and there exist left-invariant 1-forms σ1 , σ2 and σ3 on SU(2) such that g S 3 = σ12 + σ22 + σ32 .

Twistor Spinors with Zero on Lorentzian 5-Space

589

On S 3 in E these left-invariant forms are explicitly given by σ1 = σ2 = σ3 =

1 (−x2 d x1 r2 1 (−x3 d x1 r2 1 (−x4 d x1 r2

+ x1 d x2 − x4 d x3 + x3 d x4 ), + x4 d x2 + x1 d x3 − x2 d x4 ), − x3 d x2 + x2 d x3 + x1 d x4 ) .

We denote the dual orthonormal frame on T S 3 by { ∂σ∂ 1 , ∂σ∂ 2 , ∂σ∂ 3 }. Eventually, we see that the Minkowski metric on R5 \{r = 0} is given in cylindrical coordinates by g0 = −d x02 + dr 2 + r 2 (σ12 + σ22 + σ32 ) . We know that this metric can be smoothly completed to the singular set {r = 0} of the cylindrical coordinate system (which is a real line in R5 ). The result is the Minkowski metric g0 on R5 . Now let us define the cone L := { (x0 , x1 , x2 , x3 , x4 ) ∈ R5 : r ≤ |x0 | } with singular point at the origin of R5 . The boundary set of the cone L in R5 is L o := { (x0 , x1 , x2 , x3 , x4 ) ∈ R5 : r = |x0 | } . Next we define the radial coordinate  ro :=

0

on L

r 2 −x02 r

on R5 \ L

.

Furthermore, let a > 0 be a real parameter. Then we set Ba := { (x0 , x1 , x2 , x3 , x4 ) ∈ R5 : 0 < ro <

1 } a

and B˜ a := Ba ∪ L. Both sets Ba and B˜ a are open in R5 for all a > 0. The set Ba is a subset of R5\L and B˜ a is simply connected. We also introduce the set Ba> := B˜ a\{r = 0}, where the real line {r = 0} is deleted. Figure 1 shows a diagram of the basic domains of definition. On B˜ a we define a family of pointwise symmetric bilinear forms ga , a > 0, as follows. Let  g0 − r 2 (aro )4 · σ32 + a 4 (rβ)−2 ro2 · α 2 on Ba> ga := , g0 on {r = 0} where we set β :=

 1 − (aro )4

and

α := (r 2 + x02 )dr − 2x0 r d x0 .

Obviously, the symmetric bilinear form ga is smoothly defined on B˜ a \L o for all a > 0, and by definition, ga restricted to L \ L o is the flat Minkowski metric. The symmetric bilinear form ga can be rewritten on Ba> as ga = −d x02 + dr 2 + r 2 ( σ12 + σ22 + β 2 σ32 ) + a 4 (rβ)−2 ro2 · α 2 .

590

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Fig. 1.

Proposition 1. The symmetric bilinear form ga is a C 1 -metric of Lorentzian signature (1, 4) on the subset B˜ a of R5 for all a > 0. The metric ga is not of class C 2 . We want to discuss some geometric aspects of the Lorentzian metric ga . The restriction of ga to the disk E ∩ Ba (with deleted origin) in E ∼ = R4 is given by h a :=

dr 2 + r 2 ( σ12 + σ22 + (1 − (ar )4 )σ32 ) . 1 − (ar )4

This is a Riemannian metric on E ∩ Ba , which admits a smooth (even analytic) extension to the origin. Off the origin, the metric ga is conformally equivalent to the Eguchi-Hanson metric   d R2 g E H := + R 2 σ12 + σ22 + (1 − (a/R)4 )σ32 4 1 − (a/R) defined on the complement of a closed ball in R4 . In fact, with R := 1/r on E \{0} we have g E H = r14 · h a . We want to point out that the complete Eguchi-Hanson metric is a Ricci-flat Kähler metric defined on the total space of the cotangent bundle of S 2 , which is asymptotically locally Euclidean at infinity and hyperkähler with irreducible holonomy group SU(2) = Sp(1). The metric g E H on R4 \{R ≤ a} which is used throughout this paper is a simply-connected Z2 cover of the complete Eguchi-Hanson metric on T ∗ S 2 with the zero section deleted. The metric g E H is asymptotically Euclidean, Ricci-flat and hyperkähler, and therefore, admits a 2-dimensional space of parallel spinors. The metric g E H is not conformally flat. In fact, g E H is half-conformally flat, i.e., the Weyl curvature tensor W = W + + W − is anti-selfdual (W + = 0) (cf. [EH78, KR96]). We set g˜ a :=

1 · ga (r 2 − x02 )2

on B˜ a \ L o .

Proposition 2. a) Let g˜ a = (r 2 − x02 )−2 · ga , a > 0, be a conformally equivalent metric to ga on B˜ a \ L o . Then

Twistor Spinors with Zero on Lorentzian 5-Space

591

Fig. 2.

(1) the metric g˜ a is flat for r < |x0 |. (2) For |x0 | < r the metric is given by g˜ a = −ds 2 + g E H , where we define the coordinate change  by s :=

−x0 r 2 −x02

and R :=

r r 2 −x02

.

In particular, g˜ a is Ricci-flat on B˜ a \ L o . b) The Weyl tensor W ga of the smooth Lorentzian metric ga on B˜ a \ L o admits a continuous extension of class C 1 to the singular set L o . For this extension it is W ga ≡ 0 on L and W ga = 0 on Ba , i.e., ga is not conformally flat. We note that the Ricci-curvature tensor of the metric ga on B˜ a \ L o does not admit a continuous extension to B˜ a . With µ := ln |r 2 − x02 | it is Ric ga = −(n − 2) · (H ess g˜a (µ) − dµ2 ) − (g˜a µ + (n − 2) · |dµ|2 ) · g˜ a . Furthermore, we note that the hypersurface {s = 0} is totally geodesic with respect to the metric −ds 2 + g E H . This implies that the disk E ∩ B˜ a is a totally umbilic hypersurface in ( B˜ a , ga ). Figure 2 shows the coordinate change  on B˜ a together with the conformally equivalent metrics ga and g˜ a . Next we define on Ba> with metric ga an orthonormal frame e = {e0 , e1 , e2 , e3 , e4 } in the following way. Let T := −(r 2 + x02 )

∂ ∂ − 2r x0 ∂r ∂ x0

be a vector field on R5 . We set ∂ 2x0 a 4 ro2 − · · T, ∂ x0 r 1+β r 2 + x02 a 4 ro2 ∂ + · T, · e1 := ∂r r2 1+β

e0 :=

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F. Leitner

∂ , ∂σ1 ∂ e3 := r −1 · , ∂σ2 ∂ e4 := (rβ)−1 · . ∂σ3

e2 := r −1 ·

Lemma 1. The orthonormal frame e = {e0 , e1 , e2 , e3 , e4 } on Ba> is of class C 1 . We proceed by introducing spinor calculus on B˜ a (cf. e.g. [Baum81]). Let Spin(1, 4) denote the spin group with universal covering map λ : Spin(1, 4) → S O(1, 4) onto the special orthonormal group and let Cl(1, 4) be the Clifford algebra. The complex spinor module 1,4 is isomorphic to C4 and a realisation of the Clifford algebra Cl(1, 4) on 1,4 ∼ = C4 is given by ⎛ ⎞ ⎛ ⎞ −1 0 0 0 0 0 −1 0 0 0 1⎟ ⎜ 0 −1 0 0 ⎟ ⎜0 , γ1 = ⎝ , γ0 = ⎝ 0 0 1 0⎠ 1 0 0 0⎠ 0 0 0 1 0 −1 0 0 ⎛

0 ⎜ 0 γ2 = ⎝ −i 0

0 0 0 −i

−i 0 0 0

⎞ 0 −i ⎟ , 0⎠ 0 ⎛

0 ⎜ 0 γ4 = ⎝ 0 −i

⎛

0 ⎜0 γ3 = ⎝ 0 1 0 0 i 0

0 i 0 0

0 0 1 0

0 −1 0 0

⎞ −1 0⎟ , 0⎠ 0

⎞ −i 0⎟ , 0⎠ 0

where γ0 · γ0 = 1, γi · γi = −1 for all i = 1, . . . , 4 and γi · γ j = −γi · γ j for all i = j. The Lorentzian manifold ( B˜ a , ga ) with C 1 -metric is simply connected and oriented. There exists a unique spin structure π : Spin( B˜ a ) → S O( B˜ a ), where Spin( B˜ a ) denotes a Spin(1, 4)-principal fibre bundle over B˜ a , the spinor frame bundle, which is a Z2 covering of the orthonormal frame bundle S O( B˜ a ) such that the fibre actions of Spin(1, 4) and SO(1, 4) are compatible with the projections π and λ. We denote the spinor bundle on ( B˜ a , ga ) by S := Spin( B˜ a ) ×Spin(1,4) 1,4 . The spinor bundle S is globally trivial on B˜ a . With respect to a C 1 -section of π : Spin( B˜ a ) → B˜ a (i.e., a global spinor frame of class C 1 ) the space C 1 ( B˜ a , S) of differentiable spinor fields is uniquely identified with the space C 1 ( B˜ a , 1,4 ) of 1,4 -valued continuously differentiable functions on B˜ a . The spinor bundle S admits an invariant inner product, which we denote by ·, · S , and · : T M ⊗ S → S denotes the Clifford multiplication of tangent vectors with spinors. The spinor derivative is ∇ S and the Dirac

Twistor Spinors with Zero on Lorentzian 5-Space

593

operator D S acting on spinor fields is given with respect to some (local) orthonormal frame {t0 , . . . , t4 } by DS :

C 1 ( B˜ a , S) → C 0 ( B˜ a , S).

4 S φ → −t0 · ∇tS0 φ + i=1 ti · ∇ti φ.

The twistor equation for spinor fields φ on B˜ a is given by 1 X · DSφ = 0 for all X ∈ T B˜ a . n This is a first order differential equation on spinors, which is well known to be conformally covariant (cf. Sect. 3). We call a spinor field φ ∈ C 1 ( B˜ a , S) a twistor spinor if it satisfies the twistor equation. Any spinor field φ = 0 defines in a unique way a non-trivial vector field Vφ (spinor square) by demanding the relation ∇ XS φ +

ga (Vφ , X ) = φ, X · φ S

for all X ∈ T B˜ a .

If φ is a twistor spinor then Vφ is a conformal Killing vector field, i.e., 2 div(Vφ ) · ga n for the Lie derivative of the metric along Vφ . The C 1 -frame e : Ba> → S O( B˜ a ) (cf. Lemma 1) admits exactly two lifts (of class 1 C ) to the spinor frame bundle Spin( B˜ a ). We choose one of these lifts and denote it by L Vφ ga =

es : Ba> → Spin( B˜ a ). Any spinor field φ on Ba> can then be uniquely represented with respect to the spinor frame es by a 1,4 -valued function. It is φ = [es , w] ∈ C 1 (Ba> , S) for some function w ∈ C 1 (Ba> , 1,4 ). Now let w(b, c) denote the constant 1,4 -valued function (b, −c, 0, 0) , where (b, c) ∈ C2 . We set > ψbc := (x0 e0 + r e1 ) · [ es , w(b, c) ]

on Ba> ,

> is an where the dot · denotes Clifford multiplication. Obviously, the spinor field ψbc 1 > element of C (Ba , S) for all (b, c). Calculating the Clifford product results in ⎛ ⎞ −x0 b ⎜ x c ⎟ > ψbc = [ es , ⎝ 0 ⎠ ] . rb rc

We denote by V := −2x0 r

∂ ∂ − (r 2 + x02 ) ∂r ∂ x0

a smooth vector field on R5 . Figure 3 shows the vector field V in a neighbourhood of the origin in R5 together with integral curves and the flow of the umbilic hypersurface from E to some E + , resp. E − .

594

F. Leitner

Fig. 3. > be a spinor on (B > , g ) with a > 0. Theorem 1. Let (b, c) ∈ C2 and ψbc a a

(1) (2) (3) (4)

> on B > admits a unique extension ψ to ( B ˜ a , ga ) of class C 1 . The spinor field ψbc bc a The unique extension ψbc is a twistor spinor on ( B˜ a , ga ). For (b, c) = 0 the twistor spinor ψbc admits exactly one zero at the origin {0} ∈ B˜ a . The zero set of the spinor length square u bc := ψbc , ψbc  S is L o . The function u bc solves the equation

−u bc · Ric0 = (n − 2) · H ess(u bc )0 on B˜ a\L o , where Ric0 and H ess(u bc )0 denote the trace-free parts of the symmetric tensors Ric ga , resp. H ess ga (u bc ). In particular, the metric g˜ a = u12 ga is Einstein bc

for u bc = 0. (5) The spinor square Vψbc is a smooth conformal vector field on ( B˜ a , ga ). The following equation holds: Vψbc = (b2 + c2 ) · V .

(6) The vector Vψbc is timelike on B˜ a \L o , lightlike on L o\{0} and zero only in the origin {0} ∈ B˜ a . Here a vector X = 0 on ( B˜ a , ga ) is called timelike if ga (X, X ) < 0 and lightlike if ga (X, X ) = 0. In short, Theorem 1 says that there exists a 2-dimensional set of twistor spinors on ( B˜ a , ga ) for all a > 0, which admit an isolated zero at the origin. There exist no further twistor spinors on ( B˜ a , ga ), since the Eguchi-Hanson metric g E H admits exactly two linearly independent (parallel) twistor spinors for a > 0. Together with Proposition 2 we obtain from Theorem 1 the following important observation for our construction. Corollary 1. There exists a family of Lorentzian C 1 -metrics ga , a > 0, in dimension 5, which admit twistor spinors and a smooth causal conformal Killing vector field, all with isolated zero at some point { p} such that ga is non-conformally flat around the zero at { p}.

Twistor Spinors with Zero on Lorentzian 5-Space

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We remark that the vector field V is complete on B˜ a , i.e., the flow of V to the time t generates a one-parameter group of conformal transformations on B˜ a . These conformal transformations are not isometries with respect to any metric in the conformal class ca := [ga ] (cf. Sect. 3). Conformal transformations with the latter property are called essential. In particular, the conformal Killing vector field V is called essential. The statement of Corollary 1 implies the existence of essential conformal Killing fields and transformations on non-compact Lorentzian spaces, which are not conformally flat. In contrast, a conjecture by A. Lichnerowicz states that essential conformal transformation groups do not exist on any compact Lorentzian manifold unless it is conformally flat (cf. [D’AG91]). In fact, we do not expect that our construction is suitable to obtain compact examples. We want to add some further comments concerning our construction. The metric ga can be considered as a completion of the metric −ds 2 + g E H , which is Ricci-flat and ‘asymptotically Minkowskian’, to the set L with infinity L o . The twistors extend to L as well with a zero at some point of infinity. In general, it is known that a Lorentzian metric with differentiable Weyl tensor has to be conformally flat in the causal past and future of a zero of a twistor spinor (cf. Sect. 3). Therefore, it is reasonable in our construction to do the conformal completion to L by using the flat Minkowski metric g0 on the ‘other side’ of the infinity set L o . There exists no extension (conformal completion) with differentiable Weyl tensor of ga on Ba to a neighbourhood of the origin, which is not conformally flat on L, but preserves the existence of a twistor spinor. This fact implies that our completion of ga can not be analytic. We want to point out again that our construction is even not smooth. However, the question remains whether there is a conformally equivalent metric to ga on B˜ a , whose regularity is better than of class C 1 . The existence of the C 1 -extension of the Weyl tensor of ga to the infinity set L o certainly does not pose an obstruction to this question. 3. Proof of Statements We prove here the statements which we made in the previous section. We start with a discussion of differentiability of certain functions on B˜ a ⊂ R5 . For some arbitrary p-tuple I p = (i 1 , . . . , i p ) ∈ {0, . . . , 4} p let us denote by ∂ I p :=

∂ ∂ ··· ∂ xi1 ∂ xi p

a partial derivative of order p. Moreover, for any 5-tuple l = (lr , l0 , . . . , l4 ) with

4 li and define the smooth function lr , l0 , . . . , l4 ∈ N ∪ {0} we set sl := −lr + i=0 fl = f (lr , l0 , . . . , l4 ) := r −lr · x0l0 · . . . · x4l4

on Ba .

We say that the rational function fl is of order sl . Remember that we defined the radial function ro to be (r 2 − x02 )/r on Ba and identically zero on L (cf. Sect. 2). For any function f on Ba we understand the product ro · f in a unique way as a function on B˜ a = Ba ∪ L, which is identically zero on L. For t > 0 a real number we denote Bat := Ba ∩ {x ∈ R5 : r ≤ t}. Notice that if a function f is continuous on Ba and its absolute value | f | is bounded on Bat for all t > 0 then ro · f is continuous on B˜ a . In fact, for this conclusion it is sufficient for | f | to be bounded on Bat˜ for all t > 0 with some a˜ > a.

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Lemma 2. Any function on B˜ a of the form rom · fl with m > 0 is of class C k−1 but not of class C k , where k := min{m, m + sl }. Proof. First, we note that |xi | < r on Ba for all i = 0, . . . , 4, and we see that the absolute value | fl | of any function of the form fl with sl ≥ 0 is bounded on Bat by t sl for all t > 0. More generally, the absolute value of the partial derivative ∂ I p fl is bounded on Bat for all t > 0 if sl − p ≥ 0. In particular, the absolute value of x0 /r is bounded on Ba . Moreover, x0 /r is continuous on (Ba ∪ L o )\{0}. This shows that the extension of the function r − x0 · x0 /r by zero to the origin in R5 is a continuous function on Ba ∪ L o and is identically zero on L o . And this implies that the coordinate ro is continuously defined on B˜ a . The function ro is not continuously differentiable. However, it is   4  xi x02 xi −2x0 d x0 + dro = + d xi , r r3 r i=1

and we see that the coefficients of dro are bounded on Ba . (This implies that d(ro2 ) = 2ro dro is continuous on B˜ a , i.e., ro2 is of class C 1 .) Eventually, since a function of the form rom · fl admits terms of lowest order m + sl , such a function is at most of class C m+sl −1 . However, a p th order derivative of rom · fl on Ba can be extended continuously to the zero function on L only if p < m. In fact, any application of a derivative ∂ Im of order m admits a non-trivial term of the form ∂ m! · fl · m k=1 dro ( ∂ xi ), which can not be extended continuously by zero to L. On the k

other side, it is ∂ Im−1 (rom · fl ) = ro · h, where |h| is bounded on Bat for all t > 0 if sl ≥ 0. This shows that rom · fl is of class C k−1 with k := min{m, m + sl }, but it is not k-times continuously differentiable.   Now we set ωa := (aro )4 · (r σ3 )2

ρa =

and

 α 2 a 4 ro2 · . 1 − (aro )4 r

With these notations it is ga = g0 − ωa + ρa on B˜ a , where g0 is the flat Minkowski metric on B˜ a . Proof of Proposition 1. The metric g0 is smooth on B˜ a . We have to discuss the differentiability of ωa and ρa . The coefficients of the 1-forms r · σ3 and α/r are of order sl = 0, resp. sl = 1. With application of Lemma 2 we conclude that ωa is of class C 3 and ρa is of class C 1 . The symmetric 2-form ρa is not of class C 2 . This implies that the symmetric bilinear form ga on B˜ a is of class C 1 for all a > 0, but it is not of class C 2 . We postpone the proof that ga is a metric of Lorentzian signature until the proof of Proposition 2. The proof of Lemma 1 about the existence of the orthonormal frame e will show the Lorentzian signature of ga as well.   For the proof of Proposition 2 we use the coordinate change  : R5 \ L o (x0 , r, ϕi )

→ →

R5 \ L o , (s, R, ϕi ) =



−x0 r , , ϕi r 2 − x02 r 2 − x02

 ,

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where the ϕi ’s are some (local) coordinates on S 3 which remain unchanged. The coordinate transformation  is smooth on R5 \ L o and we have s2 + R2 ds (R 2 − s 2 )2 2s R ds dr = 2 (R − s 2 )2 ∂ ∂ = −2s R ∂r ∂s ∂ ∂ 2 = −(s + R 2 ) ∂ x0 ∂s

d x0 = −

This shows also T =

∂ ∂R

and V =

2s R d R, (R 2 − s 2 )2 2 2 s +R − d R, (R 2 − s 2 )2 ∂ − (s 2 + R 2 ) , ∂R ∂ − 2s R . ∂R +

∂ ∂s .

Proof of Proposition 2. First, we calculate the symmetric bilinear form ga on Ba with respect to the coordinate transformation . Remember that α = (x02 + r 2 )dr − 2x0 r d x0 . It is −d R , (R 2 − s 2 )2 −ds 2 + d R 2 . −d x02 + dr 2 = (R 2 − s 2 )2 α=

With R 2 = ro−2 on Ba and r 2 − x02 = (R 2 − s 2 )−1 we obtain 1 ( −ds 2 + d R 2 + R 2 (σ12 + σ22 + (1 − (a/R)4 )σ32 ) ) (R 2 − s 2 )2 R2 · d R2 − 2 r (1 − (R/a)4 ) · (R 2 − s 2 )4  1 − ds 2 + d R 2 + R 2 (σ12 + σ22 + (1 − (a/R)4 )σ32 ) = (R 2 − s 2 )2  1 2 − d R 1 − (R/a)4   1 d R2 2 2 2 2 4 2 = − ds + + R (σ1 + σ2 + (1 − (a/R) )σ3 , (R 2 − s 2 )2 1 − (a/R)4

ga =

and we can conclude that g˜ a =

(r 2

1 · ga = −ds 2 + g E H − x02 )2

on Ba . The corresponding (even simpler) calculation on L \ L o , where ro ≡ 0, shows that g˜ a =

(r 2

1 ga = −ds 2 + d R 2 + R 2 (σ12 + σ22 + σ32 ) , − x02 )2

i.e., g˜ a is the flat metric for r < |x0 |. In particular, since g˜ a on B˜ a \ L o is a metric of Lorentzian signature, we have shown that the conformally equivalent symmetric bilinear

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form ga of class C 1 on B˜ a is a metric and admits Lorentzian signature as well, which completes the proof of Proposition 1. Next we review curvature properties of the Eguchi-Hanson metric g E H . This discussion will provide us with all the information that we need to prove our claims about the curvature properties of the Lorentzian metrics g˜ a and ga . Let us fix the orthonormal frame   ∂ ∂ ∂ ∂ , , R −1 , R −1 , (Rβ)−1 { f 1 , f 2 , f 3 , f 4 } := −β ∂R σ1 σ2 σ3  where β := 1 − (a/R)4 . We denote by { f i : i = 1, . . . , 4} the dual frame. The connection 1-form ω and the curvature 2-form  of the Levi-Civita connection ∇ g E H are determined by the structure equations dfi =

4 

ωki ∧ f k

and

ij = dωij −

k=1

4 

ωki ∧ ωkj .

k=1

We have ωij = g E H (∇ E H f i , f j )

and

ij = g E H (R(ei , e j )·, ·) ,

where R(ei , e j ) = ∇eEi H ∇eEj H − ∇eEj H ∇eEi H − ∇[eEiH,e j ] . The components are explicitly calculated to = −β R −1 · f 2 = −β · σ 1 ,

ω21 = ω43

ω31 = −ω42 = −β R −1 · f 3 = −β · σ 2 , ω41 = ω32

= −γ ·

where γ = β R −1 + β  and β  =

∂β ∂R

f 4 = −γ Rβ · σ 3 ,

= 2(a/R)4 (Rβ)−1 , and

2a 4 1 λ , R6 − 2a 4 13 = −24 = − 6 λ2− , R 4a 4 3 1 2 4 = 3 = 6 λ− , R

12 = 34 = −

where the λi− ’s build a basis of the anti-selfdual 2-forms for g E H and are defined as λ1− = f 1 ∧ f 2 − f 3 ∧ f 4 , λ2− = f 1 ∧ f 3 − f 4 ∧ f 2 , λ3− = f 1 ∧ f 4 − f 2 ∧ f 3 . It follows that the Riemannian curvature tensor R E H of g E H is anti-selfdual. This implies that g E H is Ricci-flat and R E H equals the Weyl tensor W E H , i.e., we have R E H = W E H = W − = 0.

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In particular, since the Weyl tensor is a complete obstruction to conformal flatness in dimension 4, we can see that g E H is nowhere conformally flat on its domain of definition (which is Ba ∩ E, resp. (Ba ∩ E)). The metric g˜ a = −ds 2 + g E H is an ordinary semi-Riemannian product. Hence the ∂ curvature components of g˜ a in the direction of the coordinate ∂s vanish, i.e., the curva2 ture tensor of −ds + g E H is entirely determined by the components of the Riemannian curvature tensor R E H . In particular, we see that the metric −ds 2 + g E H is Ricci-flat ∂ and the components of the Weyl tensor W g˜a of g˜ a in the direction of the coordinate ∂s do vanish as well. Since, by construction, the metric g˜ a = −ds 2 + g E H is conformally equivalent to ga on Ba , we know the Weyl tensor W ga of ga on Ba as well. It is simply a rescaling of W g˜a . Obviously, the metric ga is not conformally flat on Ba . On L \ L o the metric ga is flat and therefore conformally flat, i.e., W ga = 0 on Ba and W ga ≡ 0 on L \ L o . Finally, on the lightcone L o the Weyl tensor of ga is not defined in the usual way, because ga is only of class C 1 at L o . We aim to show that the Weyl tensor of ga on B˜ a \ L o admits a continuous extension to L o . For this we note that the Weyl tensor rescales explicitly by W ga = ro4 r 4 · W g˜a . Then calculating the components of W ga with respect to the coordinate system u := { ∂∂x0 , . . . , ∂∂x4 } using our formulae for W E H from above results in expressions of the form    ∂ ∂ ∂ ∂ A · ro2 + B · ro6 /r 4 on Ba = W ga , , , 0 on L \ L o ∂ x i ∂ x j ∂ x k ∂ xl for all i, j, k, l ∈ {0, . . . , 4}, where A, B are sums of functions of the form fl , β · fl and β −1 · fl with order sl = 4, i.e., the extensions of all components to L o by zero are C 1 -functions on B˜ a . We conclude that the Weyl tensor W ga has a continuous extension  of class C 1 on B˜ a .  Now we consider the frame e = {e0 , . . . , e4 }, which we have defined in Sect. 2 and which was claimed there to be orthonormal for ga on Ba> and of class C 1 . Proof of Lemma 1. First, we show that the frame e is orthonormal in every point of (Ba> , ga ). Obviously, this is true on L \{r = 0}, since ga is the flat Minkowski metric thereon. It is also obvious that the vectors e2 , e3 and e4 are orthonormal for ga on Ba and that they are orthogonal to the remaining basis vectors e0 and e1 . For the latter we find with

a 4 ro2 1+β

= R 2 (1 − β) and T =

∂ ∂R

the expressions

∂ ∂ − 2s Rβ ∂s ∂R ∂ ∂ e1 = −2s R − (s 2 + R 2 )β , ∂s ∂R

e0 = −(s 2 + R 2 )

and

from which we see that e0 and e1 are orthonormal with respect to ga = (R 2 −s 2 )−2 (−ds 2 + g E H ) on Ba as well. We conclude that the frame e is a pointwise orthonormal basis on Ba . It remains to discuss the differentiability of the coefficients of the vectors {e0 , . . . , e4 }. a4r 2

For this we notice that the function 1+βo is only of class C 1 on Ba> . The function β −1 is of class C 3 and all other functions, which are involved in the coefficients are smooth on Ba> .  

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Let us introduce the vectors ∂ −1 ∂ + 2s Rβ ) ( (s 2 + R 2 ) R2 − s2 ∂s ∂R −1 ∂ ∂ e˜1 := 2 ( 2s R + (s 2 + R 2 )β ) 2 R −s ∂s ∂R e˜0 :=

and

with respect to the -transformed coordinates, and let us denote   ∂ ∂ ∂ , , R −1 · , (Rβ)−1 · e˜ := e˜0 , e˜1 , R −1 · ∂σ1 ∂σ2 ∂σ3 which is an orthonormal frame with respect to g˜ a =

1 · (r 2 −x02 )2 2 2 (R − s ) · e˜i , i

ga on Ba> \ L o . As we

know from the proof of Lemma 1, it is ∗ (ei ) = = 0, . . . , 4. Moreover, we set   ∂ ∂ ∂ ∂ ∂ −1 −1 −1 f := − , −β , R · , (Rβ) · , R · ∂s ∂R ∂σ1 ∂σ2 ∂σ3 on Ba> \ L o . On Ba this is just the extension by f 0 of the frame { f 1 , f 2 , f 3 , f 4 } that we introduced already for the Eguchi-Hanson metric g E H . The matrix ⎛ 2 ⎞ s + R2 2s R 0 0 0 ⎜ 2s R ⎟ s2 + R2 0 0 0 ⎜ ⎟ 1 2 2 ⎜ ⎟ κ= 2 0 0 R −s 0 0 ⎜ ⎟ 2 R −s ⎝ 2 2 ⎠ 0 0 0 R −s 0 2 2 0 0 0 0 R −s gives the transformation e˜ = f · κ. With t := ln R−s R+s and ⎞ ⎛ 0 −1 0 0 0 ⎜ −1 0 0 0 0 ⎟ ⎟ ⎜ 0 0 0 0⎟ E 01 := ⎜ 0 ⎝ 0 0 0 0 0⎠ 0 0 0 0 0 we have κ = exp(t E 01 ). The elements in the preimage of κ by the group covering λ : Spin(1, 4) → SO(1, 4) are given by ± exp( 2t γ0 γ1 ), where we use the γ -matrices introduced in Sect. 2. We choose in the following κ˜ := exp( 2t γ0 γ1 ), which is given by ⎛ ⎞ R 0 −s 0 1 R 0 s⎟ ⎜ 0 κ˜ = √ ⎝ −s 0 ⎠. R 0 2 2 R −s 0 s 0 R Before we start with the proof of Theorem 1, let us recall the conformal covariance of the twistor equation in explicit terms. In general, let g˜ = e2σ g be a rescaled metric in the conformal class of a given metric g and let ϕ = [vs , w] be a twistor with respect to g, where vs denotes the lift of some orthonormal frame v. We set ϕ˜ := [v˜s , w], where v˜s denotes the lift of the rescaled frame v˜ := v · (e−σ id), which naturally corresponds to the lift vs . Then the spinor field eσ/2 · ϕ˜ is a twistor spinor with respect to the rescaled metric g˜ (cf. [BFGK91]).

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Proof of Theorem 1. The verification of the first two statements of Theorem 1 is the > is a main part of the proof. We will show this in some few steps. First, we prove that ψbc > twistor on Ba and also on L\(L o ∪ {r = 0}), which already implies that ψbc is a twistor > , but first use the on Ba> . Thereby, we will not directly check the twistor equation for ψbc conformal transformation from ga to the Ricci-flat metric g˜ a . In the next step we show > extends to a C 1 -spinor on B ˜ a \{0}. This spinor will still be a twistor. Finally, we that ψbc show that the latter spinor can be extended to the origin by a zero. The resulting spinor > , which is of class C 1 and solves the twistor ψbc is a unique continuous extension of ψbc equation everywhere on B˜ a . > on B > \L . The spinor ψ > is given with respect to To start with, let us consider ψbc o a bc the spinor frame es by [ es , (−x0 b, x0 c, r b, r c) ]. It is e = e˜ · ((R 2 − s 2 )id), where e˜ 1 is orthonormal with respect to g˜ a = (r 2 −x ˜s be the corresponding lift of the 2 )2 ga . Let e 0

rescaled frame e. ˜ Then the spinor νbc :=



> R 2 − s 2 · ψ˜ bc = [ e˜s ,



R 2 − s 2 · (−x0 b, x0 c, r b, r c) ]

is a twistor with respect to g˜ a (by conformal covariance). Further, we have νbc =



R 2 − s 2 · [ f s , κ˜ · (−x0 b, x0 c, r b, r c) ] ,

where f s denotes the lift of the frame f , with ⎛ ⎞ ⎛ −x0 b R  ⎜ x0 c ⎟ ⎜ 0 2 2 R − s · κ˜ ⎝ = r b ⎠ ⎝ −s rc

which corresponds to the lift e˜s . Eventually,

0

0 R 0 s

−s 0 R 0

⎞⎛ ⎞ ⎛ ⎞ 0 −x0 b 0 s ⎟ ⎜ x0 c ⎟ ⎜ 0 ⎟ = 0 ⎠ ⎝ rb ⎠ ⎝ b ⎠ R rc c

we find that νbc = [ f s , (0, 0, b, c) ] . The spinor derivative of νbc with respect to g˜ a is given by 1 ∇˜ S νbc = [ f s , · 2



ωij ⊗ γi γ j · (0, 0, b, c) ],

0≤i< j≤4

where the ωij ’s are the components of the Levi-Civita connection of g˜ a . On Ba it is ω0j = 0 and the other ωij ’s are just the components that we calculated in the proof of Proposition 2 for the Eguchi-Hanson metric g E H . Notice also that on L\(L o ∪ {r = 0}) the components ωij admit the same expressions (with β ≡ 1) as on Ba with respect to the frame f . The relations for the ωij ’s immediately prove that νbc is a parallel spinor with respect to g˜ a on Ba> \ L o for any (b, c) ∈ C2 \0. (In fact, the spinors of the form νbc restricted to the Eguchi-Hanson metric g E H , which is a hyperkähler metric for any a > 0, form the space of all parallel spinors thereon.) Any parallel spinor is a twistor spinor. In particular, νbc is a twistor spinor for g˜ a . Hence, by conformal covariance and > is of class C 1 on B > , it follows that ψ > is a twistor on (B > , g ). the fact that ψbc a a a bc

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Now let

⎛

r ⎜0 ⎜ G = r −1 · ⎜ 0 ⎝0 0

0 x1 −x2 −x3 −x4

0 x2 x1 x4 −x3

0 x3 −x4 x1 x2

⎞ 0 x4 ⎟ ⎟ x3 ⎟ −x2 ⎠ x1

be a matrix valued function on Ba> . It is e · G = { ∂∂x0 , . . . , ∂∂x4 } on L\(L o ∪{r = 0}). The standard frame u is orthonormal on L \(L o ∪ {r = 0}) and admits a smooth extension to L \ L o . (Of course, the matrix G is singular for r = 0.) A transformation matrix for corresponding spinor frames is given by ⎞ ⎛ r 0 0 0 0 0 ⎟ ⎜0 r . G˜ = r −1 · ⎝ 0 0 x1 + i x2 x3 + i x4 ⎠ 0 0 −x3 + i x4 x1 − i x2 This form of the matrix is due to the fact that Spin(4) is isomorphic to SU(2) × SU(2). > = [e , (−x b, x c, r b, r c) )] is presented with respect to the spinor The spinor ψbc s 0 0 frame u s on L \(L o ∪ {r = 0}) by > ψbc = [ u s , G˜ −1 (−x0 b, x0 c, r b, r c) ) ] .

Obviously, the vector valued function ⎛ ⎞ ⎛ ⎞ −x0 b −x0 b x0 c ⎜ x c ⎟ ⎜ ⎟ G˜ −1 ⎝ 0 ⎠ = ⎝ rb (x1 − i x2 )b − (x3 + i x4 )c ⎠ rc (x3 − i x4 )b + (x1 + i x2 )c > on B > admits a C 1 -extension is non-singular and smooth on L\L o . Hence the spinor ψbc a o , which is by continuity reasons a twistor on to B˜ a \{0}. We denote this extension by ψbc B˜ a \{0}. o extends further to a C 1 -spinor ψ on B ˜ a . For this We still have to show that ψbc bc purpose, we improve our change of frame from above and introduce a non-singular C 1 o with respect to a frame around the origin. Then we show that the components of ψbc 1 corresponding non-singular spinor frame are of class C . So let ⎞ ⎛ k q 0 0 0 ⎜q k 0 0 0⎟ ⎟ ⎜ Q = ⎜0 0 1 0 0⎟ ⎝0 0 0 1 0⎠ 0 0 0 0 1

with

 k :=  q :=

1 + ro2 ρ · 1 − 4x02 ρ



(r 2 + x02 )β 2 1 − ·ρ r 2 (1 + β)

2x0 (r 2 + x02 )β 2 1 + ro2 ρ · ρ, · r (1 + β) 1 − 4x02 ρ

 and

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603

where ρ = a 4 β −2 ro2 . It is Q ≡ 1l on L and 4x02 ρ < 1 on an open neighbourhood of L. For 4x02 ρ < 1 the function k is well defined and of class C 1 (cf. Lemma 2). It follows that on a certain open neighbourhood of L in B˜ a the function k is positive. We denote this set by Ca . In fact, it is k 2 − q 2 ≡ 1 on Ca , i.e., the transformation matrix Q takes values in SOo (1, 4) and is of class C 1 on Ca . The matrix Q is useful, because the transformed frame {h 0 , . . . , h 4 } := e · Q is given by h 0 = (1 − 4x02 ρ)−1/2 ·

∂ , ∂ x0

∂ ∂ + ρ2 (1 + ρ1 )−1/2 · (1 − 4x02 ρ)−1 · , ∂r ∂ x0 for i = 2, 3, 4

h 1 = (1 + ρ1 )−1/2 · h i = ei

on Ba> ∩ Ca with ρ1 := r −2 (r 2 + x02 )2 ρ and ρ2 := −4x0 r −1 (r 2 + x02 )ρ, i.e., the first basis vector h 0 admits now a continuous extension to {r = 0}. The remaining basis vectors are still singular at {r = 0}. However, a straightforward calculation shows that the frame h˜ = {h˜ 0 , . . . , h˜ 4 } := e · (QG) admits a C 1 -extension to {r = 0}, i.e., h˜ is a non-singular C 1 -frame on Ca , which is an open neighbourhood of the origin. A corresponding transformation matrix to Q for spinor frames is given by ⎞ ⎛
k+1 √ −q 0 0 2 2(k+1)
⎟ ⎜ ⎟ ⎜ k+1 √ q 0 ⎟ ⎜ 0 2 2(k+1) ⎟ ⎜ ˜
Q=⎜ ⎟ . −q k+1 ⎜√ 0 0 ⎟ 2 2(k+1) ⎠ ⎝
k+1 √ q 0 0 2 2(k+1) This matrix is again of class C 1 on Ca . In particular, it is non-singular and equal to the ˆ where Qˆ is some identity on L. In fact, the matrix Q˜ can be written as Q˜ = 1l + ro2 · Q, matrix valued function on Ca whose components are sums of functions of the form fl o is expressed with respect to the corresponding spinor frame with sl ≥ 0. The spinor ψbc h˜ s by o ψbc = [ h˜ s , G˜ −1 · Q˜ −1 (−x0 b, x0 c, r b, r c) ] .

We have

⎛
⎞ k+1 √ qr b − · x b − 0 ⎛ ⎞ 2(k+1) ⎟ ⎜
2 −x0 b ⎜ ⎟ qr c k+1 √ ⎜ 2 · x 0 c +
2(k+1) ⎟ −1 ˜ −1 ⎜ x 0 c ⎟ −1 ⎜ ⎟ . ˜ ˜  := G · Q ⎝ =G ⎜ ⎟ rb ⎠ k+1 ⎜ √q x0 b ⎟ + · r b 2(k+1) ⎝ ⎠ rc
2 k+1 √q x0 c + · r c 2 2(k+1)

From Lemma 2 we know that the function qrx0 is of class C 1 , since it behaves like ro2 · xr0 , where xr0 has order zero. This observation is sufficient to conclude that the vector valued function  extends to a C 1 -function on Ca . Obviously, the extended C 1 -function  is o extends to a C 1 -spinor ψ on B ˜ a with zero zero at the origin. We can conclude that ψbc bc o is a at the origin. If (b, c) = 0 the origin is the only zero of ψbc . Moreover, since ψbc

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twistor and ψbc is C 1 , the expression ∇ XS ψbc + n1 X · D S ψbc is continuous on B˜ a and zero on B˜ a \{0} for all differentiable vector fields X . This shows that ψbc satisfies the twistor equation in the origin. Altogether we have proven yet the first three statements of Theorem 1. > = [e , (−x b, x c, r b, r c) ] is by definition (cf. The length square u bc of ψbc s 0 0 [Baum81]) equal to ( γ0 · (−x0 b, x0 c, r b, r c) , (−x0 b, x0 c, r b, r c) )C4 = (r 2 − x02 ) · (b2 + c2 ) . Obviously, the function u bc is smooth on B˜ a and its zero set is L o . We know already ˜ from Proposition 2 that u −2 bc · ga is a Ricci-flat metric on Ba \ L o , i.e., the function u bc provides a rescaling to an Einstein metric in the conformal class. It is well known that such a rescaling function satisfies the partial differential equation −u bc · Ric0 = (n − 2) · H ess(u bc )0 . It is interesting to note that the function u bc has a non-trivial zero set (cf. [Gov04]). Furthermore, using the definition ga (Vψbc , X ) = ψbc , X · ψbc  S and calculating the products (γ0 · (−x0 b, x0 c, r b, r c) , γi · (−x0 b, x0 c, r b, r c) ) for i = 0, . . . , 4 shows easily that the spinor square of the twistor ψbc is equal to   Vψbc = (b2 + c2 ) · −(x02 + r 2 )e0 − 2x0 r e1 = (b2 + c2 ) · V. For (b, c) = 0 the vector field Vψbc is smooth with unique zero at the origin. Finally, since α(V ) = 0, we obtain ga (V, V ) = −(r 2 −x02 )2 . This shows that Vψbc for (b, c) = 0 is everywhere timelike except on L o where the spinor square is lightlike, resp. zero, only at the origin.   Corollary 1 is a simple conclusion using Theorem 1 and Proposition 2. We add some remarks about the vector field V . Although the twistor spinor ψbc is not smooth, the spinor square Vψbc is smooth. Since ψbc is a twistor we immediately know that V is a conformal Killing vector field for ga on B˜ a . However, we simply reprove this statement here directly. Namely, the following equations hold: L V g0 = −4x0 · g0 , L V r m = −2mx0 · r m , L V ro2 = 0, L V σ32 = L V (−r (aro )

0

= −4x0 · (−r 2 (aro )4 · σ32 ), L V α = −4x0 · α, L V (a 4 (rβ)−2 ro2 · α 2 ) = (4x0 − 2 · 4 · x0 )(a 4 (rβ)−2 ro2 · α 2 ), 2

4

· σ32 )

= −4x0 · (a 4 (rβ)−2 ro2 · α 2 ). This proves that L V ga = −4x0 · ga on B˜ a , i.e., V is a conformal Killing vector with div ga (V ) = −10 · x0 . An interesting property of V is the fact that it is an essential conformal Killing vector field. The reason is that the spinor square VDψbc of D S ψbc is given in {0} ∈ B˜ a by VDψbc =

n · grad(div(Vψbc )) , 2

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which does not vanish, since D S ψbc = 0 in the origin. This argument is true for any metric in the conformal class ca = [ga ], i.e., the divergence of Vψbc , resp. V , does not vanish identically with respect to any metric in ca . Finally, we want to state a reason why an extension of the metric ga on Ba to L with differentiable Weyl tensor has to be conformally flat in order to preserve twistor spinors and the conformal Killing vector V . One observes the following facts. All integral curves of V on L converge in one flow direction to the origin, i.e., the origin is in the closure of any integral curve on L. The length square |W 2,2 |2 of the Weyl (2, 2)-tensor is constant along integral curves of V . Moreover, with our assumptions we know that in the origin W ga has to vanish (cf. [Baum99]), i.e., |W 2,2 |2 is identically zero on the closure L of L \ L o . Then, since V inserted into the Weyl tensor W ga produces zero (cf. [Baum99]) and V is timelike on L \ L o , it follows that the length square of W ga is non-negative on L \ L o and it is zero if and only if the Weyl tensor vanishes. With the argument from before we can conclude that the Weyl tensor of the extension has to vanish on L. References [Oba71]

Obata, M.: The conjectures of conformal transformations of Riemannian manifolds. Bull. Amer. Math. Soc. 77, 265–270 (1971) [LF71] Lelong-Ferrand, J.: Transformations conformes et quasi-conformes des varietes riemanniennes compactes. Acad. Roy. Belg. Cl. Sci. Mem. Coll. 39(5), 1–44 (1971) [Ale72] Alekseevskii, D.: Groups of conformal transformations of Riemannian spaces. Mat. Sbornik 89 131, (1972) (in Russian), English translation Math. USSR Sbornik 18, 285–301 (1972) [Yos75] Yoshimatsu, Y.: On a theorem of Alekseevskii concerning conformal transformations. J. Math. Soc. Japan 28, 278–289 (1976) [EH78] Eguchi, T., Hanson, A.J.: Asymptotically flat self-dual solutions to euclidean gravity. Phys. Lett. B. 74, 249–251 (1978) [Baum81] Baum, H.: Spin-Strukturen und Dirac-Operatoren über pseudo-Riemannschen Mannigfaltigkeiten. No. 41 of Teubner-Texte zur Mathemtik. Leipzig, Teubner, 1981 [Lic88] Lichnerowicz, A.: Killing spinors, twistor spinors and Hijazi inequality. J. Geom. Phys. 5, 2–18 (1988) [Lic89] Lichnerowicz, A.: On the twistor spinors. Lett. Math. Phys. 18, 333–345 (1998) [Lic90] Lichnerowicz, A.: Sur les zeros des spineurs-twisteurs. C.R. Acad. Sci. Paris, Serie I, 310, 19–22 (1990) [D’AG91] D’Ambra, G., Gromov, M.: Lectures on transformation groups: geometry and dynamics. In: Surveys in differential geometry (Cambridge, MA, 1990), Bethlehem, PA: Lehigh Univ., 1991, pp. 19–111 [BFGK91] Baum, H., Friedrich, Th., Grunewald, R., Kath, I.: Twistor and Killing spinors on Riemannian manifolds, Teubner-Text No. 124, Stuttgart-Leipzig, Teubner-Verlag, 1991 [KR95] Kühnel, W., Rademacher, H.-B.: Twistor spinors with zeros. Int. J. Math. 5, 877–895 (1994) [KR96] Kühnel, W., Rademacher, H.-B.: Twistor spinors and gravitational instantons. Lett. Math. Phys. 38, 411–419 (1996) [KR98] Kühnel, W., Rademacher, H.-B.: Asymptotically euclidean manifolds and twistor spinors. Comm. Math. Phys. 196(1), 67–76 (1998) [Baum99] Baum, H.: Lorentzian twistor spinors and CR-geometry. J. Diff. Geom. and Its Appl. 11(1), 69–96 (1999) [Lei99] Leitner, F.: Zeros of conformal vector fields and twistor spinors in Lorentzian geometry. SFB288 e-print no. 439, Berlin 1999, available at http://www-sf6288.math.tu-berlin.de/Publications/ Preprints.html [Lei01] Leitner F.: The twistor equation in Lorentzian geometry. Dissertation, HU Berlin, 2001 [Lei04] Leitner, F.: A note on twistor spinors with zeros in Lorentzian geometry. http://arXiv.org/list/math. DG/0406298, 2004 [Gov04] Gover, A.R.: Almost conformally Einstein manifolds and obstructions. electronic preprint, http://arXiv.org/list/math.DG/0412393, 2004 Communicated by G.W. Gibbons

Commun. Math. Phys. 275, 607–658 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0325-0

Communications in

Mathematical Physics

SUSY Vertex Algebras and Supercurves Reimundo Heluani Department of Mathematics, MIT, Cambridge, MA 02139, USA. E-mail: [email protected] Received: 19 April 2006 / Accepted: 3 June 2007 Published online: 21 August 2007 – © Springer-Verlag 2007

Abstract: Given a strongly conformal SUSY vertex algebra V and a supercurve X , we construct a vector bundle V Xr on X , the fiber of which is isomorphic to V . Moreover, the state-field correspondence of V canonically gives rise to (local) sections of these vector bundles. We also define chiral algebras on any supercurve X , and show that the vector bundle V Xr , corresponding to a SUSY vertex algebra, carries the structure of a chiral algebra. 1. Introduction 1.1. Vertex algebras were introduced about 20 years ago by Borcherds [Bor86]. They provide a rigorous definition of the chiral part of 2-dimensional conformal field theory, intensively studied by physicists. Since then, they have had important applications to string theory and conformal field theory, and to mathematics, by providing tools to study the most interesting representations of infinite dimensional Lie algebras. Since their appearance, they have been extensively studied in many papers and books (for the latter we refer to [FLM88, FHL93, Kac96, Hua97, FBZ01, BD04]). Vertex algebras also appeared in algebraic geometry as Factorization Algebras on complex curves [BD04, FBZ01]. In the last five years, numerous applications of this deep connection between factorization algebras and vertex algebras have been exploited, notably in the study of the moduli spaces (of curves, vector bundles, principal bundles, etc.) arising in algebraic geometry. There are also connections between the theory of vertex algebras and the geometric Langlands conjecture [FBZ01, Ch. 17]. Vertex algebras have also given new invariants of manifolds [KV04, MSV99] and applications to mirror symmetry [Bor01]. Even though these approaches have been successful in formalizing 2-dimensional conformal field theories, it has been known for some time to physicists, that in order to describe supersymmetric theories, similar objects should be defined on supercurves instead of simply curves (cf. [DRS90, Coh87, BDFM88]). With this motivation,

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mathematicians have studied in detail the supergeometry of manifolds, and in particular supercurves (cf. [DM99, Man91, Man97, Vai90, CR88] among others). A supersymmetric version of the above mentioned vertex algebras appeared and were studied in detail in a series of papers by K. Barron [Bar96, Bar00, Bar03, Bar04], as “N = 1 Neveu-Schwarz vertex operator superalgebras”. In those articles, the author systematically develops the theory of N K = 1 (in the language used below) SUSY vertex algebras, finding identities like Skew-Symmetry and the Jacobi Identity. The author also describes in [Bar04] a changes of variables formula for these super vertex algebras. These algebras were recently generalized and studied from a different perspective in [HK07]. The purpose of this article is to generalize the above objects to describe chiral algebras over supercurves. To accomplish this, we will use the above mentioned supersymmetric (SUSY) vertex algebras, following [HK07]. Roughly speaking, SUSY vertex algebras are vertex algebras with a state-field correspondence that includes the odd coordinates of a supercurve as formal parameters, that is, to any vector a in a SUSY vertex algebra, we associate a superfield Y (a, z, θ 1 , . . . , θ N ), such that structural properties, similar to those of ordinary vertex algebras, hold. The use of superfields instead of ordinary fields was first formalized by Barron in [Bar96] (see also [Bar00]). Given a SUSY vertex algebra V and a supercurve X , we want to assign a vector bundle V over X in such a way that the fiber at a point x ∈ X is identified with V . Moreover, we would like Y to canonically define sections of this vector bundle (more precisely, its restricted dual). Here we find the first difference with the classical theory, namely, supercurves come in different flavors: general 1|n dimensional supercurves and superconformal 1|n supercurves. The latter are to the former what holomorphic curves are to compact connected 2-manifolds. In [HK07], two different versions of SUSY vertex algebras are defined, one which localizes to give vector bundles on a general 1|n-dimensional supercurve (called N W = n SUSY vertex algebras) and another which gives vector bundles on superconformal supercurves (called N K = n SUSY vertex algebras). There are several relations between these different SUSY vertex algebras. As a basic example, let us consider the cases with low odd dimensions. Roughly speaking, a general N = 1 supercurve is the data of a curve X and a line bundle L over it, sections of this line bundle are considered to be the values of a coordinate in the odd direction. Similarly, an (oriented) superconformal N = 2 supercurve consists of a curve X and two line bundles L and H over it such that L ⊗ H is the canonical bundle ω of X . It follows that an N = 1 supercurve (call it X ) gives rise canonically to another N = 1 supercurve (interchanging L with ω ⊗ L −1 ) and to a superconformal N = 2 supercurve (by taking H = ω ⊗ L −1 ). Let us call these curves Xˆ (the “dual curve” to X ) and Y respectively. On the algebraic side, any (conformal) N W = 1 SUSY vertex algebra gives rise to a (conformal) N K = 2 SUSY vertex algebra (this corresponds to the isomorphism between the superconformal Lie algebras K (1|2) and W (1|2)) and both of them correspond to vertex algebras with N = 2 superconformal structure. It follows that any such vertex algebra gives vector bundles in both N = 1 supercurves X and Xˆ , and in the corresponding superconformal N = 2 supercurve Y . These three vector bundles are intimately related as we will see in Sect. 3.3. This phenomenon occurs even for the algebra of free fields (see Example 3.17). For instance, consider the vertex algebra generated by two charged fermions ϕ ± and two charged bosons α ± . This vertex algebra comes equipped with an embedding of the N = 2 superconformal vertex algebra. We

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obtain three vector bundles V X , V Xˆ and VY on X , Xˆ and Y respectively. Each of these bundles is filtered by conformal weights, looking at the conformal weight ≤ 1/2 part in VY , we obtain a rank 1|2 vector bundle V≤1/2 on Y , “generated” by the vacuum vector |0 and the two fermions ϕ ± . To construct the corresponding vector bundle in X we need to consider a different Virasoro element, and consequently, a different filtration by conformal weights. This is accomplished by performing the well known topological twist on the N = 2 vertex algebra. With respect to this new Virasoro, the fermion ϕ − has now conformal weight 0. It follows that looking at the filtered part of conformal weight 0 in V X we obtain a rank 1|1 vector bundle V X,0 on X “generated” by the vacuum vector and the fermion ϕ − . Finally, applying the well known mirror involution to the N = 2 vertex algebra, we obtain yet another Virasoro element. With respect to this last Virasoro, the fermion ϕ + has conformal weight 0. Applying the construction mentioned above, we obtain a rank 1|1 vector bundle V Xˆ ,0 on Xˆ “generated” by the vacuum vector and the fermion ϕ + . The new phenomenon is that the bundle VY,≤1/2 appears to be “built” from the bundles V X,0 and V Xˆ ,0 , with the fermion ϕ − “living” in X and the fermion ϕ + in Xˆ (see the short exact sequences (3.12) for a rigorous statement). The vector bundles we construct (more precisely quotients of them) are extensions of (powers of) the Berezinian bundle of X (a super analog of the canonical bundle). The algebraic properties of V reflect in geometric properties of V as in the ordinary vertex algebra case. We obtain thus superprojective structures, affine structures, global differential operators, etc. as splittings of these extensions. In particular, the state-field correspondence itself gives such splittings (locally). 1.2. After constructing these vector bundles, it is natural to ask if they carry the structure of a chiral algebra on a supercurve. It is shown that the usual definitions carry over to the super case with minor difficulties, and that the vector bundles obtained from V are indeed chiral algebras. This allows us to define the coinvariants and conformal blocks of a SUSY vertex algebra in a coordinate independent way. 1.3. The organization of this article is as follows: In Sect. 2 we recall some well known notions about vertex algebras and supercurves. We also summarize here some results on the structure theory of SUSY vertex algebras. In Sect. 3 we construct a vector bundle with a flat connection associated to an N W = n SUSY vertex algebra, over any N = n supercurve. We also construct vector bundles associated to N K = n SUSY vertex algebras over oriented superconformal N = n supercurves. In this section we follow closely [FBZ01], showing the main subtleties that arise specially in the N K = n case. The new phenomena can be found in Sect. 3.3 where we analyze in detail examples with supersymmetry. In Sect. 3.1 we define the groups AutO of changes of coordinates and the AutO-torsor Aut X for a supercurve. In Sect. 3.2 we construct the vector bundles themselves and their sections. In particular we show that the state-field correspondence for a SUSY vertex algebra is a section of the dual of the corresponding vector bundle. To construct this vector bundle, we need to prove a supersymmetric version of Huang’s formula for changes of coordinates. This was first done in the N K = 1 case in [Bar04]. The general case is an easy generalization of that result. In Sect. 3.3 we compute explicitly some examples of vector bundles over supercurves of low odd dimension. In Sect. 4 we define chiral algebras over supercurves and we prove that the vector bundles constructed from SUSY vertex algebras are examples of chiral algebras. We also define the spaces of coinvariants in a coordinate independent way.

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2. Preliminaries 2.1. SUSY vertex algebras. In this section we collect some results and examples of SUSY vertex algebras from [HK07]. The general theory of vertex algebras has been widely studied in several papers and books (for the latter we refer to [FLM88, DL93, FHL93, Kac96, Hua97, FBZ01, LL04, BD04]). In the N K = 1 case, vertex superalgebras over Grassman algebras were defined in [Bar00]. 2.1.1. Let N be a non-negative integer. We will denote Z = (z, θ 1 , . . . , θ N ), where z is an even indeterminate, θ i ’s are odd anticommuting indeterminates commuting with z. For J = ( j1 , . . . , js ) ordered subset of {1, . . . , N }, and j ∈ Z, we will denote θ J = θ j1 . . . θ js ,

Z j|J = θ J z j ,

and we will denote by N \ J the ordered complement of J in {1, . . . , N }. For two disjoint subsets I, J ⊂ {1, . . . , N } define σ (I, J ) = ±1 by θ I θ J = σ (I, J )θ I ∪J , and σ (J ) = σ (J, N \ J ). Finally, define ei = {i}. Given a vector superspace V , we will denote by V [[Z ]] (resp. V ((Z ))) the space of formal power series (resp. formal Laurent series) in Z with values in V , namely, formal sums of the form ⎛ ⎞

j|J j|J Z v j|J ⎝ resp. Z v j|J ⎠ , j≥0,J

j≥N0 ,J

where N0 is some integer number and v j|J ∈ V . Finally, we will denote by V [Z , Z −1 ] the space of Laurent polynomials with coefficients in V , namely, elements of V ((Z )) which are finite sums. Let HW (resp. H K ) be the associative superalgebra generated by an even element T and N odd elements S i subject to the relations [T, S i ] = 0, [S i , S j ] = 0 (resp. 2δi, j T ). An N W = N (resp. N K = N ) SUSY vertex algebra (V, |0, Y ) is the data of a HW -module (resp. H K -module) V (the space of states), an even vector |0 ∈ V (the vacuum vector) and a parity preserving C-bilinear product with values in Laurent series over V :
V ⊗ V → V ((Z )), a ⊗ b → Y (a, Z )b = Z −1− j|N \J a( j|J ) b, j∈Z,J

subject to the following axioms (a, b ∈ V ): • • •

(vacuum axioms) Y (a, Z )|0| Z =0 = a, T |0 = S i |0 = 0, for i = 1, . . . , N , (translation invariance) [T, Y (a, z)] = ∂z Y (a, Z ), [S i , Y (a, Z )] = ∂θ i Y (a, Z ) (resp. (∂θ i − θ i ∂z )Y (a, Z )), (locality) (z − w)n [Y (a, Z ), Y (b, W )] = 0 for some n ∈ Z+ .

Remark 2.1. Note that when N = 0, this definition agrees with the usual definition of vertex algebra.

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2.1.2. Denote  = (λ, χ 1 , . . . , χ N ), where λ is an even indeterminate and χ i ’s are odd indeterminates, subject to the relations: [λ, χ i ] = 0, [χ i , χ j ] = 0 ( resp. − 2δi, j λ), N i i θ χ . and write Z  = zλ + i=1 Let V be an N W = N (resp. N K = N ) SUSY vertex algebra. For a, b ∈ V we define [a b] = res Z e Z  Y (a, Z )b,

(2.1.1)

where res Z stands for the coefficient to the right of Z −1|N . We note that the right hand side of (2.1.1) is a finite sum of monomials in  times elements of V (cf. [HK07]). This operation is called the -bracket. As in the usual vertex algebra case, it encodes the singular part of the operator product expansion (OPE) in V . Define the normally ordered product :: as a C-bilinear product on V : V ⊗ V → V, a ⊗ b →: ab := a(−1|N ) b. The action of HW (resp. H K ) on V is by derivations of both, the -bracket, and the normally ordered product. As proved in [HK07], these two operations encode all the structure of the SUSY vertex algebra V . Example 2.2. (Virasoro) This is an ordinary vertex algebra generated by one even field L satisfying λ3 [L λ L] = (T + 2λ)L + c, (2.1.2) 12  where c ∈ C is the central charge. Expanding this field as L(z) = n∈Z z −2−n L n , we obtain that the operators L n satisfy the commutation relations of the Virasoro algebra of central charge c, namely: [L m , L n ] = (m − n)L m+n + δm,−n

m3 − m c. 12

A field L in an ordinary vertex algebra V satisfying (2.1.2) will be called a Virasoro field of central charge c. Definition 2.3. Let V be an ordinary vertex algebra with a Virasoro field L. We will say that a vector a ∈ V that satisfies [L λ a] = (T + λ)a + O(λ2 ) has conformal weight . If moreover, a satisfies [L λ a] = (T + λ)a, we will say that a is primary. Example 2.4. (Neveu-Schwarz) This vertex algebra is generated by a Virasoro field as in Example 2.2 and an odd field G, primary of conformal weight 3/2. The remaining λ-bracket is given by: λ2 [G λ G] = 2L + c. 3 If we expand the corresponding fields as:

L(z) = L n z −2−n , G(z) = G n z −3/2−n , n∈Z

n∈1/2+Z

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then the coefficients of such expansions satisfy the following commutation relations: [L m , L n ] = (m − n)L m+n + δm,−n

m3 − m c, 12

 m 2 − 1/4 n [G m , L n ] = m − G m+n , [G m , G n ] = 2L m+n + δm,−n c. 2 3

(2.1.3)

Remark 2.5. Let V be a vertex algebra with an N = 1 superconformal vector τ (cf. [Kac96, Def. 5.9]). Namely, the Fourier modes of the fields
G(z) = Y (τ, z) G n z −n−3/2 , n∈1/2+Z


1 L n z −2−n , L(z) = Y (G −1/2 τ, z) = 2 n∈Z

satisfy the relations (2.1.3) of a Neveu-Schwarz algebra for some c ∈ C, L −1 (= G 2−1/2 ) = T and the operator L 0 is diagonalizable with eigenvalues bounded below. Then V carries a structure of an N K = 1 SUSY vertex algebra with S = G −1/2 and the superfields are defined as: Y (a, z, θ ) = Y (a, z) + θ Y (G −1/2 a, z).

(2.1.4)

Remark 2.6. Equation (2.1.4) was first used in this vertex operator notation by Barron in [Bar96] (see also [Kac96, Eq. (5.9.5)]). Example 2.7. [Kac96, Ex. 5.9a] Let B1 be the vertex algebra generated by an even vector (free boson) α and an odd vector (free fermion) ϕ, namely: [αλ α] = λ, [ϕλ ϕ] = 1, [αλ ϕ] = 0. Then B1 is a (simple) vertex algebra with a family of N = 1 superconformal vectors τ = (α(−1) ϕ(−1) + mϕ(−2) )|0, m ∈ C, of central charge c =

3 2

− 3m 2 .

Example 2.8. [Kac96, Thm 5.10] The N = 2 vertex algebra is generated by a Virasoro field L of central charge c, an even field J , primary of conformal weight 1, and two odd fields G ± , primary of conformal weight 3/2. The remaining λ-brackets are: c ± ± ± λ, [G ± λ G ] = 0, [Jλ G ] = ±G , 3 c 1 [G +λ G − ] = L + ∂ J + λJ + λ2 . 2 6 [Jλ J ] =

This vertex algebra contains an N = 1 superconformal vector: τ = G +(−1) |0 + G − (−1) |0. Also, this vertex algebra admits a Z/2Z × C∗ family of automorphisms. The generator of Z/2Z is given by L → L, J → −J and G ± → G ∓ . The C∗ family is given by

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G + → µG + and G − → µ−1 G − . Applying these automorphisms, we get a family of N = 1 superconformal structures. By expanding the corresponding fields:
L(z) = L n z −2−n , n∈Z




±

G (z) =

−3/2−n G± , nz

J (z) =

n∈1/2+Z




Jn z −1−n ,

n∈Z

we get the commutation relations of the Virasoro operators L n , and the following remaining commutation relations: m ± δm,−n c, [Jm , G ± n ] = ±G m+n , 3  n ± G m+n , [L m , Jn ] = −n Jm+n , , L ] = m − [G ± n m 2

m−n c 1 2 J m δm,−n . ] = L + + − [G +m , G − m+n m+n n 2 6 4 [Jm , Jn ] =

Sometimes it is convenient to introduce a different set of generating fields for this ˜ vertex algebra. We define L(z) = L(z) − 1/2∂z J (z). This is a Virasoro field with central charge zero, namely: ˜ = (∂ + 2λ) L. ˜ [ L˜ λ L] With respect to this Virasoro element, G + is primary of conformal weight 2 and G − is primary of conformal weight 1; J has conformal weight 1 but is no longer a primary field. To summarize the commutation relations, we write:
Q(z) = G + (z) = Q n z −2−n , n∈Z −

H (z) = G (z) =




Hn z −1−n ,

n∈Z

˜ L(z) =




Tn z −2−n .

(2.1.5)

n∈Z

The corresponding λ-brackets of these fields are given by: ˜ = (∂ + 2λ) L, ˜ [ L˜ λ L]

[ L˜ λ J ] = (∂ + λ)J −

[ L˜ λ Q] = (∂ + 2λ)Q, [ L˜ λ H ] = (∂ + λ)H, c [Hλ Q] = L˜ − λJ + λ2 . 6

λ2 c, 6 (2.1.6)

The commutation relations of the coefficients in (2.1.5) are: [Tm , Tn ] = (m − n)Tm+n ,

[Q m , Q n ] = [Hm , Hn ] = 0,

c δm,−n , 12 c [Tm , Q n ] = (m − n)Q m+n , [Hm , Q n ] = Tm+n − m Jm+n + m(m − 1) δm,−n . 6 (2.1.7) Finally, defining G (1) = G + + G − and G (2) = i(G + − G − ), we obtain another set of generators for this vertex algebra. We note that, with respect to L, the fields G (i) are [Tm , Hn ] = −n Hm+n ,

[Tm , Jn ] = −n Jm+n − m(m + 1)

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primary of conformal weight 3/2, and J is primary of conformal weight 1. The other commutation relations between the generating fields L , J, G (i) are cλ2 , [G (1) λ G (2) ] = −i (∂ + 2λ) J, 3 [Jλ G (1) ] = −i G (2) , [Jλ G (2) ] = i G (1) ,

[G (i) λ G (i) ] = 2L +

or, equivalently:

1 c (i) 2 [G (i) δm,−n , , G ] = 2L + m − m+n m n 4 3 (2.1.8)

(2) [G (1) m , G n ] = i (n − m) Jm+n , (2)

(1)

(2) [Jm , G (1) n ] = −i G m+n , [Jm , G n ] = i G m+n .

Remark 2.9. As in the N = 1 case, given an N = 2 superconformal vertex algebra, namely a vertex algebra with a vector j and two operators S 1 , S 2 satisfying [T , S i ] = 0,

[S i , S j ] = 2δi, j T,

and such that the corresponding fields: 1 Y (S 2 S 1 j, z), 2 G (1) (z) ≡ G + (z) + G − (z) = −Y (S 2 j, z),  G (2) (z) ≡ i G + (z) − G − (z) = Y (S 1 j, z), J (z) = −iY ( j, z),

L(z) =

(i)

satisfy the λ-brackets of Example 2.8, L −1 = T , G −1/2 = S i , and L 0 is diagonalizable with eigenvalues bounded below, we obtain an N K = 2 SUSY vertex algebra by letting Y (a, Z ) = Y (a, z) + θ 1 Y (S 1 a, z) + θ 2 Y (S 2 a, z) + θ 2 θ 1 Y (S 1 S 2 a, z). Similarly, given a vertex algebra with two vectors j, h and an odd operator S such that [T, S] = 0, S 2 = 0 and the associated fields: J (z) = −Y ( j, z), Q(z) = Y (S j, z),

H (z) = Y (h, z), ˜ L(z) = Y (Sh, z) − ∂z J (z),

satisfy the commutation relations (2.1.6), T−1 = T , Q −1 = S, J0 is diagonalizable, and T0 is diagonalizable with eigenvalues bounded below, we obtain an N W = 1 SUSY vertex algebra by letting: Y (a, Z ) = Y (a, z) + θ Y (Sa, z). Example 2.10. [Kac96, Ex. 5.9d] Consider the vertex algebra generated by a pair of free charged bosons α ± and a pair of free charged fermions ϕ ± , where the only non-trivial commutation relations are: [α ± λ α ∓ ] = λ,

[ϕ ± λ ϕ ∓ ] = 1.

This vertex algebra contains the following family of N = 2 vertex subalgebras: G ± =: α ± ϕ ± : ±m∂ϕ ± , J =: ϕ + ϕ − : −m(α + + α − ), m ∈ C, 1 1 m L =: α + α − : + : ∂ϕ + ϕ − : + : ∂ϕ − ϕ + : − ∂(α + − α − ). 2 2 2

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The vector j = i J−1 |0 provides this vertex algebra with the structure of an N K = 2 (1) = G + + G − SUSY vertex algebra, by letting T = L −1 and S i = G (i) −1/2 , where G (2) + − and G = i(G − G ). Example 2.11. (W N series) Now we define an N W = N SUSY vertex algebra for each non-negative integer N . When N = 0, W0 is the Virasoro vertex algebra of central charge c. W1 is the N W = 1 SUSY vertex algebra generated1 by an odd superfield L and an even superfield G satisfying: [L  L] = (T + 2λ)L ,

[Q  Q] = S Q +

λχ c, 3

(2.1.9)

λ2 [L  Q] = (T + λ)Q − χ L + c, 6 where c ∈ C is the central charge. Expanding these superfields as:

1 + − L(Z ) = G (z) + θ L(z) + ∂z J (z) , Q(Z ) = −J (z) + θ G (z), 2

we find that the fields L , J, G ± generate an N = 2 vertex algebra of central charge c as in Example 2.8. W2 is the N W = 2 SUSY vertex algebra generated by an even superfield L and two odd superfields Q 1 and Q 2 satisfying: [L  L] = (T + 2λ)L , [Q 1  Q 2 ] = (S 1 + χ 1 )Q 2 − χ 2 Q 1 +

λ c, 6

[Q i  Q i ] = S i Q i , [L  Q i ] = (T + λ)Q i + χ i L .

(2.1.10)

Finally, for N ≥ 3 we let W N be the N W = N SUSY vertex algebra generated by a superfield L of parity N mod2 and N superfields Q i , i = 1, . . . , N of parity N + 1mod2, satisfying: [L  L] = (T + 2λ)L ,

[Q i  Q j ] = (S i + χ i )Q j − χ j Q i ,

[L  Q i ] = (T + λ)Q i + (−1) N χ i L .

(2.1.11)

It is proved in [HK07] that the Lie superalgebra W (1|N ) of derivations on C[Z , Z −1 ] acts on W N . We let W (1|N )− (resp. W (1|N )< ) be the Lie subalgebras of regular vector fields (resp. regular vector fields vanishing at the origin). Definition 2.12. An N W = N SUSY vertex algebra V is called conformal if there exists N + 1 vectors ν, τ 1 , . . . , τ N in V such that their associated superfields L(Z ) = Y (ν, Z ) and Q i (Z ) = Y (τ i , Z ) satisfy (2.1.11) for N ≥ 3, (2.1.10) for N = 2, (2.1.9) for N = 1 or (2.1.2) for N = 0, and moreover: i • ν(0|0) = T , τ(0|0) = Si , • The operator ν(1|0) acts diagonally with eigenvalues bounded below and with finite dimensional eigenspaces. 1 See [HK07] for the definition of generating fields of a SUSY vertex algebra.

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If moreover, the action of W (1|N )< on V can be exponentiated to the group of automorphisms of the 1|N dimensional superdisk D 1|N (see 2.2.6 for a definition), we will say that V is strongly conformal. This amounts to the following extra condition: N i • The operators ν(1|0) and i=1 σ (ei )τ(0|e have integer eigenvalues. i) Remark 2.13. The notion of strongly conformal N K = 1 SUSY vertex algebra is the same as the notion of N = 1 Neveu-Schwarz vertex operator superalgebra first given by Barron in [Bar96] and [Bar00]. Example 2.14 (K N series). For N ≤ 3, let K N be the N K = N SUSY vertex algebra generated by one superfield G of parity N mod2 satisfying:

 N
λ3−N χ N i i c, (2.1.12) χ S G+ [G  G] = 2T + (4 − N )λ + 3 i=1

where c ∈ C is called the central charge. Let K 4 be the N K = 4 SUSY vertex algebra generated by an even superfield G satisfying:

 4
S i χ i G + λc. [G  G] = 2T + (2.1.13) i=1

In the case N = 1, if we expand the corresponding superfield as G(z, θ ) = G(z) + 2θ L(z), we find that the fields G(z) and L(z) generate a Neveu-Schwarz vertex algebra of central charge c as in Example 2.4. When N = 2, expanding the corresponding superfield as: √ G(z, θ 1 , θ 2 ) = −1J (z) + θ 1 G (2) (z) − θ 2 G (1) (z) + 2θ 1 θ 2 L(z), where G (1) = G + + G − and G (2) = i(G + − G − ), we find that the corresponding fields J, L , G ± satisfy the commutation relations of the N = 2 vertex algebra as in Example 2.8. It was proved in [HK07] that the Lie superalgebra K (1|N ) of vector fields on the 1|N -dimensional superdisk D 1|N preserving the differential 1-form ω = dz +

N


θ i dθ i ,

i=1

up to multiplication by a function, acts on K N . We let K (1|N )− (resp. K (1|N )< ) be the Lie subalgebra of regular vector fields (resp. regular vector fields vanishing at the origin). Definition 2.15. Let N ≤ 4, an N K = N SUSY vertex algebra V is called conformal if there exists a vector τ ∈ V (called the conformal vector) such that the corresponding field G(Z ) = Y (τ, Z ) satisfies (2.1.12) for N ≤ 3 or (2.1.13) for N = 4, and moreover • τ(0|0) = 2T , τ(0|ei ) = σ (N \ ei , ei )S i ,

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• the operator τ(1|0) acts diagonally with eigenvalues bounded below and finite dimensional eigenspaces. If moreover, the representation of K (1|N )< can be exponentiated to the group of automorphisms of the disk D 1|N preserving the differential form ω up to multiplication by a function, we will say that V is strongly conformal. This amounts to the extra condition √ • the operator τ(1|0) has integer eigenvalues, and if N = 2, the operator −1τ(0|N ) has integer eigenvalues. Example 2.16. (Boson-fermion system) Let B1 be the strongly conformal N K = 1 SUSY vertex algebra generated by one odd superfield  satisfying: [ ] = χ . Expanding the corresponding superfield as: (Z ) = ϕ(z) + θ α(z), we find that the ordinary fields ϕ and α generate the well known Boson-fermion system as in Example 2.7. 2.1.3. Now we summarize some basic results in the structure theory of SUSY vertex algebras. The proofs can be found in [HK07]. We will denote ∇ = (T, S 1 , . . . , S N ), and for each ( j|J ) as above, we define: ∇ j|J = T j S J ,

∇ ( j|J ) =

J (J +1) 2

(−1) j!

∇ j|J .

Let Z = (z, θ 1 , . . . , θ N ) be as before, and W = (w, ζ 1 , . . . , ζ N ) be such that w is an even indeterminate, commuting with z, θ i ’s and ζ i ’s, and ζ i are odd anticommuting indeterminates, commuting with z, w and anticommuting with θ i ’s. In the N W = N case we will write: Z − W = (z − w, θ 1 − ζ 1 , . . . , θ N − ζ N ), and in the N K = N case:

Z −W = z−w−

N


(2.1.14)

 θ ζ ,θ − ζ ,...,θ i i

1

1

N

−ζ

N

.

(2.1.15)

i=1

We define the formal super delta-function to be:  δ(Z , W ) = i z,w − i w,z (Z − W )−1|N , where i z,w denotes the expansion in the domain |z| > |w|. We note that this definition is independent of the definition of Z − W as (2.1.14) or (2.1.15). Put Z ∇ = zT + i θ i S i . Similarly, in the N W = N (resp. N K = N ) case, we put DW = (∂w , ∂ζ i ) (resp. DW = (∂w , ∂ζ i + ζ i ∂w )).

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Proposition 2.17. Let V be a N W = N or N K = N SUSY vertex algebra. Then
(−1) J N +I N +I J σ (J )σ (I ) [a(n|I ) , Y (b, W )] = ( j|J ), j≥0

 ( j|J )  × DW W n|I Y a( j|J ) b, W . If, moreover, n ≥ 0, this becomes: [a(n|I ) , Y (b, W )] = Y (e−W ∇ a(n|I ) e W ∇ b, W ). Theorem 2.18. In an N W = N (resp. N K = N ) SUSY vertex algebra the following identities hold (see [HK07] for the definition of ( j|J )-th product of superfields): (1) Y (a( j|J ) b, Z ) = σ (J )Y (a, Z )( j|J ) Y (b, Z ) (the ( j|J )-th product identity), (2) Y (a(−1|N ) b, Z ) =: Y (a, Z )Y (b, Z ) :, (3) Y (T a, Z ) = ∂z Y (a, Z ), (4) Y (S i a, Z ) = ∂θ i Y (a, Z ) (resp. (∂θ i + θ i ∂z )Y (a, Z )), (5) we have the following OPE formula:
( j|J ) σ (J )(DW δ(Z , W ))Y (a( j|J ) b, W ) [Y (a, Z ), Y (b, W )] = ( j,J ), j≥0

=




(i z,w − i w,z )(Z − W )−1− j|N \J Y (a( j|J ) b, W ),

( j|J ), j≥0

(2.1.16) where the sum is finite and the operator i z,w denotes the expansion in the domain |z| > |w|. Remark 2.19. Property (4) of the theorem in the case N K = 1 is the G(−1/2)-derivative property first given by Barron as an axiom for N = 1 Neveu-Schwarz vertex operator superalgebras in [Bar96] and [Bar00]. Property (5) is equivalent to the commutator formula (31) in [Bar00]. Lemma 2.20. The following identity is true (note that DW and Z −W here have different meanings in the N W = N and N K = N case): ( j|J )

DW

δ(Z , W ) = σ (J, N \ J )(i z,w − i w,z )(Z − W )−1− j|N \J .

(2.1.17)

Theorem 2.21 (Skew-symmetry). In a SUSY vertex algebra the following identity, called skew-symmetry, holds Y (a, Z )b = (−1)ab e Z ∇ Y (b, −Z )a.

(2.1.18)

Remark 2.22. This theorem, in the N K = 1 case, was formulated in [Bar00, Eq. (54)] as “skew-supersymmetry”. Theorem 2.23 (Cousin property). For any SUSY vertex algebra V and vectors a, b, c ∈ V , the three expressions: Y (a, Z )Y (b, W )c ∈ V ((Z ))((W )), (−1) Y (b, W )Y (a, Z )c ∈ V ((W ))((Z )), Y (Y (a, Z − W )b, W ) c ∈ V ((W ))((Z − W )) ab

are the expansions, in the domains |z| > |w|, |w| > |z| and |w| > |w − z| respectively, of the same element of V [[Z , W ]][z −1 , w −1 , (z − w)−1 ]. Remark 2.24. This theorem, in the N K = 1 case, was first proved by Barron in [Bar00, §8].

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2.1.4. Let V be an N W = N (resp. N K = N ) SUSY vertex algebra. It was proved in [HK07] that Lie(V ) = V˜ /∇˜ V˜ is naturally a Lie algebra2 , where V˜ = V ⊗C C[Z , Z −1 ] and ∇˜ V˜ is the space spanned by vectors of the form: T a ⊗ f (Z ) + a ⊗ ∂z f (Z ), S i a ⊗ f (Z ) + (−1)a N a ⊗ ∂θ i f (Z ), (resp.S a ⊗ f (Z ) + (−1) i

aN

(2.1.19)

a ⊗ (∂θ i + θ ∂z ) f (Z )), i

for a ∈ V , f (Z ) ∈ C[Z , Z −1 ]. Let ϕ : Lie(V ) → End(V ) be the linear map defined by a = a ⊗ Z n|I → (−1)a I σ (I )a(n|I ) ,

a ∈ V.

(2.1.20)

Similarly, we construct V ⊗C C((Z )) and consider its quotient Lie (V ) by the vector space generated by vectors of the form (2.1.19). Then (2.1.20) defines a map ϕ  : Lie (V ) → End(V ). Theorem 2.25. The maps ϕ, and ϕ  are Lie algebra homomorphisms. 2.2. Supercurves 2.2.1. For a general introduction to the theory of supermanifolds and superschemes, the reader should refer to [Man97]. We will follow [BR99] for the theory of supercurves over a Grassmann algebra . The deformation theory of superspaces and sheaves over them can be found in [Vai90]. The relations between superconformal Lie algebras and the moduli spaces of supercurves was stated in [Vai95]. The reader may also find useful the notes [DM99]. Definition 2.26. A superspace is a locally ringed space (X, O X ) where X is a topological space and O X is a sheaf of supercommutative rings. A morphism of superspaces is a graded morphism of locally ringed spaces. We will use X to denote such a superspace when no confusion should arise. A superscheme is a superspace such that (X, O X,0¯ ) is ¯ 1. ¯ a scheme, where from now on O X,i denotes the ith graded part of O X , i = 0, 2 . J is a sheaf of ideals 2.2.2. Given a superspace (X, O X ) define J = O X,1¯ + O X, 1¯ in (X, O X ); the corresponding subspace (X, O X /J ) will be denoted (X rd , O X rd ).

Example 2.27. Let R be a supercommutative ring, and let J = R1 + R12 be the ideal generated by R1¯ as above, then (Spec R, R) is a superscheme. Note that as topological spaces Spec R = Spec R/J since every element in J is nilpotent (we consider only homogeneous ideals with respect to the Z/2Z-grading). Definition 2.28 (cf. [Man97]). A supermanifold is a superspace (X, O X ) such that for every point x ∈ X there exists an open neighborhood U of x and a locally free sheaf E of O X rd |U -modules, of (purely odd) rank 0|q such that (U, O X |U ) is isomorphic to (Urd , SO X rd (E )|U ). Here S(E ) denotes the symmetric algebra of a vector bundle. 2 Here we need to change the parity of V˜ /∇ ˜ V˜ if N is odd, see [HK07] for details.

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2.2.3. An open sub-supermanifold of (X, O X ) consists of an open subset U ⊂ X and the restriction of the structure sheaf, namely (U, O X |U ). 2.2.4. In the analytic setting, the situation is easier to describe. The supermanifold C p|q is the topological space C p endowed with the sheaf of supercommutative algebras O[θ1 , . . . , θq ], where O is the sheaf of germs of holomorphic functions on C p and θi are odd anticommuting variables. A complex supermanifold is a topological space |X | with a sheaf of supercommutative algebras O X locally isomorphic to C p|q . Morphisms of supermanifolds are continuous maps σ : |X | → |Y | together with morphisms of sheaves σ  : σ ∗ OY → O X . Let  = C[α1 , . . . , αn ] be a Grassmann algebra. The 0|n-dimensional superscheme Spec  has as underlying topological space a single point. We will work in the category of superschemes over , namely superschemes S together with a structure morphism S → Spec . In the case when S is a proper, smooth of relative dimension 1|q superscheme, we say that S is a N = q supercurve (over ). Definition 2.29. More explicitly (cf. [BR99]), a smooth compact connected complex supercurve over  of dimension 1|N is a pair (X, O X ), where X is a topological space and O X is a sheaf of supercommutative -algebras over X , such that: (1) (X, O Xred ) is a smooth compact connected algebraic curve. Here O Xred is the reduced sheaf of C-algebras on X obtained by quotienting out the nilpotents in O X . (2) For some open sets Uα ⊂ X and some linearly independent odd elements θαi of O X (Uα ) we have O X (Uα ) = O Xred ⊗ [θα1 , . . . , θαN ]. The Uα above are called coordinate neighborhoods of (X, O X ) and Z α = (z α , θα1 ,. . ., θαN ) are called local coordinates for (X, O X ) if z α (mod nilpotents) are local coordinates for (X, O Xred ). On overlaps Uα ∩ Uβ we have: i z β = Fβα (z α , θαj ), θβi = βα (z α , θαj ),

(2.2.1)

where Fβα are even and βα are odd. We will write such a change of coordinates as Z β = ρβ,α (Z α ) with ρ = (F,  i ), where no confusion should arise. 2.2.5. A -point of a supercurve (X, O X ) is a morphism ϕ : Spec  → (X, O X ) over , namely the composition of ϕ with the structure morphism (X, O X ) → Spec  is the identity. Locally, a  point is given by specifying the images of the local coordinates under the even -homomorphism ϕ  : O X (Uα ) → . These local parameters ( pα = ϕ  (z α ), παi = ϕ  (θαi )) transform as the coordinates do in (2.2.1). 2.2.6. The N = q formal superdisk is an ind-superscheme as in the non-super situation, namely, let R = C[t, θ 1 , . . . , θ q ] and let m be the maximal ideal generated by (t, θ 1 , . . . , θ q ). We define the superschemes D (n) = Spec R/mn+1 and we have embeddings D (n+1) → D (n) . The formal disk is then D = lim D (n) . − → n→∞ If we want to emphasize the dimensions of these disks we will denote them by D 1|q .

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2.2.7. Vector bundles of rank ( p|q) over a supermanifold (X, O X ) are locally free sheaves E of O X -modules over X , of rank p|q. That is, locally, E is isomorphic to p O X ⊕ (O X )q , where  is the parity change functor. An example is the tangent bundle to a p|q-dimensional supermanifold (X, O X ); it is a rank p|q vector bundle. Its fiber at the point x ∈ X is given as in the non-super case as the subset of morphisms in Hom(D (1) , X ) mapping the closed point in D (1) to x. The cotangent bundle 1X of (X, O X ) is the dual of the tangent bundle. Another example is the Berezinian bundle of a supermanifold (X, O X ). We will define this bundle by giving local trivializations. Recall [DM99, §1.10] that given a free module L of finite type over a supercommutative algebra A, the superdeterminant is a homomorphism sdet : GL(L) → GL(1|0) = A× 0, definedin coordinates as follows: for a parity preserving automorphism T of A p|q with K L matrix M N we put: sdet(T ) = det(K − L N −1 M) det(N )−1 . With this definition we can now define the Berezinian of the module L as the following A-module denoted Ber(L). Let {e1 , . . . , e p+q } be a basis of L where the first p elements are even and the last q are odd. This basis defines a one-element basis of Ber(L) denoted by [e1 . . . e p+q ] of parity qmod2. Given an automorphism T of L we put: [T e1 . . . T e p+q ] = sdet(T )[e1 . . . e p+q ]. This makes Ber(L) a well defined rank 1|0 A-module when q is even and a rank 0|1 A-module when q is odd. Now we can define the Berezinian bundle of (X, O X ) as Ber X = Ber(1X ). The definition of coherent and quasi-coherent sheaves is exactly the same as in the non-super case, in particular for super manifolds it follows that the structure sheaf is coherent [Vai90]. 2.2.8. Given an N = 1 supercurve (X, O X ) and an extension of O X by an invertible sheaf E : 0 → O X → Eˆ → E → 0, (2.2.2) we can construct an N = 2 supercurve (Y, OY ) canonically. Its local coordinates are given by (z α , θα , ρα ), where (z α , θα ) are local coordinates of X and ρα are local sections of E . In each coordinate patch Uα we can construct the form dz α − dθα ρα . We say that the N = 2 supercurve (Y, OY ) is superconformal if this form is globally defined up to multiplication by a function. This happens if on overlaps Uα ∩ Uβ we have (see (2.2.1)):

∂θ F ∂ F ∂z  ρα + . (2.2.3) ρβ = sdet z ∂θ F ∂θ  ∂θ  Here sdet is the superdeterminant of an automorphism defined above, which can be written as



DF ∂ F ∂z  sdet z =D , ∂θ F ∂θ  D where D = ∂θ + θ ∂z . Conversely, if (2.2.3) is satisfied on overlaps, the cocycle condition is satisfied and we have an extension as in (2.2.2).

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Therefore to each N = 1 supercurve (X, O X ), we canonically associate a N = 2 superconformal curve (Y, OY ). From (2.2.3) we see that we have an exact sequence of sheaves on Y : 0 → O X → OY → Ber X → 0, where Ber X is the Berezinian bundle on (X, O X ). The last map Dˆ : OY → Ber X is given in the above local coordinates, by the differential operator ∂ρα . Introducing new coordinates zˆ α = z α − θα ρα , θˆα = θα , ρˆα = ρα , we obtain on overlaps Uα ∩ Uβ : zˆ β = F(ˆz α , ρˆα ) +

D F(ˆz α , ρˆα ) D F(ˆz α , ρˆα ) (ˆz α , ρˆα ), ρˆβ = , D(ˆz α , ρˆα ) D(ˆz α , ρˆα )

(2.2.4)

where D = ∂θ + θ ∂z in local coordinates (z, θ, ρ) as above. We see from (2.2.4) that OY contains the structure sheaf of another N = 1 supercurve ( Xˆ , O Xˆ ), whose local coordinates are (ˆz α , ρˆα ). We call ( Xˆ , O Xˆ ) the dual curve of (X, O X ). Finally, we define an N = 1 superconformal curve as an N = 1 supercurve (X, O X ) which is self-dual. We see from (2.2.4) that the transition functions F,  must satisfy: D F =  D,

(2.2.5)

for (X, O X ) to be superconformal. In this case the operator Dα = ∂θα + θα ∂z α transforms as: (2.2.6) Dβ = (D)−1 Dα , hence in this situation the supercurve (X, O X ) carries a 0|1-dimensional distribution D such that D 2 is nowhere vanishing (since D 2 = ∂z in local coordinates). Remark 2.30. An equivalent definition of N = 1 and N = 2 superconformal curves was given by Manin [Man91] (under the name SUSY curves). Let X be a complex supermanifold of dimension 1|N (N = 1 or 2). When N = 1 we say that a locally free direct subsheaf T 1 ⊂ T X (T X is the tangent sheaf of X ) of rank 0|1 for which the Frobenius form (T 1 )⊗2 → T 0 := T X /T 1 , t1 ⊗ t2 → [t1 , t2 ] mod T 1 , is an isomorphism, is a SUSY structure on X . When N = 2, a SUSY structure consists of two locally free direct subsheaves T  , T  of T X of rank 0|1 whose sum in T X is direct; they are integrable distributions and the Frobenius form T  ⊗ T  → T X /(T  ⊕ T  ), t1 ⊗ t2 → [t1 , t2 ] mod (T  ⊕ T  ), is an isomorphism. Let (X, O X ) be an N = 1 supercurve and Dα be a family of vector fields in Uα , such that Dα and Dα2 form a basis for T X on Uα and Dα = G αβ Dβ on Uα ∩ Uβ , where G αβ is a family of invertible even functions. The sheaf defined by T 1 |Uα = O X Dα is a SUSY structure in (X, O X ) [Man91]. In local coordinates as above, the vector fields

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623

Dα = ∂θα + θα ∂z α satisfy these conditions when X is an N = 1 superconformal curve (see (2.2.6)).   The N = 2 case is similar. Let (X, O X ) be an N = 2 supercurve and Dα1 , Dα2 be a family of vector fields such that Dαi , [Dα1 , Dα2 ] generate T X in Uα and, moreover, we have: (Dα1 )2 = f α1 Dα1 , (Dα2 )2 = f α2 Dα2 , 1 Dβ1 , Dα1 = Fα,β

2 Dα2 = Fαβ Dβ2 on Uα ∩ Uβ ,

i are even functions. Putting T  | 1  2 where f αi and Fαβ Uα = O X Dα and T |Uα = O X Dα we  obtain an N = 2 superconformal structure on (X, O X ). If the two distributions T and T  can be distinguished globally, the N = 2 superconformal curve is called orientable and a choice of one of these distributions is called its orientation. It is clear that the construction given in 2.2.8 gives an oriented N = 2 superconformal curve; conversely, given such a curve, we can consider the functor X → X/T  (recall that T  is integrable, therefore this quotient makes sense). The duality that was explained in 2.2.8 corresponds to the duality X/T  ↔ X/T  .

2.2.9. Recall [BR99] that a -point of an N = 1 supercurve X transforms as an irreducible divisor of the dual curve Xˆ . Indeed, an irreducible divisor of X is given in local coordinates (z α , θα ) by expressions of the form Pα = z α − zˆ α − θα ρα . Two divisors Pα and Pβ are said to correspond to each other in the intersection Uα ∩ Uβ if in this intersection we have Pβ (z β , θβ ) = Pα (z α , θα )g(z α , θα ) for some even invertible function g(z α , θα ) (we consider Cartier divisors). It is easy to see that the parameters zˆ α , ρα transform as in (2.2.4), namely as the parameters of a -point of Xˆ . 2.2.10. We can define a theory of contour integration on an N = 1 superconformal curve as in [Fri86, McA88, Rog88]. We describe briefly a generalization to arbitrary N = 1 supercurves due to Bergvelt and Rabin (cf. [BR99]). For simplicity, we will work in the analytic category. Let us define a super contour to be a triple  = (γ , P, Q) consisting of an ordinary contour γ on the reduction |X | and two Cartier divisors as in 2.2.9 such that their reductions to |X | are the endpoints of γ . If in local coordinates P = z − pˆ − θ π, ˆ

Q = z − qˆ − θ ξˆ ,

then the corresponding -points of the dual curve Xˆ are given by ( p, ˆ πˆ ) and (q, ˆ ξˆ ). Let z = pˆ rd and z = qˆrd be the equations for the reductions of these points, i.e. the endpoints for γ . We define the integral of a section ωα = D fˆα of the Berezinian sheaf of X (recall that D : O Xˆ
Ber X ) along  by: 

Q P

 ω=

Q

D fˆ = fˆ(q, ˆ ξˆ ) − fˆ( p, ˆ πˆ ),

P

where we assume that the contour connecting P and Q lies in a single simply connected open set Uα . If the contour traverses several open sets then we need to choose intermediate divisors on each overlap and we have to prove that the resulting integral

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is independent of these divisors. In what follows we will only need the integration in a sufficiently “small” open set Uα (the formal disk around a point). Dually, we can integrate sections of Ber Xˆ along contours in X . Indeed, let γ be a path in the topological space |X | and two -points P, Q of X whose reduced parts are the end-points of γ . Let ωˆ ∈ Ber Xˆ (Uα ) and suppose that γ lies in a simply connected open Uα . Then ωˆ = Dˆ f for some function f ∈ O X (Uα ), and we put 

Q

ωˆ = f (Q) − f (P).

P

As it is shown in [BR99], this theory of integration can be understood in terms of a theory of contour integration on the corresponding N = 2 superconformal curve (cf. [Coh87]). For this let X and Xˆ be an N = 1 supercurve and its dual, and let Y be the corresponding N = 2 superconformal curve. We have two short exact sequences of sheaves in Y : D−

0 → O X → OY −−→ Ber Xˆ → 0, D+

0 → O Xˆ → OY −→ Ber X → 0. We can define a sheaf operator on OY⊕2 by the component-wise action of the differential operators (D − , D + ). It is shown in [BR99] that for U a simply connected open in |Y | = |X | and ( f, g) a section of OY⊕2 (U ) such that (D − , D + )( f, g) = 0, there exists a section H ∈ OY (U ), unique up to an additive constant, such that ( f, g) = (D − H, D + H ). Let M be the subsheaf of OY⊕2 consisting of closed sections ( f, g) as above. It follows that M = Ber X ⊕ Ber Xˆ . A super contour in Y consists of a triple (γ , P, Q), where P and Q are -points of Y such that their reduced points are the endpoints of γ . If γ is supported on a simply connected open set U , then any section ω ∈ M (U ) can be written as (D − H, D + H ) and we put: 

Q

ω = H (Q) − H (P).

P

The extension to contours not lying in a single simply connected U is straightforward but we will not need it. 2.2.11. We will define in general a superconformal N = n supercurve to be a supercurve such that in some coordinate system Z α = (z α , θαi ), the differential form ω = dz α +




θαi dθαi

(2.2.7)

i

is well defined up to multiplication by a function. It is easy to show that this definition agrees with the definition above in the N = 1 and N = 2 cases (cf. §3.1.3 and §3.1.4). A set of coordinates Z = (z, θ i ) such that the form ω has the form (2.2.7) (up to multiplication by a function) will be called SU SY coordinates (or coordinates compatible with the superconformal structure). Let (z, θ ) and (z  , θ  ) be two local coordinates compatible with a (local) superconformal structure on an N = 1 supercurve (X, O X ). Denote D = ∂θ + θ ∂z and

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D  = ∂θ  + θ  ∂z  . Let G be the invertible function such that D = G D  (cf. (2.2.6)). We define the Schwarzian derivative of (z  , θ  ) with respect to (z, θ ) to be the (odd) function DG D 2 G D3 G −2 . (2.2.8) G G2 Definition 2.31. A superprojective structure on an N = 1 superconformal curve over  is a (maximal) atlas consisting of coordinates (z α , θα ) compatible with the superconformal structure, and such that its transition functions are fractional linear transformations in PC(2|1) (see [Man97]), that is, changes of coordinates of the form: σ (G) =

z =

az + b + αθ , cz + d + βθ

θ =

γ z + δ + eθ , cz + d + βθ

for some even constants a, b, c, d and e ∈  and some odd constants α, β, γ and δ ∈ , such that ⎛ ⎞ a bα sdet ⎝ c d β ⎠ = 1, γ δ e and that the following equations hold: ad − bc − γ δ = 1, 

e2 + 2αβ = 1, γ e = aβ − cα, δe = bβ − dα.

Proposition 2.32 ([Man97] Proposition 4.7). Let (z, θ ) and (z  , θ  ) be two local coordinates on (X, O X ). The following statements are equivalent: (1) (z, θ ) and (z  , θ  ) are compatible with a common superconformal structure and σ = 0. (2) (z, θ ) and (z  , θ  ) define the same superprojective structure. 3. The Associated Vector Bundles 3.1. The groups AutO 3.1.1. We start this section by describing the groups of changes of coordinates in the formal superdisk D 1|N . We analyze in detail their corresponding Lie superalgebras in the cases N = 1 and N = 2. We then define principal bundles for these groups over any smooth supercurve. In this section, we let  be a Grassmann, algebra over C. We will work in the category of superschemes over  unless explicitly stated. When we work with a supergroup G, we will be interested in its -points. Let SSch/k be the category of superschemes over a field k and let Set be the category of sets. Fix a non-negative integer N and a separated superscheme X of finite type over k (cf. 2.26). Let D (m) be as in 2.2.6 and D 1|N be the formal superdisk. Define a family of contravariant functors Fm : SSch/k → Set, Fm (Y ) = Homk (Y ×k D (m) , X ). The proof of the following proposition is standard:

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Proposition 3.1. The functors Fm are representable by superschemes X m . Note in particular that X 0 = X , and when N = 1 we see that X 1 is the total tangent space of X . The embeddings D (m) → D (m+1) induce projections X m+1 → X m and we define the Jet superscheme of X as J X = lim X m . ← − m→∞ 3.1.2. Let us analyze first the case N = 1. Consider the group of continuous (even) automorphisms of the topological commutative superalgebra [[Z ]], where Z = (z, θ ) are topological generators. Such an automorphism is given by a pair of power series z → a1,0 z + a0,1 θ + a1,1 zθ + . . . , θ → b1,0 z + b0,1 θ + b1,1 zθ + . . . ,

 a0,1 is in G L(1|1)3 . Denote this supergroup by AutO 1|1 . In what where the matrix ab1,0 1,0 b0,1 follows we will analyze its C-points. This supergroup is a semidirect product of G L(1|1) and a pro-unipotent super group, a0,1 namely, the subgroup Aut + O 1|1 of automorphisms, where ab1,0 = Id. In fact, 1,0 b0,1 Aut+ O 1|1 = lim Spec C[a1,1 , b1,1 , a2,0 , b2,0 , . . . , an,1 , bn,1 ]. ← − n→∞

Let m be the maximal ideal of C[Z ] generated by (z, θ ). We have Aut + O 1|1 = lim Aut(C[Z ]/mn ). ← − n→∞

Similarly for its Lie superalgebra Der + O 1|1 , we have Der + O 1|1 = lim Der(C[Z ]/mn ), ← − n→∞ where for each C-superalgebra R, we denote Der(R) the Lie superalgebra of derivations of R. The exponential map is an isomorphism at each step, giving an isomorphism exp : Der + O 1|1 → Aut + O 1|1 . The linearly compact Lie superalgebra Der 0 O 1|1 = Lie(AutO 1|1 ) has the following topological basis: z n ∂z (n ≥ 1), z θ ∂z (n ≥ 0), n

z n ∂θ (n ≥ 1), z n θ ∂θ (n ≥ 0),

or the following one (n ≥ 0): Tn = −z n+1 ∂z − (n + 1)z n θ ∂θ ,

Jn = −z n θ ∂θ ,

Q n = −z n+1 ∂θ ,

Hn = z n θ ∂z .

(3.1.1)

These elements satisfy the commutation relations of the N = 2 algebra (2.1.7) for n ≥ 0. In particular, we see that Der 0 O 1|1 is the formal completion of the Lie algebra W (1|1)< . The Lie subalgebra Der + O is topologicaly generated by the same vectors with n ≥ 1. 3 Here and further, G L( p|q) is the group of even automorphisms of a p|q dimensional module over .

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3.1.3. We now turn our attention to the superconformal N = 1 case (cf. [Bar04]). Consider the differential form ω = dz + θ dθ on the formal superdisk D 1|1 , and the supergroup Aut ω O 1|1 of automorphisms of D 1|1 preserving this form, up to multiplication by a function. This is a subgroup of AutO 1|1 whose Lie superalgebra Der ω0 O 1|1 consists of derivations X in Der 0 O 1|1 such that L X ω = f ω for some formal power series f (here L X denotes the Lie derivative). More explicitly, the linearly compact Lie superalgebra Der ω0 O 1|1 is topologically generated by Ln = −

n+1 n z θ ∂θ − z n+1 ∂z , 2

n ∈ Z+ ,

(3.1.2) 1 G n = −z (∂θ − θ ∂z ), n ∈ + Z+ . 2 These generators satisfy the commutation relation of the Neveu-Schwarz algebra as defined in (2.1.3). In particular, we see that Der ω0 O 1|1 is the formal completion of the Lie superalgebra K (1|1)< . An automorphism of the formal superdisk is determined by two power series F(Z ), (Z ) which are the images of the generators Z = (z, θ ). Under this transformation we have (recall ∂θ is an odd derivation) n+1/2

dz + θ dθ → ∂z Fdz − ∂θ Fdθ + (∂z dz + ∂θ dθ ) = (∂z F + ∂z )dz − (∂θ F − ∂θ )dθ, therefore we get that, in order for ω to be preserved up to multiplication by a function, we need (∂θ F − ∂θ ) = −θ (∂z F + ∂z ), and this is equivalent to (2.2.5). 3.1.4. Finally we turn our attention to the (oriented) superconformal N = 2 case. For this we consider the differential form ω = dz + θ 1 dθ 1 + θ 2 dθ 2 on the formal superdisk D 1|2 . We want to analyze the group of automorphisms of D 1|2 preserving this form in the sense of the previous paragraph 3.1.3. Such an automorphism is determined by an even power series F(Z ) and two odd power series  1 (Z ) and  2 (Z ), where Z = (z, θ 1 , θ 2 ) are the coordinates on D 1|2 . Under such a change of coordinates, the differential form ω changes to:   ∂z Fdz − ∂θ 1 Fdθ 1 − ∂θ 2 Fdθ 2 +  1 ∂z  1 dz + ∂θ 1  1 dθ 1 + ∂θ 2  1 dθ 2   + 2 ∂z  2 dz + ∂θ 1  2 dθ 1 + ∂θ 2  2 dθ 2     = ∂z F +  1 ∂z  1 +  2 ∂z  2 dz + −∂θ 1 F +  1 ∂θ 1  1 +  2 ∂θ 1  2 dθ 1   + −∂θ 2 F +  1 ∂θ 2  1 +  2 ∂θ 2  2 dθ 2 . (3.1.3) Collecting terms, imposing that the form ω is preserved up to multiplication by a function, and defining the differential operators D i = ∂θ i +θ i ∂z , we obtain that the automorphisms we are considering satisfy the equations: D i F =  1 D i  1 +  2 D i  2 , i = 1, 2.

(3.1.4)

Note also that a particular case of (3.1.3) when F = z − 21 θ 1 θ 2 ,  1 = 2i (θ 2 − θ 1 ) and  2 = 21 (θ 1 + θ 2 ) transforms the form ω → dz + θ 2 dθ 1 = dz − dθ 1 θ 2 ,

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and the supergroup of automorphisms of D 1|2 preserving the latter form is the supergroup of changes of coordinates preserving an N = 2 superconformal structure as in 2.2.8. The linearly compact Lie superalgebra Der ω0 O 1|2 = Lie(Autω O 1|2 ) is topologically generated by:  n+1 n 1 L n = −z n+1 ∂z − z θ ∂θ 1 + θ 2 ∂θ 2 , n ∈ Z+ , 2

  1 n−1/2 1 2 1 (2) n+1/2 2 z θ ∂z − ∂θ 2 − n + θ θ ∂θ 1 , n ∈ + Z+ , G n = +z 2 2 (3.1.5)

  1 1 (1) n+1/2 1 n−1/2 1 2 z θ ∂z − ∂θ 1 + n + θ θ ∂θ 2 , n ∈ + Z+ , G n = +z 2 2   n 2 1 Jn = −i z θ ∂θ 1 − θ ∂θ 2 n ∈ Z+ . These operators satisfy the commutation relations of the N = 2 generators as in (2.1.8) for n ≥ 0. We see that the Lie superalgebra Der ω0 O 1|2 is the formal completion of the Lie superalgebra K (1|2)< . It is useful to consider complex coordinates θ ± = θ 1 ± iθ 2 , and derivations D ± = 1 1 ± i D 2 ). In the coordinates (z, θ + , θ − ), these derivations are expressed as: (D 2 1 D ± = ∂θ ∓ + θ ± ∂z . 2 If we change coordinates by ρ = (F,  + ,  − ), with  ± =  1 ± i 2 , the superconformal condition (3.1.4) reads D± F =

1 + ± − 1 − ± +  D  +  D  . 2 2

(3.1.6)

Therefore, under a change of coordinates (z α , θα± ) → (z β , θβ± ), the operators D ± transform as − + )Dβ+ + (Dα± β,α )Dβ− . (3.1.7) Dα± = (Dα± β,α In the following sections, we will consider only oriented superconformal N = 2 supercurves (cf. Remark 2.30), namely those for which there exists a coordinate atlas (Uα , z α , θα± ) such that on overlaps we have [Coh87]: ± Dα± β,α = 0.

(3.1.8)

In these coordinates, the topological generators of the Lie superalgebra Der ω0 O 1|2 are expressed as: n+1 n + z (θ ∂θ + + θ − ∂θ − ), n ∈ Z+ , 2 Jn = −z n (θ + ∂θ + − θ − ∂θ − ), n ∈ Z+ , (3.1.9)

n + 1/2 1 1 n+1/2 z n−1/2 θ ± θ ∓ ∂θ ± , n ∈ + Z+ , ∂θ ± − θ ∓ ∂ z − G± n = −z 2 2 2 L n = −z n+1 ∂z −

where as before we have G ± = 21 (G (1) ∓ i G (2) ).

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Recall from 2.2.8 that an oriented superconformal N = 2 supercurve (Y, OY ) projects onto two N = 1 supercurves X and its dual Xˆ . Defining new coordinates (u, θ + , θ − ), where u = z − 21 θ + θ − , we see that Eqs. (3.1.6), for a change of coordinates ρ = (G = F + 21  +  − ,  + ,  − ) are expressed in these coordinates as: D− G =  − D− +, Moreover, the operators

D±

D + G = 0.

(3.1.10)

D − = ∂θ + + θ − ∂u .

(3.1.11)

are expressed as:

D = ∂θ − , +

Note that the coordinate θ − does not appear in the transition functions for u, θ + , therefore

these coordinates give the topological space |Y | the structure of an N = 1 supercurve. Let us call this curve X . Similarly, if we define u  = z − 21 θ + θ − we obtain that u  , θ − defines the dual curve ( Xˆ , O Xˆ ). It follows from the above discussion that given a change of coordinates ρ = (G,  + ) ∈ AutO 1|1 , we obtain uniquely a change of coordinates ρ = (G,  + ,  − ) ∈ Aut ω O 1|2 , where  − = D − G/D −  + . This map induces an isomorphism of supergroups from AutO 1|1 to the identity component of Aut ω O 1|2 . This isomorphism corresponds to the isomorphism of Lie superalgebras K (1|2) ≡ W (1|1) (cf. [KvdL89]), and has a geometric counterpart (cf. [Vai95]) relating the moduli space of (oriented) superconformal N = 2 supercurves and the moduli space of N = 1 supercurves. Remark 3.2. Let X be a superconformal N = n supercurve. Then for some coordinate n atlas Z α = (z α , θα1 , . . . , θαn ), the form ω = dz + i=1 θ i dθ i is globally defined up to multiplication by a function. Let ω be that form on the superdisk D 1|N and on D (m) as well. Define the functors Fmω : SSch/k → Set by: Fm (Y ) = Homωk (Y ×k D (m) , X ), where Homω denotes the set of morphisms preserving the form ω (up to multiplication by a function). It follows in the same way as in Proposition 3.1 that the functors Fm are ω . This allows us to define the superscheme representable by superschemes X m ω , J X ω = lim X m ← − m→∞

parametrizing maps D → X preserving the superconformal structure. 3.1.5. Let X be an N = n supercurve and let x ∈ X . If Z = (z, θ 1 , . . . , θ n ) are local coordinates at x and Ox denotes the completion of the local ring at x, we have an isomorphism Ox ≡ C[[Z ]], where we should replace C by  if X is defined over . For the purposes of this section it is enough to consider curves over C, the relative case follows easily. Let Aut x denote the set of local coordinates Z = (z, θ i ) at x. In the algebraic setting we mean by coordinates an étale map Z : X → A1|n . The set Aut x is a torsor for the group AutO 1|n . The torsors Aut x glue to form an AutO 1|n -torsor Aut X . Indeed AutX consists of pairs (x, Z ), where x is a point in X and Z = (z, θ i ) is a local coordinate at x. The action of AutO 1|n on the fibers is by changes of coordinates. The torsor Aut X may be described as an open subscheme of J X consisting of jets of maps D 1|n → X such that their 1-jet is in G L(1|n). Since we can cover X by Zariski open subschemes Uα and étale maps f α : Uα → A1|n we see that the AutO 1|n -torsor Aut X is locally trivial in the Zariski topology (cf. [FBZ01, §5.4.2]).

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3.1.6. Similarly, let X be a (oriented) superconformal N = n supercurve and x ∈ X . Let Aut ωx be the set of SUSY coordinates Z at x (that is, compatible with the superconformal structure). It follows that this set is an Aut ω O 1|n -torsor. Moreover these torsors glue to form an Aut ω O 1|n -torsor Aut ωX → X . As in the previous paragraph, Aut ωX is an open sub superscheme of J X ω (cf. 3.2) consisting of jets of maps D 1|n → X compatible with the superconformal structure and with the invertible 1-jet. Remark 3.3. Let V be a finite rank AutO-module (resp. a finite rank Aut ω O-module), and let X be an N = n supercurve (resp. a superconformal N = n supercurve). We define a vector bundle on X by AutO

V X = Aut X × V

(resp. Aut ωX

Autω O

× V ),

consisting of pairs (x, ˜ v) with x˜ in Aut X (resp. Aut ωX ) and v ∈ V with the identification (x˜ · g, v) ∼ (x, ˜ g · v) for g ∈ AutO (resp. g ∈ Autω O ). We call V X the Aut X (resp. ω Aut X ) twist of V . 3.2. Vector bundles, sections and connections 3.2.1. In this section we construct vector bundles on supercurves associated with SUSY vertex algebras. Briefly, a strongly conformal N W = n SUSY vertex algebra is a module for the Harish-Chandra pair (DerO 1|n , AutO 1|n ), therefore we can apply the BeilinsonBernstein localization construction [BB93] to get a vector bundle with a flat connection over any N = n supercurve. Similarly, a strongly conformal N K = n SUSY vertex algebra (n ≤ 4), is a module for the Harish-Chandra pair (Der ω O 1|n , Aut ω O 1|n )4 , therefore we can construct vector bundles with flat connections over any oriented superconformal N = n curve. It turns out that the state-field correspondence in all these cases can be seen as a (local) section of the corresponding bundles. The corresponding change of coordinates formula (a generalization of Huang’s formula [Hua97] in the non-super case and Barron’s change of coordinates [Bar04] in the N K = 1 case) is proved in this section. 3.2.2. Let V be a strongly conformal N W = n SUSY vertex algebra. Therefore we have j N +1 vectors ν and τ 1 , . . . , τ N such that their Fourier modes ν(m,I ) and τ(m,I ) with m ≥ 0 generate a Lie superalgebra isomorphic to DerO 1|n . The derivation ∂z (corresponding to ν(0,0) ) cannot be exponentiated to the group AutO 1|n and the Lie superalgebra spanned j by ν(m,I ) , and τ(m,I ) for m ≥ 1 if I = 0 is isomorphic to Der 0 O 1|n . In order to exponentiate the representation V of Der 0 O 1|n to a representation of the group AutO 1|n we note as before that this Lie algebra is a semidirect product of gl(1|n) with the pro-nilpotent Lie subalgebra Der + O 1|n . Namely, the subalgebra spanned by z∂z , θ i ∂θ j , z∂θ i and θ j ∂z is isomorphic to gl(1|n). It follows from the definition of strongly conformal N W = N SUSY vertex algebras in 2.12, that we can exponentiate this representation of gl(1|n) (the fact that the nilpotent part of the Lie superalgebra exponentiates follows easily from the OPE formula and the locality axiom). 4 From now on, we will abuse notation and denote by Aut ω O 1|n its identity component.

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3.2.3. Let X be an N = n supercurve over a Grassmann algebra , let x ∈ X and Ox be the completion of the local super-ring at x. Let Z = (z, θ i ) be local coordinates at x (recall that in the formal setting, Z is an étale map X → A1|n ). With such a choice of coordinates we get an isomorphism Ox ≡ [[Z ]], and the set of coordinates at x, Aut x , is an AutO 1|n -torsor. Let us work in the analytic setting first for the sake of simplicity. Let Dx be a small disk around x. Let p be a -point given in the local coordinates Z = (z, θ i ) by Y = (y, α i ). The coordinates Z induce coordinates Z − Y = (z − y, θ i − α i ) at p. Now let ρ ∈ AutO 1|n be a change of coordinates. Recall that this change of coordinates is given by power series (F(Z ),  i (Z )), where F(Z ) ∈ [[Z ]] is even and  i ∈ [[Z ]] are odd. This change of coordinates induce new coordinates at p, given by: ρ(Z ) − ρ(Y ) = (F(Z ) − F(Y ),  i (Z ) −  i (Y )).

(3.2.1)

The coordinates Z − Y = (z − y, θ i − α i ) and (3.2.1) at p are related by a change of coordinates ρY = (FY , Yi ) satisfying: ρY (Z − Y ) = ρ(Z ) − ρ(Y ). Therefore, letting W = (w, ζ i ) = Z − Y , we get: ρY (W ) = ρ(W + Y ) − ρ(Y ).

(3.2.2)

In the formal setting we can not consider a small disk, but given a point x and coordinates Z at x, we can still define ρ Z ∈ AutO 1|n for any ρ ∈ AutO 1|n by formula (3.2.2) with Y replaced by Z . Let V be a strongly conformal N W = n SUSY vertex algebra, so that V is an AutO 1|n -module. We will call this representation R. Theorem 3.4. Let V be a strongly conformal N W = n SUSY vertex algebra. Let ρ = (F,  j ) ∈ AutO 1|n and a ∈ V . The following change of coordinates formula is true:   (3.2.3) Y (a, Z ) = R(ρ)Y R(ρ Z )−1 a, ρ(Z ) R(ρ)−1 , where by ρ(Z ) we understand the images of z, θ j under ρ, namely F(z, θ i ),  j (z, θ i ). Proof. The proof is similar to the analogous formula in the ordinary vertex algebra case. Namely, the state-field correspondence Y (·, Z ) is an element in the vector space Hom(V, F (V )), where F (V ) is the space of all End(V )-valued superfields. For each ρ ∈ AutO 1|n consider the linear operator in Hom(V, F (V )) given by (Tρ X )(a, Z ) = R(ρ)X (R(ρ Z )−1 a, ρ(Z ))R(ρ)−1 . It is easy to check that Tρ X ∈ Hom(V, F (V )). Moreover, this action defines a representation of AutO 1|n in Hom(V, F (V )). Recall that the group structure in AutO 1|n is given by composition, namely, if ρ = (F,  j ) and τ = (G,  j ) then ρ  τ is given by H,  j , where H (z, θ j ) = G(F(z, θ j )),  k (z, θ j ),

 i (z, θ j ) = i (F(z, θ j ),  k (z, θ j )).

It follows that ρ Z  τρ(Z ) = (ρ  τ ) Z . Indeed, the LHS, when evaluated in W is given by τρ(Z ) (ρ(W + Z ) − ρ(Z )) = τ (ρ(W + Z ) − ρ(Z ) + ρ(Z )) − τ (ρ(Z )), which is the RHS.

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It follows from this formula that ρ → Tρ defines a representation of AutO 1|n . In fact, we have: (Tρτ X )(a, Z ) = R(ρ  τ )X (R((ρ  τ ) Z )−1 a, τ (ρ(Z )))R(ρ  τ )−1 = R(ρ)R(τ )X (R(ρ Z  τρ(Z ) )−1 a, τ (ρ(Z )))R(τ )−1 R(ρ)−1   = R(ρ) R(τ )X (R(τρ(Z ) )−1 R(ρ Z )−1 a, τ (ρ(Z )))R(τ )−1 R(ρ)−1  = Tρ (Tτ X )](a, Z ). We have reduced the proof of the theorem to show that Y (·, Z ) is fixed under this action. Since the exponential map exp : Der 0 O 1|n → AutO 1|n is surjective, we need only to show that Y (·, Z ) is stable under the induced infinitesimal action of Der 0 O 1|n . For this we let ρ = exp(εv), where v = v(Z )∂ Z ∈ Der 0 O 1|n , v(Z ) = ( f (Z ), g 1 (Z ), . . . , g n (Z )) with f (Z ) an even function and g i (Z ) odd functions of Z . As before, ∂ Z = (∂z , ∂θ 1 , . . . , n ∂θ n ) and the product v(Z )∂ Z denotes the scalar product f (Z )∂z + i=1 g i (Z )∂θ i . We want to compute ρ Z . For this we put ρ Z = exp(εu). Expanding ρ Z (W ) in powers of ε, we get   u = v(Z + W )∂W − v(Z )∂W = e Z ∂W v(W ) ∂W − v(Z )∂W . (3.2.4) Noting that the operators corresponding to ∂W = (∂w , ∂ζ 1 , . . . , ∂ζ n ) are −∇ = (−T, −S 1 , . . . , −S n ), we obtain: R(u) = e−Z ∇ R(v)e Z ∇ + v(Z )∇.

(3.2.5)

The (infinitesimal) action of Tρ on Y (a, Z ) is given by Y (a, Z ) plus the linear term in ε, which in turn is: [R(v), Y (a, Z )] − Y (R(u)a, Z ) + v(Z )∇ Z Y (a, Z ). The first term comes from the adjoint action of R(ρ), the second term is the ε-linear term in R(ρ Z )−1 , and the last term comes from the Taylor expansion of the change of coordinates. The result follows from (3.2.4), (3.2.5), Proposition 2.17 and Theorem 2.18.   3.2.4. Now we can define a vector bundle associated to an N W = n SUSY vertex algebra over any N = n supercurve. Moreover, we will define a canonical section of this bundle and a flat connection on it. First recall that from any finite dimensional AutO 1|n module we can construct a vector bundle over an N = n supercurve X by twisting this AutO 1|n -module by the AutO 1|n -torsor Aut X (see Remark 3.3). Given a strongly conformal N W = n SUSY vertex algebra V , we have a filtration V≤i by finite dimensional submodules, namely, V≤i is the span of fields of conformal weight less than or equal to i. By our assumptions, these are finite dimensional AutO-submodules of V . Let V≤i be the corresponding Aut X twist. These vector bundles come equipped with embeddings V≤i → V≤i+1 . The limit of this directed system is a O X -module V X 5 : V X = lim V≤i . − → i→∞

This O X -module is quasi-coherent by definition. 5 When there is no possible confusion, we will denote this bundle simply by V .

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∗ come equipped with surjections V ∗ On the other hand, the dual modules V≤i ≤i+1
∗ ∗ . The inverse therefore we get a projective system of O X -modules V≤i+1
V≤i limit of this system is by definition V X∗ , namely: ∗ , V≤i

∗ V X∗ = lim V≤i . ← − i→∞

Thus, we have defined O X -modules associated with the SUSY vertex algebra V . We will call these modules the SUSY vertex algebra bundle and its dual. By construction, the fiber of the bundle V at a point x ∈ X is isomorphic as a vector space, to V . Similar constructions can be applied when X is replaced by a formal superdisk near a point x ∈ X . Namely, let Dx be such a formal superdisk; we have as before an AutO 1|n torsor Aut Dx over Dx . Then V Dx is the twist of V by this torsor. It is easy to see that in this case we get V X | Dx = V Dx . Let Aut x be the torsor of coordinates at x as before. Then the fiber of V at x is given by: AutO

Vx = Aut x × V. Let Dx× be the punctured disk at x, that is the formal completion: Dx× = lim Spec(K˜x /mi+1 ), − → i→∞

where K˜x is the ring of fractions of the local ring at x and m is the maximal ideal defining x. If Z = (z, θ i ) are coordinates at x, this is isomorphic to the formal spectrum of ((Z )). We will define an End Vx -valued section of V ∗ on Dx× . In order to define such a section it is enough to give its matrix coefficients, namely, for each ϕ ∈ Vx∗ , v ∈ Vx and s a section of V | Dx we assign a function on Dx× , that is an element of Kx , the ring of fractions of Ox . This assignment is denoted by: ϕ, v, s →< ϕ, Yx (s) · v >, and should be linear in v and ϕ and Ox linear in s. Let Z = (z, θ i ) be coordinates at ∼ x; we obtain a trivialization i Z : V [[Z ]] → (Dx , V ), of V | Dx . This induces isomor∼ ∼ phisms V → Vx and V ∗ → Vx∗ , where V ∗ is the restricted dual of V . Let v ∈ V and ∗ ϕ ∈ V . Denote their images in Vx and Vx∗ , under these isomorphisms by (Z , v) and (Z , ϕ) respectively. Let s ∈ V [[Z ]]; its image under the isomorphism i Z is a regular section of V in Dx . By Ox linearity, we may assume that s = a ∈ V . To this data, we assign the function: < (Z , ϕ), Yx (i Z (a)) · (Z , v) >=< ϕ, Y (a, Z )v > .

(3.2.6)

Theorem 3.5. The assignment (3.2.6) is independent of the coordinates Z = (z, θ 1 , . . . , θ n ) chosen, i.e. Yx is a well defined End(Vx )-valued section of V ∗ on Dx× . Proof. The proof follows the lines of the ordinary vertex algebra case in [FBZ01]. Let W = (w, ζ i ) be another set of coordinates at x. Then W and Z are related by ρ ∈ AutO, ρ(Z ) = W . Given these new coordinates, we construct another assignment by the same formula (3.2.6), namely < (W, ϕ), Y˜ (i W (a)) · (W, v) >=< ϕ, Y (a, W )v > .

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We need to show that this assignment coincides with Y . By the definition of the bundle V we have (Z , v) = (ρ −1 (W ), v) = (W, R(ρ)−1 v), where R(·) is the representation of AutO 1|n in V . Similarly (Z , ϕ) = (W, ϕ R(ρ)). We need to find how the section i Z (a) transforms by this change of coordinates. Recall from 3.2.3 that in the analytic setting, if we trivialize V | Dx with the coordinates Z , we can use the coordinates (Z − Y ) := (z − y, θ i − α i ) at Y = (y, α i ) to identify V y with V . We obtain: (Z − Y, a) = (W − ρ(Y ), R(ρY )−1 a), (3.2.7) therefore the section i Z (a) is i W (R(ρ Z )−1 a) in the W -trivialization. In the formal setting, we can replace the coordinates by their n-jets, but these in turn can be extended by definition to a small Zariski open neighborhood of x, in this case, the formula (3.2.7) is true as we have shown. We have reduced the problem to prove: < ϕ, R(ρ)Y (R(ρ Z )−1 a, W )R(ρ)−1 v >=< ϕ, Y (a, Z )v >, thus, the theorem follows from Theorem 3.4.

 

3.2.5. In the superconformal case, the situation is slightly more complicated. Roughly, the only changes that we have to make in the above prescription are the induced coordinates at a -point and consequently the definition of ρ Z . Like in the N K = n SUSY vertex algebra situation, given two set of coordinates Z = (z, θ 1 , . . . , θ n ) and W = (w, ζ 1 , . . . , ζ n ) we will write

 n
i i 1 1 n n Z −W = z−w− θ ζ ,θ − ζ ,...,θ − ζ . i=1

Let V be a strongly conformal N K = n SUSY vertex algebra (n ≤ 4), hence V is an Aut ω O 1|n -module. Moreover, V has a filtration by finite dimensional submodules V≤i given by conformal weight as above. Let X be an oriented superconformal N = n supercurve over . We constructed an Aut ω O-torsor Aut ωX over X (see 3.1.6). As above we can define the vertex algebra bundles V and V ∗ . Similarly, we can define the N K = n SUSY vertex algebra bundles over the superconformal disks Dxω . The fibers Vx of these bundles are the Aut ωx -twists of V , where Aut ωx is the torsor of coordinates at x, compatible with the superconformal structure (see Remark 3.3). We define an End(Vx )-valued section Yx of V ∗ on the punctured disk Dx× by formula (3.2.6). Theorem 3.6. The assignment Yx is independent of the coordinates Z = (z, θ i ) chosen as long as they are compatible with the superconformal structure on X . Proof. Let us first work in the analytic setting. If p is a -point in Dx (now a small analytic disk near x ∈ X ) given by local parameters Y = (y, α i ), then Z induces local coordinates at T = (t, ηi ) = Z − Y near p. The coordinates T are compatible with the superconformal structure. Indeed, we have: dt = dz +

n
i=1

α i dθ i ,

dηi = dθ i ,

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therefore: dt +

n


ηi dηi = dz +

i=1

n


α i dθ i + (θ i + α i )dθ i = dz +

n


i=1

θ i dθ i .

i=1

If W = (w, ζ i ) = ρ(Z ) is another set of coordinates compatible with the superconformal structure at x, with ρ = (F,  i ) ∈ Aut ω O 1|n , then W induces another set of coordinates at p, namely

 n
ρ(Z ) − ρ(Y ) = F(z, θ ) − F(y, α) −  i (z, θ ) i (y, α),  j (z, θ ) −  j (y, α) . i=1

These are related with the coordinates T by a change of coordinates ρY = (FY , Yi ) ∈ Aut ω O 1|n . We have: ρY (T ) = ρ(T + Y ) − ρ(Y ), where, as in the N K = n SUSY vertex algebra case, we write T + Y = T − (−Y ). The theorem will follow if we prove formula (3.2.3) for ρ ∈ Autω O 1|n . This is achieved as in the proof of Theorem 3.4 by first showing that the action Tρ is a representation of Autω O 1|n in Hom(V, F (V )). For this we first note that (ρ  τ ) Z = ρ Z  τρ(Z ) in exactly the same way as in the N W = n case. Again we just have to prove that Y is fixed under this action, and we check this at the level of Lie alge1 , . . . , D n ), where D i = ∂ + ζ i ∂ . Similarly, denote bras. Denote DW = (∂w , DW w ζi W W n i 1 i ¯ ¯ ¯ ¯ DW = (∂w , DW , . . . , DW ), where DW = ∂ζ i − ζ ∂w . Let ρ = exp(εv), where v = v(W ) D¯ W ∈ Der ω0 O 1|n , and put ρ Z = exp(εu). Expanding ρ Z in powers of ε we find: u = v(W + Z ) D¯ W − v(Z ) D¯ W . Note that in this context we have two different Taylor expansions6 : ¯

e Z DW f (W ) = f (Z + W ), e Z DW f (W ) = f (W + Z ); using the second, we see that   ¯ u = e Z DW v(W ) D¯ W − v(Z ) D¯ W . From this and the fact that the operators corresponding to D¯ W are: −∇ = (−T, −S 1 , . . . , −S n ), we obtain:

R(u) = e−Z ∇ R(v)e Z ∇ + v(Z )∇.

The theorem now follows as in the N W = n case. 6 Note that Z + W  = W + Z .

 

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Remark 3.7. The N K = 1 version of Theorem 3.4 was first proved in [Bar04]. The approach we present here follows closely [FBZ01]. The formula in [Bar04] is stronger as the author deals with actual changes of coordinates on the annulus (not just the formal disk). The subtleties involved are treated carefully there. To construct the vector bundles we are interested in, we only need to generalize this weaker version to the N K = N case. Now we construct connections on the vector bundles V from the previous paragraphs. Theorem 3.8. Let X be a (1|N ) dimensional supercurve. Let U ⊂ X be open and Z be coordinates in U defining the vector fields ∂z and ∂θ i . Let V be a strongly conformal N W = N SUSY vertex algebra and V the associated bundle. Define the connection operators ∇χ : V|U → V|U for each vector field χ in U by ∇∂z = ∂z + T,

∇∂θ i = ∂θ i + S i .

Then ∇ is a well defined (left) connection on V (independent of the coordinates chosen). Moreover, this connection is flat. Proof. The proof is verbatim the proof of the analogous statement in [FBZ01, 16.1]. Indeed, strongly conformal SUSY vertex algebras are modules for the Harish Chandra pair (DerO 1|N , AutO 1|N ) and this in turn acts simply transitively on the torsor Aut X → X . The localization procedure of formal geometry applies without difficulties.   Remark 3.9. Note that this connection endows V with a structure of a left D X -module for any supercurve X and any strongly conformal N W = N SUSY vertex algebra V . Let V be a strongly conformal N K = N SUSY vertex algebra, and let V be the associated vector bundle over an oriented superconformal curve X . For an open U as before, and superconformal coordinates Z in U we will define the superconformal differential operators D X (U ) to be the super ring of differential operators generated by all the D iZ . This defines a sheaf of algebras of superconformal differential operators D X over any (oriented) superconformal curve X . The assignment D iZ · f (Z )a = (D iZ f (Z ))a + (−1) f f (Z )S i a

(3.2.8)

gives V the structure of a left D X -module.

3.3. Examples. 3.3.1. In this section we give the first non-trivial examples of the super vector bundles that arise with the construction of the previous sections. To simplify the notation, we will use the ordinary description of the involved vertex algebras. For example, when we analyze the boson-fermion system (cf. Example 3.10) we will work with the fermion ϕ and the boson α instead of the superfields  and S. Note that the Grassmann algebra  is a SUSY vertex algebra (either N W = N or N K = N ) with T = S i = 0 and |0 = 1. In this section, given a SUSY vertex algebra V , we will consider the tensor product W =  ⊗ V (either of N W = N or N K = N SUSY vertex algebras), therefore

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we can view W as a SUSY vertex algebra over , namely, W is a -module and the vertex operators are -linear7 . Let us start with N K = 1 bundles. For this let X be a super conformal N = 1 supercurve over . Let Uα and Uβ be open in X and p = (t, ζ ) a -point in the intersection. Let V be a strongly conformal N K = 1 SUSY vertex algebra, so that V carries a representation of Der0ω O 1|1 that exponentiates to a representation of Aut ω O 1|1 . Suppose we have coordinates (z α , θα ) in Uα and (z β , θβ ) in Uβ that are compatible with the superconformal structure. They are related by a change of coordinates ρβα = (F(z α , θα ), (z α , θα )) satisfying D F =  D, where D = ∂θα + θα ∂z α . These coordinates define coordinates at the point p, therefore we obtain different trivializations of the bundle V . The transition functions for the structure sheaf give us transition functions for V , in particular, they act in the fiber at the point p as R(ρ p )−1 (cf. 3.2.7). In order to compute R(ρ p ) we need to look only at the odd coordinate, namely, expand in Taylor series: z,θ (t, ζ ) = (t + z + ζ θ, ζ + θ ) − (z, θ ) t2 = ζ D + t D 2  + ζ t D 3  + D 4  + . . . 2⎞ ⎛
= exp ⎝− (vi L i + wi G (i) )⎠ A−2L 0 · ζ,

(3.3.1)

i≥1

where as in (3.1.2) we have Ln = −

n+1 n t ζ ∂ζ − t n+1 ∂t , G (n+1/2) = G n = −t n+1/2 (∂ζ − ζ ∂t ), 2

and vi = vi (z, θ ) are even functions and wi = wi (z, θ ) are odd functions. Truncating the series in (3.3.1) at order 2 we have:   (z,θ) (t, ζ ) = A ζ + tw1 + ζ tv1 + t 2 (w2 + v1 w1 ) + . . . , from where we get the equations: A = D,

w1 A = D 2 , 1 v1 A = D 3 , (w2 + v1 w1 )A = D 4 . 2

(3.3.2)

We can solve this system to get: D3 D2 v1 = , w1 = , D D

1  D3 D2 1 D4 w2 = = σ D , −2 2 2 D (D) 2

(3.3.3)

where σ is the N = 1 super-schwarzian defined in (2.2.8). 7 This approach of defining vertex algebras over Grassmann algebras was first used in [Bar00], where the connection with the geometry of superconformal Riemann surfaces, from a differentiable point of view, is explained.

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Example 3.10. (Free Fields) Recall the strongly conformal N K = 1 SUSY vertex algebra B1 defined in Example 2.7. We will denote this vertex algebra as B(1). As an ordinary vertex algebra, it is graded with respect to conformal weight. The fermion ϕ is primary of conformal weight 1/2 and the boson α has conformal weight 1 but it is not primary unless m = 0. It follows easily from the non-commutative Wick formula, that the only non-trivial relations with the fermion ϕ are given by: G (1) ϕ = −m|0,

L 0ϕ =

1 ϕ, 2

therefore the subspace B(1)≤1/2 of B(1) spanned by {|0, ϕ} is an Aut ω O 1|1 submodule. For a given change of coordinates ρ = (F, ) we can compute the action of R(ρ(z,θ) )−1 . For this we write in the basis {|0, ϕ}: ⎛ ⎞
R(ρ(z,θ) )−1 = A2L 0 · exp ⎝ (vi L i + wi G (i) )⎠ =

1 −mw1 0 A



i≥1



D2  D

= 1 −m 0 D

(3.3.4)

.

Hence, if B(1) is the vector bundle associated to the Autω O 1|1 module B(1) and B(1)≤1/2 is the vector bundle corresponding to B(1)≤1/2 we see that the transition functions that define B(1)≤1/2 are given on the intersections Uα ∩ Uβ by the functions (3.3.4). Dually, sections of the bundle B(1)∗≤1/2 transform by (note that we use the supertranspose instead of the transpose, as defined in [Man97, § 3.1]):

1 0 2 . (3.3.5) m DD D Note that we have a section Y of B(1)∗ which projects to a section of B(1)∗≤1/2 . In the basis {|0, ϕ} this section is given by:

Id , ϕ(z, θ ) where, according to (2.1.4), we have ϕ(z, θ ) = Y (ϕ(−1) |0, z) + θ Y (G −1/2 ϕ(−1) |0, z). According to (3.3.5) and Theorem 3.6 we see that the field ϕ(z, θ ) transforms as: ϕ(z, θ ) = R(ρ)ϕ(ρ(z, θ ))R(ρ)−1 D + m

D2 Id, D

(3.3.6)

where ρ = (F, ). In particular, since X is a superconformal N = 1 curve we have:



DF Fz z D = D = sdet . Fθ θ D Therefore when m = 0, ϕ(z, θ )[dzdθ ] transforms as an End B(1) p -valued section of the Berezinian bundle of X on the punctured disk D × p for any -point p ∈ X . When

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639

m = 0 this bundle is not split and ϕ(z, θ ) gives rise to an End B(1) p -valued section of B(1)∗≤1/2 that projects onto the section 1 ⊗ Id of the quotient O X ⊗ End B(1) p and transforms according to (3.3.6) with changes of coordinates. In other words, the bundle B≤1/2 (1)∗ is an extension: 0 → Ber X → B≤1/2 (1)∗ → O X → 0

(3.3.7)

which is non-split unless m = 0. In the case when m = 0 the section Y projects into the constant section 1 of O X 8 . We want to understand the geometric meaning of these sections. Equivalently, we want to find the set of splittings of the extension (3.3.7). This set, if non-empty, is a torsor over the space of even sections of Ber X . Recall also that the operator D = ∂θ + θ ∂z takes values in Ber X for a superconformal N = 1 curve. Theorem 3.11. The superfield ϕ(z, θ ) transforms as an odd differential operator ∇ : Ber X → Ber ⊗2 X locally of the form ∇ = −m Dα + gα (z α , θα ), where on the open subset Uα with coordinates (z α , θα ) we have Dα = ∂θα + θα ∂z α and gα is an odd function. Proof. Recall that in a superconformal N = 1 curve the generators [dz α dθα ] of the Berezinian bundle transform as: [dz β dθβ ] = (Dα β,α )[dz α dθα ], where the change of coordinates is θβ = β,α (z α , θα ). Since ∇ : Ber X → Ber ⊗2 X we have:   ∇α f α = (Dα β,α )2 ∇β (Dα β,α )−1 f α . Therefore we get: ∇α f α = −m Dα f α + gα f α = −m(Dα β,α )Dβ f α + gα f α    −1   −1  = (Dα β,α )2 −m Dβ Dα β,α f α + gβ Dα β,α fα   2 −1   = −m Dα β,α Dβ f α − m Dα β,α Dβ Dα β,α fα  + gβ Dα β,α f α   −1 2  = −m Dα β,α Dβ f α + m Dα β,α Dα β,α f α + gβ Dα β,α f α ,  D 2 β,α gα = Dα β,α gβ + m α . Dα β,α Hence we find:

thus proving the theorem.



1 gα



=

m

1 D2  D

0 D



1 , gβ

 

8 From now one, we abuse notation and forget the fact that the section Y is End(V )-valued. x

640

R. Heluani

3.3.2. Given that we can integrate a section of Ber X along a super contour as in 2.2.10, we can state [FBZ01, 7.1.9] in this situation. We define an affine structure on a superconformal N = 1 curve to be a (equivalence class of) coordinate atlas Uα with coordinates (z α , θα ) such that the transition functions on overlaps satisfy9 : z β = Fβ,α (z α , θα ) = a 2 z α + θα ξ a + b, θβ = β,α (z α , θα ) = θα a + ξ,

(3.3.8)

where a, b are even constants with a invertible and ξ is an odd constant (these constants may change with α and β). Given such an atlas, we can define ∇α = −m Dα and we get from: −m Dα = −m Dα β,α Dβ , and the fact that D 2  = 0 for these transition functions, that ∇α is a well defined operator as in Theorem 3.11. On the other hand, suppose we have such a differential operator ∇α = −m Dα +gα . Let f α [dz α dθα ] be a section of Ber X in Uα such that f α is an even function and ∇α · f α = 0. Choose a -point P = (x, π ) of Uα and, for any other point Q in Uα , we define the function ξα to be  Q ξα (Q) = fα . P

From the definition of this integral we see that ξ is an odd function, indeed, to compute this integral we need to solve Dω = f and then this integral becomes ω(Q) − ω(P). By shrinking if necessary the open cover Uα we may assume that f α does not vanish everywhere (it is an even function), it follows that Dξ is invertible everywhere. We now solve the differential equation Dw = ξ Dξ (we may need to shrink Uα even more) and obtain thus a coordinate atlas Uα with new coordinates (wα , ξα ). We claim that this atlas is indeed an affine structure on X . We have made some choices. One is the reference point P which shifts the function ξα by an odd constant. The other choice was the solution ξ , which is unique up to an invertible even multiple (for this we can apply a version of Cauchy’s theorem in super geometry). Therefore ξ is well defined up to affine transformations of the form ξ → aξ + ζ . This forces w to change to w˜ with D w˜ = (aξ + ζ )D(aξ + ζ ) = a 2 ξ Dξ + aζ Dξ, hence w˜ = a 2 w + w  with Dw  = Daζ ξ . Finally we see that w  = aζ ξ + w  , where Dw = 0, namely the choices made combine into changes of the form: ξ → aξ + ζ, w → a 2 w + aζ ξ + b, where a, b are even constants (a is invertible) and ζ is odd. Since these changes of coordinates are of the form (3.3.8), we have proved: Theorem 3.12. Let X be an N = 1 superconformal curve. For every m = 0 the set of differential operators ∇ : Ber X → Ber ⊗2 X locally defined as ∇α = −m Dα + gα for odd functions gα are in one to one correspondence with the set of affine structures on the curve X . These in turn are in one to one correspondence with the set of splittings of the extension (3.3.7). 9 These are SUSY changes of coordinates where the odd coordinate changes by affine transformations.

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Example 3.13. (The Neveu-Schwarz vertex algebra) Recall the strongly conformal N K = 1 SUSY vertex algebra K 1 defined in Example 2.4 (see also Example 2.14). Denote this vertex algebra by K (1). Note that the sub-vector space spanned by the primary elements of conformal weight less than or equal to 3/2, namely the vacuum vector and the N = 1 superconformal vector τ , is Aut ω O 1|1 -invariant. In order to compute the transition functions we use the relevant relations in this case: L (1) τ =

3 2 τ, G (2) τ = c. 2 3

Therefore we can compute in the basis {|0, τ }: ⎛ ⎞




|0 |0 = A2L 0 exp ⎝ R(ρ(z,θ ))−1 vi L i + wi G (i) ⎠ τ τ i≥1 2



|0 1 3 cw2 = . 3 τ 0 A It follows from (3.3.3) and (3.3.2) that the transition functions for the corresponding bundle K≤3/2 (1) are given by: R(ρ(z,θ) )

−1



1 3c σ D , = 0 (D)3

(3.3.9)

 where, as before, σ D is the super shwarzian derivative. Dualizing, we obtain an extension: ∗ 0 → Ber ⊗3 (3.3.10) X → K≤3/2 (1) → O X → 0. This extension is not split if c = 0 and, as for the free fields, we see that the section Y of K≤3/2 (1)∗ projects onto the section 1 ∈ O X in this case. Denote by τ (z, θ ) = G(z) + 2θ L(z) the superfield Y (τ, z, θ ). By taking the super transpose of (3.3.9) we find that τ (z, θ ) transforms as: c  τ (z, θ ) = R(ρ)τ (ρ(z, θ ))R(ρ)−1 (D)3 − σ D , 3 which in turn implies, according to Proposition 2.32 the following: Theorem 3.14. The set of splittings of (3.3.10) is in one to one correspondence with the set of superprojective structures in X . 3.3.3. Now we turn our attention to the oriented superconformal N = 2 case. We will use the coordinates (z, θ ± = θ 1 ± iθ 2 ) and the change of coordinates ρ = (F,  ± =  1 ± i 2 ) (cf. 3.1.4). It follows that: ± + − (z,θ + ,θ − ) (t, ζ , ζ )



1 + − − + + + − − t + z + (ζ θ + ζ θ ), ζ + θ , ζ + θ = 2 ± + − − (z, θ , θ ), ±

642

R. Heluani

which we want to expand in Taylor series (here  denotes either  + or  − ): (z,θ ± )

1 + − 1 + − − + − + 2 2 = 1 + (ζ θ + ζ θ )∂z + (ζ θ + ζ θ ) ∂z 2 8 · (t + z, ζ + + θ + , ζ − + θ − ) − (z, θ + , θ − )

1 + − + − 2 1 + − − + = 1 + (ζ θ + ζ θ )∂z + ζ ζ θ θ ∂z 2 4 + + − − · (t + z, ζ + θ , ζ + θ ) − (z, θ + , θ − )



 1 − 1 + + − + ∂θ − + θ ∂ z = ζ ∂θ + θ ∂ z + ζ 2 2



1 − 1 + + − + + t∂z + ζ t ∂θ + θ ∂z ∂z + ζ t ∂θ − + θ ∂z ∂z 2 2

 1 2 2 1 − 1 + 1 + − 2 + − + ∂θ − ,θ + − θ ∂z,θ − + θ ∂z,θ + θ θ ∂z + t ∂z  + . . . +ζ ζ 2 2 4 2  = ζ + D − + ζ − D + + t∂z + ζ + t D − ∂z + ζ − t D + ∂z  1 1 + ζ + ζ − (D + D − − ∂z ) + t 2 ∂z2  + . . . , 2 2

where D ± = ∂θ ∓ + 21 θ ± ∂z . Since the curve is oriented (cf. (3.1.8)), this reduces to:  + + − + t D+ D− + ζ +t D− D+ D− (z,θ ±) = ζ D  1 1 + ζ + ζ − (D + D − ) + t 2 (D + D − )2  + + . . . , 2 2  − − + − + (z,θ ± ) = ζ D + t D D + ζ − t D + D − D +  1 1 + ζ − ζ + (D − D + ) + t 2 (D − D + )2  − + . . . . 2 2 We want to express these as the exponential of a vector field. For this we compute: ⎛ exp ⎝−




⎞

 ⎠ B −J0 A−2L 0 · ζ ± = B ±1 A ζ ± + tw ± vi L i + u i Ji + w ± G ± 1 (i)

i≥1

 1 1 1 + ζ ± t (v1 ± u 1 + w1∓ w1± ) + t 2 (w2± + w1± (2v1 ± u 1 )) + ζ ± ζ ∓ w1± + . . . , 2 2 2 where we have used (3.1.9). We get the equations: B ±1 A = D ∓  ± ,

w1± =

z± z± D± D∓ ± = = , D∓ ± D∓ ± θ±±

1 D∓ D± D∓ ± 1 ± 1 (D ± D ∓ )2  ± ± w , w + (2v ± u ) = . v1 ± u 1 + w1± w1∓ = 1 1 2 1 2 D∓ ± 2 2 D∓ ±

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643

We can solve this system to get:



D ∓ z± 1 D − z+ D + z− 1 1 D ± z∓ ± ± , w  , v1 = + = − + 3 z,z 2 2 D− + D+ − 2D ∓  ± 2 D± ∓ D∓ ±

D + z− z+ z− 1 D − z+ 1 u1 = − = −σ2 ( + ,  − ), − 2 D− + D+ − 2 D− + D+ − where σ2 is the N = 2 schwarzian derivative (cf. [Coh87]). Example 3.15. (Free Fields) With the results of the previous sections we can compute now explicitly some vector bundles over oriented superconformal N = 2 curves. Let Y be such a curve and let B(2) be the strongly conformal N K = 2 SUSY vertex algebra described in Example 2.10. Let B(2) be the associated vector bundle over Y . The vector subspace spanned by the vacuum vectors and the two fermions (namely the fields with conformal weight less than or equal to 1/2) is an Aut ω O 1|2 -submodule. Let us denote these vectors, as in 2.10, by {|0, ϕ ± } respectively, and let B≤1/2 (2) be the associated rank 1|2 vector bundle over Y . In order to compute its transition functions explicitly we note that the only nontrivial relations (for our purposes) are: ∓ G± (1) ϕ = ∓m|0,

J0 ϕ ± = ±ϕ ± ,

L 0ϕ± =

1 ± ϕ . 2

We compute the transition functions as: ⎛ ⎛ ⎞ ⎞⎛ ⎞ |0 |0
± ± v1 L i + u i Ji + wi G (i) ⎠ ⎝ ϕ + ⎠ R(ρ)−1 ⎝ ϕ + ⎠ = A2L0 B J0 exp ⎝ ϕ− ϕ− i≥1 ⎛ ⎞⎛ ⎞ |0 1 mw1− −mw1+ BA 0 ⎠ ⎝ ϕ+ ⎠ = ⎝0 ϕ− 0 0 B −1 A ⎛ ⎞⎛ ⎞ z− z+ |0 1 m D+  −m D −  − + ⎠ ⎝ ϕ+ ⎠ . = ⎝0 D −  + 0 ϕ− 0 0 D+ −

(3.3.11)

Recall now that an oriented superconformal N = 2 curve projects onto two N = 1 supercurves: X and its dual Xˆ (cf. 2.2.8). Using the coordinates (cf. 3.1.4):

1 + − + − , u = z + θ θ ,θ ,θ 2 we obtain from (3.1.11) and (3.1.10) that −



D G 1 + + − + − + + θ Gu D+ − = D+ = D ( − θ  )(G + θ u D− + (θ++ )2 θ

 ++ G u − u+ G θ + G u u+ , (3.3.12) = θ = sdet + G θ + θ++ ( + )2 θ

where G = F +

1 + − 2 

as in 3.1.4. Similarly, we find

 G u  u− − + . D  = sdet G θ − θ−−

(3.3.13)

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Let us call π and πˆ the projections from Y onto X and Xˆ respectively. We see from (3.3.13) and (3.3.12) that taking the super-transpose in (3.3.11) we obtain an extension (of sheaves of OY -modules): 0 → π ∗ Ber X ⊕πˆ ∗ Ber Xˆ → B(2)∗≤1/2 → OY → 0.

(3.3.14)

As in the B(1) case, this extension is not split unless m vanishes. It follows in the same way as in the N = 1 case that the set of splittings of this extension corresponds to affine structures on the N = 2 superconformal curve Y . Indeed, we see in the same way as in Theorem 3.11, that the pair of fields (ϕ + , ϕ − ) transforms as a differential operator ∇ : + + − − ⊕ Ber ⊗2 Ber Xˆ ⊕ Ber X → Ber ⊗2 X which is locally of the form (m D + g , −m D + g ) Xˆ for g ± odd functions of (u, θ + ) and (u  , θ − ) respectively. We note that according to 2.2.10 sections of Ber X ⊕ Ber Xˆ can be integrated in Y up to an additive constant. The argument in the proof of Theorem 3.12 generalizes to this setting without difficulty. We will return to this example below (cf. 3.17). Example 3.16. (The N = 2 vertex algebra) Let K (2) := K 2 be the strongly conformal N K = 2 SUSY vertex algebra described in Example 2.8 (see also Example 2.14), and let K (2) be the associated vector bundle over an oriented superconformal N = 2 curve Y . The vector subspace spanned by primary fields of conformal weight 0 and 1 is an Autω O 1|2 submodule. Let us denote these vectors as above by {|0, J } respectively, and let K (2)≤1 be the associated rank 2|0 vector bundle over Y . To compute the transition functions we note that the only non-trivial relations we need are: c L 0 J = J, J1 J = |0. (3.3.15) 3 Therefore the transition functions are given by:



1 3c u 1 |0 |0 R(ρ)−1 = J J 0 A2



c 1 − 3 σ2 ( + ,  − ) |0 = . J 0 D+ − D− + It follows as before, by taking the super-transpose, that when c = 0, the superfield J (z, θ + , θ − ) transforms as a section of π ∗ Ber X ⊗πˆ ∗ Ber Xˆ , namely in this case we get an extension: 0 → π ∗ Ber X ⊗πˆ ∗ Ber Xˆ → K (2)∗≤1 → OY → 0, (3.3.16)

which is split if and only if c = 0. When c = 0, the superfield J (z, θ + , θ − ) transforms as: c J (z, θ + , θ − ) = (D +  − )(D −  + )J (ρ(z, θ + , θ − )) + σ2 ( + ,  − ). 3 We see that the section Y is an even section projecting onto 1 ∈ OY , therefore giving a splitting of (3.3.16). The set of such splittings, if non-empty, is a torsor for the even part of π ∗ Ber X ⊗πˆ ∗ Ber Xˆ . Analyzing this algebra further, we can consider the space K (2)≤3/2 spanned by vectors of conformal weight less than or equal to 3/2. This space is spanned by {|0, J, G ± }. In addition to (3.3.15) we have the following relations: 3 − G , 2 3 L 0 G+ = G+, 2

L0G− =

J0 G − = −G − ,

G +(1) G − = J,

J0 G + = G + ,

+ G− (1) G = −J,

c |0, 3 c + G− (2) G = 3 |0.

G +(2) G − =

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With these we can compute the transition functions in the basis {|0, J, G − , G + } explicitly: ⎞ ⎛ c + c − 1 3c u 1 3 w2 3 w2 ⎜0 A2 A2 w1+ −A2 w1− ⎟ ⎟, (3.3.17) R(ρ)−1 = ⎜ ⎝0 0 A3 B −1 0 ⎠ 0 0 0 A3 B the first three by three block being: ⎛ ⎞   + − − +  + − 1 D z + 3 D z 1 − 3c σ2 ( + ,  − ) 6D −c  + z,z +− −+ 2 D D ⎜ ⎟ ⎝0 (D +  − )(D −  + ) ⎠, (D +  − )z+ − + + − 2 0 0 (D  )(D  ) and the 4, 4 entry in (3.3.17) is (D +  − )(D −  + )2 . Taking the super-transpose of (3.3.17) it follows that K (2)∗Y,≤3/2 fits in a short exact sequence of the form: ⊗2  0 → π ∗ Ber X ⊗ πˆ ∗ Ber Xˆ → K (2)∗Y,≤3/2 → N The bundle N in turn fits in the exact sequence: ⊗2  ⊗ πˆ ∗ Ber Xˆ → N 0 → π ∗ Ber X

∗

∗

→ 0.

→ K (2)∗Y,≤1 → 0.

In a more “symmetric” fashion, if we look at the lower two by two block in (3.3.17), we see that we have an extension: ⊗2  ∗ ⊗2  0 → π ∗ Ber X ⊗ πˆ ∗ Ber Xˆ ⊕ π Ber X ⊗ πˆ ∗ Ber Xˆ → K (2)∗≤3/2 → K (2)∗≤1 → 0.

3.3.4. We turn our attention now to the N W = 1 case. For this let X be a general N = 1 supercurve. As before, given a change of coordinates ρ = (F, ), we expand in Taylor series: F(z,θ) (t, ζ ) = F(t + z, ζ + θ ) − F(z, θ ) t2 Fz,z + . . . , 2 (z,θ) (t, ζ ) = (t + z, ζ + θ ) − (z, θ ) = t Fz + ζ Fθ + ζ t Fθ,z +

= tz + ζ θ + ζ tz,θ + We need to express these as:

F(z,θ) (z,θ)



⎛

= exp ⎝−




t2 z,z . 2 ⎞

vi Ti + u i Ji + qi Q i + h i Hi ⎠

i≥1

× exp(−q0 Q 0 ) exp(−h 0 H0 )B −J0 A−T0



t , ζ

where, as in (3.1.1), we have: Tn = −t n+1 ∂t − (n + 1)t n ζ ∂ζ ,

Jn = −t n ζ ∂ζ ,

Q n = −t n+1 ∂ζ ,

Hn = t n ζ ∂t .

(3.3.18)

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Expanding (3.3.18) up to second order, we find: F(z,θ) = t A(1 + q0 h 0 ) + ζ Ah 0 + t 2 (v1 (A + Aq0 h 0 ) + Aq1 h 0 ) + ζ t (A(1 + q0 h 0 )h 1 + 2 Av1 h 0 + Au 1 h 0 ) + . . . , (z,θ) = ζ B A + tq0 B A + tζ B A(2v1 + u 1 + h 1 q0 ) + t 2 B A(q1 + v1 q0 ) + . . . , and we get the equations: A(1 + q0 h 0 ) = Fz , Ah 0 = Fθ , 1 v1 Fz + q1 Fθ = Fz,z , 2

B A = θ , q0 B A =  z , h 1 z + (2v1 + u 1 )θ = θ,z ,

h 1 Fz + (2v1 + u 1 )Fθ = Fz,θ ,

v1 z + q1 θ =

1 z,z . 2

From this we find: Fz θ − z Fθ , θ Fθ θ , h0 = Fz θ − z Fθ 1 Fz,z θ − z,z Fθ , v1 = 2 Fz θ − z Fθ Fz,θ θ − z,θ Fθ , h1 = Fz θ − z Fθ A=

θ2 , Fz θ − z Fθ z q0 = , θ (3.3.19) 1 Fz,z z − z,z Fz q1 = , 2 Fθ z − θ Fz Fz,θ z − z,θ Fz z,z Fθ − Fz,z θ u1 = + . Fθ z − θ Fz Fz θ − z Fθ B=

Example 3.17. (Free Fields) Consider the vertex algebra B(2) as in Example 3.15 but as a N W = 1 SUSY vertex algebra. As such, for each N = 1 supercurve X we obtain a ˜ the vector ϕ − has vector bundle B(2) X . Recall that with respect to the Virasoro field L, conformal weight 0. Therefore the vector space spanned by |0 and ϕ − is an AutO 1|1 submodule. We obtain then a rank 1|1 vector bundle over X , to be denoted B(2) X,≤0 . Let us compute explicitly the transition functions for this bundle. The relevant relations are in this case: J0 ϕ − = −ϕ − , Q 0 ϕ − = −m|0. Hence we obtain:

1 |0 −1 |0 T0 J0 = A B exp(h 0 H0 ) exp(q0 Q 0 ) − = R(ρ) ϕ− ϕ 0 which implies:

R(ρ) Noting that



−1

Fz sdet Fθ

z −m  θ

1 = 0 z θ

Fz θ −z Fθ θ2

=

−mq0 B −1



|0 , ϕ−

 .

Fz θ − z Fθ , θ2

(3.3.20)

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we see that by taking the super-transpose in (3.3.20) we obtain an extension 0 → Ber X → B(2)∗X,≤0 → O X → 0.

(3.3.21)

This short exact sequence is split if and only if m = 0. In that case, we see that ϕ − (z, θ )[dzdθ ] transforms as a section of Ber X . On the other hand, when m = 0, (3.3.21) is not split and the section Y projects into 1 ∈ O X , giving a splitting of (3.3.21). In order to analyze the splittings of these sequences, recall from 2.2.8 that to the N = 1 supercurve X we have associated another “dual” curve Xˆ and an oriented superconformal N = 2 curve Y . Introduce maps of sheaves on Y , ∇ : Ber Xˆ → Ber X ⊗ Ber Xˆ which are locally of the form ∇α = −m Dα+ + gα , for an odd function gα = gα (u, θ + ). Here we consider X with coordinates u, θ + and Xˆ with coordinates u  , θ − as in 3.1.4. We will write fˆ to denote a function of u  , θ − . It follows from (3.3.13), (3.3.14) and the fact that ∇ maps Ber Xˆ → Ber X ⊗ Ber Xˆ that on overlaps we must have:   ∇α fˆα = (D +  − )(D −  + )∇β (D −  + )−1 fˆα . Replacing ∇ in both sides by its local form and using (3.1.7) and (3.1.8), we get: − m Dα+ fˆα + gα fˆα = −m D + α fˆα + m(D −  + )−1 Dα+ Dα−  + fˆα + (D +  − )gβ f α . Now noting that D + D −  + = u+ and that: u+ u+ u+ = = , + D− + θ + + θ − u+ θ++ we get

gα = sdet

u+ G u u+ g + m , β + G θ + θ +  ++

(3.3.22)

θ

therefore proving the following Theorem. The set of splittings of (3.3.21) for m = 0 is in one to one correspondence with operators ∇ : Ber Xˆ → Ber X ⊗ Ber Xˆ locally of the form −m Dα+ + gα . Let ∇ be such an operator, and let 0 = ψα ∈ Ber Xˆ (Uα ) be a flat even section, namely ∇α ψα = 0. As a section of Ber Xˆ , it can be integrated along any contour in X (cf. 2.2.10), namely, let P be a reference -point in Uα , then for any other -point in Uα we put:  Q ζα (Q) = ψα . P

The solution ζα is unique up to an even multiplicative constant, while changing the reference point P changes ζα by an additive odd constant, shrinking Uα we may assume that Dα ζ is invertible. Choosing any other even function tα with invertible differential, we obtain charts Uα , (tα , ζα ). The transition functions between these charts are clearly affine functions for the odd coordinates, namely ζβ = aβ,α ζα + εβ,α for some even constants a and odd constants ε. Conversely, given such a covering of X , we define ∇α = −m Dα+ , where we take ζ instead of θ + and t instead of u in the definition of D + . It follows from (3.3.22) that ∇ is well defined globally since the second term in the right-hand side of (3.3.22) vanishes. Combining the above paragraph with the previous theorem we have

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Theorem. The set of splittings of (3.3.21) for m = 0 is in one to one correspondence with (equivalence classes of) atlases Uα , z α , θα , such that the transition functions are affine in the odd coordinate, namely θβ = aθα + ε for some even constant a and some odd constant ε. Note that from (3.3.20) and (3.3.11) it follows that the following sequences of OY modules are exact: 0 → πˆ ∗ Ber Xˆ → B(2)∗Y,≤1/2 → π ∗ B(2)∗X,≤0 → 0,

0 → π ∗ Ber X → B(2)∗Y,≤1/2 → πˆ ∗ B(2) Xˆ ,≤0 → 0.

The bundle B(2)Y is the corresponding bundle constructed in Example 3.15 from this vertex algebra, but viewed as an N K = 2 SUSY vertex algebra. These two extensions show how the different vector bundles constructed from the same vertex algebras in these three different curves (X , Xˆ and Y ) are related. Example 3.18. (The N = 2 vertex algebra) Let, as before K (2) be the N = 2 super vertex algebra defined in Example 2.8, but considered as an N W = 1 SUSY vertex algebra. Let X be an N = 1 supercurve. The vector space spanned by the vacuum vector, the current J , and the fermion H , is an AutO 1|1 -submodule. Indeed, with respect to the ˜ the fermion H has conformal weight 1. Denote the corresponding rank Virasoro field L, 2|1 vector bundle over X by K (2) X,≤1 . We claim that the dual of this vector bundle fits in a short exact sequence of the form: 0 → 1X ⊗ Ber X → K (2)∗X,≤1 → O X → 0.

(3.3.23)

Indeed, the relevant relations are in this case: T0 J = J, T0 H = H,

T1 J =

−c |0, 3

Q 0 H = J,

J1 J =

c |0, 3

J0 H = −H,

H0 J = H, Q1 H =

c |0, 3

therefore we compute these transition functions explicitly as before by exponentiating vector fields: ⎛ ⎞ |0 R(ρ)−1 ⎝ J ⎠ = A T0 B J0 exp(h 0 H0 ) exp(q0 Q 0 ) H ⎛ ⎞ ⎛ ⎞ |0
× exp ⎝ vi L i + u i Ji + qi Q i + h i Hi ⎠ · ⎝ J ⎠ H i≥1 ⎛ ⎞ |0 = A T0 B J0 exp(h 0 H0 ) exp(q0 Q 0 ) · ⎝ J + (u 1 − v1 ) 3c |0⎠ H + q1 3c |0 ⎛ ⎞⎛ ⎞ c 1 3c (v1 − u 1 ) |0 3 q1 ⎠⎝ J ⎠, A Aq0 = ⎝0 H 0 B −1 Ah 0 B −1 A(1 − h 0 q0 )

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which, according to (3.3.19), implies:   ⎛ F z −z,θ Fz 3 z,z Fθ −Fz,z θ 1 3c z,θ + Fθ z −θ Fz 2 Fz θ −z Fθ ⎜ Fz θ −z Fθ 0 R(ρ)−1 = ⎜ θ ⎝ Fθ 0 θ

⎞ c Fz,z z −z,z Fz 6 Fθ z −θ Fz ⎟ Fz z ⎟. θ ⎠ 2 Fz θ −z Fθ Fz θ2

(3.3.24)

Taking the super-transpose of the lower two by two block we easily see that this block corresponds to the transition functions in Ber X ⊗1 , proving thus that K (2)∗X,≤1 is given by an extension as in (3.3.23)10 . This extension is non-split unless c = 0, in which case the pair of fields {J (z, θ ), H (z, θ )} transforms as a section of Ber X ⊗1X . In order to study the splittings of this extension we need to understand the differential operators appearing in the first row of (3.3.24). We leave this to the reader. 4. Chiral Algebras on Supercurves In this section we define chiral algebras over supercurves and show that the bundles V X constructed in the previous sections from a SUSY vertex algebra V carry the structure of a chiral algebra. We note that most definitions carry over to the “super” case with minor technical changes. In particular we give a sheaf theoretical interpretation of the OPE formula (2.1.16). We define the superconformal blocks in Sect. 4.2. We will restrict our analysis to the (1|1) dimensional case for simplicity. All the results in this section can be generalized to arbitrary odd dimensions without difficulty. For the definitions of chiral algebras over ordinary curves, the reader is referred to [BD04] and [FBZ01]. For the theory of D-modules, we refer to [Ber], and [Pen83] in the supermanifold case. 4.1. Chiral algebras 4.1.1. When trying to define chiral algebras on supercurves the first problem that we encounter is that given a (1|N ) dimensional supercurve X over S, the diagonal embedding  → X × S X has relative codimension (1|N ). In particular, the diagonal is not a divisor in X × S X unless N = 0. The situation is much simpler in the superconformal case (corresponding to N K = N SUSY vertex algebras). In this case, we can define canonically a divisor in X × S X . Basically, all the arguments in the classical case work without change in the superconformal case, given that we have replaced the diagonal by a super diagonal. Since we can carry explicitly the computations in the N = 1 case, without introducing extra notation, we will assume that this is the case in the following. Lemma 4.1 (6.3 [Man91]) (cf. 4.1.4 below). Let X be a superconformal N = 1 supercurve. Let J be the ideal defining the diagonal i :  → X × S X . In local coordinates J is defined by (z − w, θ − ζ ). Let (1) be defined by J 2 . Let I be the kernel of the natural map 1X/S → Ber X/S . Finally we define s by: Os = O(1) /i ∗ (I ). 10 One can easily avoid computations and prove (3.3.23) by analyzing the representation of gl(1|1) in the space spanned by J and H .

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Then s is a (1|0) codimensional divisor in X × S X , locally defined by the equation 0 = z − w − θ ζ. This divisor will be called the super diagonal and we will simply call it the diagonal when no confusion should arise. 4.1.2. Given an O X -module M , we define two extensions of M along the super diagonal: extension by principal parts in the transversal direction and extension by delta functions in the transversal direction. The former is given by s+ M :=

O  M (∞s ) , O M

and the latter by

ω  M (∞s ) , ωM where ω is the Berezinian bundle of X defined in 2.2.7. s! M :=

4.1.3. As in the non-super case, we have a sheaf-theoretical interpretation of the OPE formula. For this we let X be a superconformal N = 1 curve over . Let V be a strongly conformal N K = 1 SUSY vertex algebra over  and let V be the associated vector bundle over X (cf. 3.2.5). Recall that, given any -point x in X , we have defined a local section Yx (cf. 3.6). Choose local coordinates Z at x compatible with the superconformal structure. Using these coordinates we trivialize the bundle V in the formal superdisk Dx around x, namely we have an isomorphism i Z : V [[Z ]] → (Dx , V ). Let W be another copy of Z , so that Dx2 is identified with Spec [[Z , W ]]. The bundle V  V (∞s ), when restricted to Dx2 , is the sheaf associated to the [[Z , W ]]-module V ⊗ V [[Z , W ]][(z − w − θ ζ )−1 ]. Similarly, the restriction of the sheaf s+ V to Dx2 is associated to the [[Z , W ]]-module V [[Z , W ]][(z − w − θ ζ )]/V [[Z , W ]]. Theorem 4.2. Define a map of O Dx2 -modules Y2,x : V  V (∞s ) → s+ V by the formula Y2,x ( f (Z , W )a  b) = f (Z , W )Y (a, Z − W )b mod V [[Z , W ]]. Then Y2,x is independent of the choice of the coordinates Z as long as they are compatible with the superconformal structure induced in Dx from that of X . Proof. Exactly as in the non-super case, we reduce the proof of this theorem to the identity:   a ∈ V, Y (a, Z − W ) = R(µW )Y R(µ Z )−1 a, µ(Z ) − µ(W ) R(µW )−1 , for any µ ∈ Aut ω O 1|1 . This identity is equivalent to (3.2.3) by substituting Z − W instead of Z and µW (Z − W ) = µ(Z ) − µ(W ) instead of ρ(Z ). Recall that in this case we have Z − W = (z − w − θ ζ, θ − ζ ).  

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Remark 4.3. In order to prove a similar statement for a general N = 1 supercurve X over , we could define a “super-diagonal” as follows. Recall that any such curve X gives rise to an oriented superconformal N = 2 super curve Y (cf. 2.2.8). Recall also that the curve Y comes equipped with two maps π : Y → X and πˆ : Y → Xˆ , where Xˆ is the dual curve. In local coordinates these maps are described by (cf. 3.16)

1 + − + + − π → z + θ θ ,θ , (z, θ , θ ) − 2

1 + − − π ˆ + − . (z, θ , θ ) − → z − θ θ ,θ 2 It is easy to show that Y embeds as a (1|0) codimensional divisor in X × Xˆ . Indeed, for a -point x in X given by local parameters Z = (z, θ ) the preimage in Y is given by local parameters (z − 21 θ ζ, θ, ζ ). Similarly, for a point W = (w, ζ ) in Xˆ we have its preimage in Y given by local parameters (w + 21 θ ζ, θ, ζ ). Then the point (Z , W ) in X × Xˆ is in the image of Y if and only if z − w − θ ζ = 0. Note in particular that when X is superconformal, namely X ≡ Xˆ this “diagonal” Y → X × Xˆ agrees with Manin’s super-diagonal given in Lemma 4.1. We could try to repeat the argument given above for superconformal curves, but the operation Y2 turns out to be coordinate-dependent.11 4.1.4. Instead of using the approach in the previous remark, note that we can define the push-forwards + and ! even when  is not a divisor. In our case these are easy to describe. Let  be the diagonal  → X × S X . Even though  is not a divisor in X × S X , its reduction || is a divisor in |X × S X | = |X | ×|S| |X |. We have then an open immersion j : X × X \  → X × X , where X × X \  is U = |X | × |X | \ || as a topological space and the structure sheaf is the restriction of O X 2 to U . We can now define the corresponding push-forwards of an O X -module M as: + M =

j∗ j ∗ (O X  M ) , OX  M

! M =

j∗ j ∗ (ω  M ) . ωM

When no confusion can arise, for any sheaf F , we will denote by F (∞) the sheaf j∗ j ∗ F . Remark 4.4. As in the non-super case, these pushforwards are in fact the push forward of left (resp. right) D X -modules along the diagonal, where in the superconformal case we understand for a D X module, a module over the ring of superconformal differential operators as in Remark 3.9 (see also Remark 4.6). 4.1.5. We construct now a morphism of O Dx2 -modules Y2,x : j∗ j ∗ (V X V X ) → s+ V X by the formula: Y2,x ( f (Z , W )a  b) = f (Z , W )Y (a, Z − W )b mod V [[Z , W ]].

(4.1.1)

As in 4.2 we have Theorem 4.5. The map Y2,x defined by (4.1.1) does not depend on the coordinates Z chosen. 11 It will be nice to find a way of describing the vertex algebra multiplication as an expression when a point x ∈ X “collides” with a point xˆ ∈ Xˆ along the “diagonal” s ⊂ X × Xˆ .

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4.1.6. We can now generalize all the results in [FBZ01, Chap. 18] on chiral algebras without difficulty. For simplicity let us assume that X is a general 1|N -dimensional supercurve. Suppose that the sheaf M on X carries a (left) action of the sheaf of differential operators D X . Let σ12 : X 2 → X 2 be the transposition of the two factors. We obtain ∗  M , given in local coordinates by a canonical isomorphism of sheaves + M  σ12 + the formula: 1⊗ψ ψ ⊗1 → e(Z −W )∇ · mod M  O X , k|K (Z − W ) (Z − W )k|K where ψ is a local section of M and ∇ is the connection that we obtain from the D-module structure in M . When M carries a right action of D X , we obtain similarly ∗  M . Note that the Berezinian bundle is of rank (0|1) if an isomorphism ! M  σ12 ! N is odd, hence in the above formula we need to multiply by (−1)ψ N in this case. Similarly, if X is a superconformal curve and M carries a (left) action of the sheaf of superconformal differential operators D X (cf. 3.9), the above formula defines isomorphisms as in the general case. 4.1.7. The Berezinian bundle ω X is a right D X -module, the action given by the Lie derivative [DM99]. Therefore for any left D X -module F we obtain a right D X -module F r := ω ⊗ F . This operation establishes an equivalence of categories between left and right D X -modules [Pen83]. The same results hold for D X -modules over superconformal curves in the sense of 3.9. Let X be a supercurve, the sheaf ω X  ω X on X 2 is isomorphic to ω X 2 . The natural map is expressed in local coordinates as: d Z  dW → [d Z dW ],

(4.1.2)

where as before d Z denotes the section [dzdθ 1 . . . dθ N ] of ω X and [d Z dW ] denotes the section [dzdwdθ 1 dζ 1 . . . dθ N dζ N ] of ω X 2 . We note the skew-symmetry in (4.1.2) since (recall the definition of the Berezinian in 2.2.7) d Z  dW → −(−1) N [dW d Z ].

(4.1.3)

We obtain thus ! ω X  ω X 2 (∞)/ω X 2 . Let µω denote the composition of the identification ω  ω(∞)  ω X 2 (∞) with the projection onto ! ω X . This map is a morphism of right D X 2 -modules satisfying the skew-symmetry condition: µω ◦ σ12 = −µω .

(4.1.4)

Note that this formula differs from (4.1.3) by a factor (−1) N . Indeed this factor appears when applying σ12 , namely the composition in the LHS of (4.1.4) is given by: σ12

µω

d Z  dW −→ (−1) N dW  d Z −→ (−1) N [dW d Z ] = −[d Z dW ] = −µω d Z  dW.

Remark 4.6. Let X be a supercurve and Z → X a closed embedding, We define the functor  Z from the category of sheaves on X to itself by letting sections of  Z (F ) be sections of F supported on Z . This functor is left exact. Let H Zi be the higher derived functors. In this sense the basic definitions of local cohomologies in [Har66] extend in a straightforward way to the super case. Similarly we can define the relative local

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cohomologies as the higher derived functors of  Z /Z  where Z  → Z is another closed embedding and  Z /Z  is defined in the usual way as the quotient of sections supported in Z modulo those supported in Z  [Har66]. From the exact sequence 0 →  Z (F ) → F → j∗ (F |U ) → H Z1 (F ) → 0, where U = X \ Z and j : U → X is the open immersion, we obtain: ! ω X = H1 (ω X 2 ). This identification of sheaves extended by delta functions on the diagonals with local cohomology sheaves shows that indeed these are push-forwards of D X -modules. 4.1.8. We have also a dictionary between D X -modules and delta functions. The space C[[Z ±1 , W ±1 ]] carries a structure of a module over the algebra of differential operators C[[Z , W ]][∇ Z , ∇W ] (here ∇ Z = (∂z , ∂θ i ) in the general case and ∇ Z = (∂z , D iZ ) in the superconformal case). The formal delta-function δ(Z , W ) satisfies the relations: (Z − W )1|0 δ(Z , W ) = 0, (Z − W )0|ei δ(Z , W ) = 0, (∇ Z + ∇W ) · δ(Z , W ) = 0. Therefore the C[[Z , W ]][∇ Z , ∇W ]-submodule of C[[Z ±1 , W ±1 ]] generated by δ(Z , W ) j|K is spanned by ∇W δ(Z , W ) with j ≥ 0. This module gives rise to a D-module on the 2 disk D = Spec C[[Z , W ]] supported on z = w (note that this is also the case in the superconformal case, where the poles are in z − w − θ i ζ i ). The assignment ( j|J )

(Z − W )−1− j|N \J dW → σ (J )∂W

δ(Z , W ),

induces an isomorphism of left D-modules on D 2 between + ω and the left D-module generated by δ(Z , W ). Similarly, tensoring with ω we obtain an isomorphism of right D-modules. In the superconformal case the situation is analogous, the proof follows from (2.1.17). 4.1.9. Recall that from Theorem 3.8 and (3.2.8), we have a natural (left) action of differential operators on V . It follows then that the push-forward + V is also a (left) D-module. Indeed, the action of vector fields locally is given by (a ∈ V ): ∂z : f (Z , W )a → (∂z f (Z , W ))a, ∂w : f (Z , W )a → (∂w f (Z , W ))a + f (Z , W )(T a), ∂θ i : f (Z , W )a → (∂θ i f (Z , W ))a, ∂ζ i : f (Z , W )a → (∂ζ i f (Z , W ))a + (−1) f f (Z , W )S i a, i ) instead of ∂ (resp. ∂ ). and similarly in the superconformal case, using D iZ (resp. DW θi ζi Also, we obtain a D-module structure on the sheaves V  V (∞), where ∂θ i acts as ∂θ i + S i and ∂ζ i acts as ∂ζ i + S i . Similarly, in the superconformal case, D iZ acts as D iZ + S i i acts as D i + S i . and DW W

Proposition 4.7. The map Y2,x commutes with the action of differential operators on Dx2 , making this map a morphism of D-modules.

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Proof. For a general supercurve X the proof is the same as in the non-super case. We sketch the proof in the superconformal case where a subtlety arises. Let X = (x, η1 , . . . , η N ). The identity Y (S i a, Z − W )b = D iX Y (a, X )b| X =Z −W = D iZ Y (a, Z − W )b translates into: Y2,x (D iZ · f (Z , W )a  b) = D iZ · Y2,x ( f (Z , W )a  b). On the other hand, consider translation invariance: [S i , Y (a, Z − W )]b = (∂ηi − ηi ∂x )Y (a, X )b| X =Z −W = (−∂ζ i + θ i ∂x − ηi ∂x )Y (a, Z − W )b| X =Z −W i = (−∂ζ i − ζ i ∂w )Y (a, Z − W )b = −DW Y (a, Z − W )b.

From here we obtain: i Y (a, Z − W )S i b = (−1)a S i Y (a, Z − W )b + (−1)a DW Y (a, Z − W )b,

and this translates into: i i Y2,x (DW · f (Z , W )a  b) = DW · Y2,x ( f (Z , W )a  b).

  Remark 4.8. Since + V is supported on the diagonal, we obtain a global version Y 2 of Y2,x by gluing these morphisms in the diagonal with the zero morphism outside of the diagonal. By the previous proposition, this morphism is a map of D-modules on X 2 . Proposition 4.9. The map Y 2 : V  V (∞) → + V satisfies Y 2 = σ12 ◦ Y 2 under ∗ V. the canonical identification +  σ12 + Proof. From the skew-symmetry property of SUSY vertex algebras (2.1.18) it follows: Y (a, Z − W )b = (−1)ab e(Z −W )∇ Y (b, W − Z )a. The sign cancels when applying σ12 and the exponential e(Z −W )∇ is the coordinate expression for the parallel translation, using the D-module structure on V , from W to Z (see 4.1.6).   4.1.10. In order to define chiral algebras over supercurves, we need to understand the composition of morphisms like Y 2 . For this we need to understand 123! A for any right D-module A over X , where 123 is the small diagonal in X 3 where the three points collide. As in the non-super case, we can write this as a composition 123! A  23! ! A .

(4.1.5)

This identity follows from the fact that the push-forward of right D-modules is exact for closed embeddings (cf. [Ber]). Now let µ : A  A (∞) → ! A be a morphism of D-modules on X 2 . We define a composition of µ: µ1{23} : j∗ A  A  A |U → 123! A ,

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where U = X 3 \ ∪i j and j : U → X 3 is the open immersion. In order to define such a composition we first apply µ to the second and third argument, and then we apply µ to the first argument and the result (cf. [FBZ01, 18.3.1]). We define other compositions of µ by changing the order in which we group the points. As in [FBZ01] we denote these compositions in the following way: given local sections a, b and c of A and a meromorphic function f (X, Y, Z ) with poles along the diagonals, we have: µ1{23} ( f (X, Y, Z )a  b  c) = µ( f (X, Y, Z )a  µ(b  c)), µ{12}3 ( f (X, Y, Z )a  b  c) = µ(µ( f (X, Y, Z )a  b)  c), µ2{13} ( f (X, Y, Z )a  b  c) = σ12 ◦ µ( f (X, Y, Z )b  µ(a  c)). With these compositions defined, we can now define a chiral algebra in the usual way: Definition 4.10. A chiral algebra on a 1|N dimensional supercurve X is a right D-module A equipped with a morphism of D-modules: µ : A  A (∞) → ! A , satisfying the following conditions: • (skew-symmetry) µ = −µ ◦ σ12 . • (Jacobi identity) µ1{23} = µ{12}3 + µ2{13} . • (Unit) We are given a canonical embedding ω X → A of the Berezinian bundle compatible with the homomorphism µω defined in 4.1.7. Remark 4.11. Note that this definition is exactly the same as in the non-super case, namely, the signs appearing when anticommuting odd-elements are taken care of by the symmetric structure of the category of modules over super-rings. Indeed the only difference with the non-super case is the fact that the unit ω is a rank (0|1)-bundle when N is odd. From the SUSY vertex algebra point of view, this is translated into the fact that the -bracket has parity N mod 2. In the superconformal case there is a subtlety. We note that the intersection of two different diagonals in the sense of 4.1 depends on the diagonals chosen, namely: s12 ∩ s23 = s13 ∩ s23 . But despite this fact, the pushforward 123! is still well defined, independent of the composition chosen as in (4.1.5). Using the equivalence between left D-modules and right D-modules, we obtain a right D-module V r = ω X ⊗ V from any strongly conformal SUSY vertex algebra. Similarly, this sheaf carries a multiplication µ = (Y 2 )r obtained from Y 2 . Theorem 4.12. The pair (V r , µ) carries a structure of a chiral algebra over X . Proof. The proof of this fact is the same as the proof in the non-super case [FBZ01, Thm 18.3.3]. This follows by considering the Cousin resolution of the Berezinian bundle in X 3 and the corresponding Cousin property of SUSY vertex algebras 2.23 proved in [HK07].   4.2. Conformal blocks. In this section we define the sheaves of coinvariants of SUSY vertex algebras. The treatment follows [FBZ01]. In fact, most results carry over without change to our situation. We only mention the major differences. 4.2.1. Recall that the polar part of a SUSY vertex algebra is naturally a SUSY Lie conformal algebra (cf. [HK07]). We can consider then the operator Yx,− which is the polar part of Yx . The notion of Lie∗ algebra over a super curve is generalized in a straightforward manner from the non-super case.

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4.2.2. Let A be a right D-module, the de Rham sequence of A is the sequence: 0→A ⊗T →A →0 placed in cohomological degrees 0 and −1, where T is the tangent sheaf of X . In the superconformal case, we do not have an action of the entire tangent sheaf, but we can act by the subsheaf T s generated by the derivations D iZ (i.e. the subsheaf T 1 of Remark 2.30 in the 1|1 dimensional case, and the sheaf T  ⊕ T  in the 1|2 dimensional case). We define then the de Rham sheaf h(A ) of A as h(A ) = A /(A · T ), whereas in the superconformal case we put h(A ) = A /(A · T s ). Proposition 4.13. Let (A , µ) be a chiral algebra. Then (1) h(A )(Dx× ) and h(A )(), for any open x ∈ /  ⊂ X are Lie superalgebras, and there is a natural homomorphism of Lie superalgebras h(A )() → h(A )(Dx× ). (2) h(A )(Dx× ) acts on the fiber Ax . (3) If (A , µ) is associated to a SUSY vertex algebra V , then there is a canonical isomorphism h(A )(Dx× )  Lie (V ) (see Theorem 2.25 for the definition of Lie (V )). Proof. We can think of A  ω ⊗ A l , where A l is a left D-module. Since we can integrate sections of the Berezinian bundle, we see immediately that we have h(! A ) = ∗ h(A ). On the other hand the map µ : A  A (∞) → ! A induces h(µ) : h(A )  h(A )(∞) → h(! A ). Restricting to regular sections and pulling back along the diagonal we obtain: [ , ] : h(A ) ⊗ h(A ) → h(A ). The fact that [ , ] satisfies the axioms of a Lie superalgebra follows from the skewsymmetry and Jacobi identity of chiral algebras. The rest of the theorem is proved in the same way as [FBZ01, Prop. 18.4.12]. (3) follows from the definitions, in formulas (2.1.19). Indeed, these formulas are the equivalent of the corresponding formulas for the action of vector fields on A l as defined in Theorem 3.8 and in (3.2.8).   Remark 4.14. As in the non-super case, for a strongly conformal SUSY vertex algebra V , we have a natural map Yx∨ : V r (Dx× ) → End(Vx )  End Vxr , on Dx× . Namely, given a section s ∈ V r (Dx× ) we obtain the endomorphism Yx∨ (s) = res X < Yx , s > on Vx . If s is a total derivative, this residue vanishes and the map Yx∨ factors through h(V r )(Dx× ). The resulting Lie superalgebra homomorphism h(V r )(Dx× ) → End(Vxr ) coincides with the homomorphism of Proposition 4.13 (2) and with the homomorphism ϕ  of Theorem 2.25.

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4.2.3. We can now define the spaces of coinvariants for a SUSY vertex algebra. For this let X be a supercurve and x ∈ X a point. We have a Lie superalgebra U = h(V r )(), where  = X \ {x} and this Lie superalgebra acts in Vx . Definition 4.15. The space of coinvariants associated to (V, X, x) is H (V, X, x) = Vx /(U · Vx ). Remark 4.16. The extension of this definition to the multiple point case with arbitrary module insertions is straightforward and we leave it for the reader. Fix N ≥ 0. Let g be the Lie superalgebra of vector fields on the 1|N dimensional punctured superdisk D × , namely g is the completion of the Lie superalgebra W (1|N ). ω be the Lie subalgebra of g consisting of vector fields preserving the form ω = Let g dt + ζ i dζ i , namely gω is the completion of the Lie superalgebra K (1|N ). Let Mg,1 be the moduli space of smooth 1|N dimensional genus g, pointed supercurves (here the genus of a supercurve X is the genus of X rd ). Let Mˆg,1 be the moduli space of triples ω and Mˆω (X, x, Z ), where (X, x) ∈ Mg,1 and Z is a coordinate system at x. Let Mg,1 g,1 be the superconformal analogue. Theorem 4.17 ([Vai95]). The Lie algebra g (resp. gω ) acts (infinitesimally) transitively ω ). This action preserves the fibers of the projection Mˆ on Mˆg,1 (resp. Mˆg,1 g,1 → Mg,1 ω ω ˆ (resp. M → M ). g,1

g,1

It follows from this theorem, by repeating the localization construction in [FBZ01, Ch. 16] that, given a strongly conformal N W = n SUSY vertex algebra (resp. a strongly conformal N K = n SUSY vertex algebra) V , we obtain a left D-module (V ) on Mg,1 ω ), whose fiber at (X, x) is the space of coinvariants H (X, x, V ). (resp. Mg,1 Acknowledgements. The author would like to thank Victor G. Kac for reading the manuscript, encouraging him, and many useful discussions. He would also like to thank David Ben-Zvi, for very useful discussions.

References [Bar96]

Barron, K.: A supergeometric interpretation of vertex operator superalgebras. Internat. Math. Res. Notices 1996, 409–430 (1996) [Bar00] Barron, K.: N = 1, Neveu Schwarz vertex operator superalgebras over Grassmann algebras with odd formal variables. In: Representations and Quantizations: Proceedings of the International Conference on Representation Theory 1998, J. Wang, Z. Lin (eds.), Beijing: China Higher Ed. Press and Springer, 2000, pp. 9–36 [Bar03] Barron, K.: The notion of n = 1 supergeometric operator superalgebra and the isomorphism theorem. Commun. Contemp. Math. 3, 481–567 (2003) [Bar04] Barron, K.: Superconformal change of variables for n = 1 Neveu-Schwarz vertex operator superalgebras. J. Algebra 277, 717–764 (2004) [BB93] Beilinson, A., Bernstein, J.: A proof of jantzen conjectures. Adv. in Soviet Math. 16, 1–50 (1993) [BD04] Beilinson, A., Drinfeld, V.: Chiral Algebras. AMS Colloquium Publications v 51, Providence, RI: Amer. Math. Soc. 2004 [BDFM88] Banks, T., Dixon, L., Friedan, D., Martinec, E.: Phenomenology and conformal field theory or can string theory predict the weak mixing angle? nucl. Phys. B 299(3), 613–626 (1988) [Ber] Bernstein, J.: Algebraic theory of D-modules, preprint 1983 [Bor86] Borcherds, R.: Vertex algebras, kac-moody algebras and the monster. Proc. Nat. Acad. Sci. USA 83(10), 3068–3071 (1986)

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Borisov, L.A.: Vertex algebras and mirror symmetry. Commun. Math. Phys. 215(3), 517–557 (2001) Bergvelt, M.J., Rabin, J.M.: Supercurves, their jacobians, and super kp equations. Duke Math. J. 98(1), 1–57 (1999) Cohn, J.D.: N = 2 Super-Riemann surfaces. Nucl. Phys. B 284, 349–364 (1987) Crane, L., Rabin, J.: Super Riemann surfaces: uniformization and teichmüller theory. Commun. Math. Phys. 113, 601–623 (1988) Dong, C., Lepowsky, J.: Generalized vertex algebras and relative vertex operators. Progress in Mathematics. Boston, MA: Birkhäuser Boston Inc., 1993 Deligne, P., Morgan, J.W.: Notes on supersymmetry. In: Quantum fields and strings: A course for mathematicians V. 1. Providence, RI: Amer. Math. Soc., 1999 Dolgikh, S.N., Rosly, A.A., Schwarz, A.S.: Supermoduli spaces. Commun. Math. Phys. 135(1), 91–100 (1990) Frenkel, E., Ben-Zvi, D.: Vertex algebras and algebraic curves. Mathematical surveys and monographs V. 88. Providence, RI: Amer. Math. Soc., 2001 Frenkel, I.B., Huang, Y., Lepowsky, J.: On axiomatic approaches to vertex operators algebras and modules. Mem. Amer. Math. Soc. 104(494), (1993) Frenkel, I., Lepowsky, J., Meurman, A.: Vertex operator algebras and the Monster. Pure and applied Mathematics V. 134. New York: Academic Press Inc., 1988 Friedan, D.: Notes on string theory and two dimensional conformal field theory. Proc. Workshop on Unified String Theories, M.B. Green, D. Gross (eds.), Singapore: World Scientific, 1986 Hartshorne, R.: Residues and duality. Number 20, Lecture Notes in Mathematics. Berlin-Heidelberg-New York: Springer-Verlag, 1966 Heluani, R., Kac, V.G.: Supersymmetric vertex algebras. Commun. Math Phys. 271(1), 103–178 (2007) Huang, Y.Z.: Two dimensional conformal geometry and vertex operator algebras. Progress in Mathematics V. 148. Boston, MA: Birkhäuser Boston Inc., 1997 Kac, V.G.: Vertex algebras for beginners. University Lecture series, V. 10 Providence, RI: Amer. Math. Soc., 1996, Second edition, 1998 Kapranov, M., Vasserot, E.: Vertex algebras and the formal loop space. Publ. Math. Inst. Hautes. Études Sci. 100, 209–269 (2004) Kac, V.G., van de Leur, J.: On classification of superconformal algebras. In: Strings-88, Singapore: World Scientific, 1989, pp. 77–106 Lepowsky, J., Li, H.: Introduction to vertex operator algebras and their representations. Progress in Mathematics. Boston, MA: Birkhäuser Boston Inc., 2004 Manin, Yu.I.: Topics in noncommutative geometry. Princeton, NJ: Princeton University Press, 1991 Manin, Yu.I.: Gauge field theory and complex geometry. Berlin-Heidelberg-New York: Springer, 1997 McArthur, I.N.: Line integrals on super riemman surfaces. Phys. Lett. B 206, 221–226 (1988) Malikov, A., Shechtman, V., Vaintrob, A.: Chiral de rham complex. Commun. Math. Phys. 204(2), 439–473 (1999) Penkov, I.B.: D-modules on super manifolds. Invent. Math. 71(3), 501–512 (1983) Rogers, A.: Contour integration on super Riemman surfaces. Phys. Lett. B 213(1), 37–40 (1988) Vaintrob, A.Yu.: Deformation of complex superspaces and coherent sheaves on them. J. Sov. Math. 51(1), 2140–2188 (1990) Yu Vaintrob, A.: Conformal lie superalgebras and moduli spaces. J. Geom. Phys. 15(2), 109–122 (1995)

Communicated by Y. Kawahigashi

Commun. Math. Phys. 275, 659–684 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0330-3

Communications in

Mathematical Physics

Regularized Determinants of the Laplacian for cofinite kleinian groups with Finite-Dimensional Unitary Representations Joshua S. Friedman Department of Mathematics and Sciences, United States Merchant Marine Academy, 300 Steamboat Road, Kings Point, NY 11024, USA. E-mail: [email protected]; [email protected]; [email protected] Received: 15 May 2006 / Accepted: 22 March 2007 Published online: 21 August 2007 – © Springer-Verlag 2007

Abstract: For cofinite Kleinian groups (or equivalently, finite-volume three-dimensional hyperbolic orbifolds) with finite-dimensional unitary representations, we evaluate the regularized determinant of the Laplacian using W. Müller’s regularization. We give an explicit formula relating the determinant to the Selberg zeta-function.

1. Introduction The regularized determinant of the Laplacian has been well studied on Riemann surfaces. In the case of compact Riemann surfaces, D’Hoker and Phong [DP86], and Sarnak [Sar87] related the regularized determinant to the Selberg zeta-function. For non-cocompact cofinite Fuchsian groups (or equivalently, finite-area non-compact Riemann surfaces with elliptic fixed points) Venkov, Kalinin, and Faddeev [VKF73] defined a regularized determinant for the Laplacian ∆ and related the determinant to the Selberg zeta-function. They regularized the trace of the resolvent kernel using the theory of Krein’s spectral shift function [Kre53, BK62, Yaf92]. Efrat [Efr88, Efr91] defined a regularized determinant for cofinite torsion-free Fuchsian groups with singular characters, and related it to the Selberg zeta-function. His regularization was essentially based on the Selberg trace formula. Efrat’s paper gave rise to an interesting question: Can the regularized determinant be defined cleanly in terms of general operator theory? In the compact case, the answer is yes. Here zeta-regularization is defined in terms of the heat kernel, which is of trace-class. In the non-compact case the heat kernel is not even Hilbert-Schmidt. W. Müller [Mül98, Mül83, Mül87, Mül92] applied Krein’s theory to define a regularization of the determinant,
a relative determinant det(H, H0 ) for two self-adjoint  operators H, H0 , satisfying tr e−H t − e−H0 < ∞. Müller’s regularization can be used for elliptic operators on non-compact manifolds. In [Mül92], Müller evaluates his determinant for the case of the Laplacian for finite-area

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surfaces with hyperbolic ends (a class of surfaces that includes Riemann surfaces), and relates the determinant to Efrat’s regularization (and hence to the Selberg zeta-function). In [Par05] J. Park studies a closely related problem. He studies eta-invariants of Dirac operators, and relates the regularized determinant of the Dirac Laplacian to the Selberg zeta-function for odd-dimensional hyperbolic manifolds with cusps. Park also uses the regularized determinant to extract information about the Selberg zeta-function. Regularized determinants have also been evaluated in the case of infinite volume Riemann surfaces, by Borthwick, Judge, and Perry [BJP]. In this paper, we evaluate Müller’s relative determinant of ∆ for the case of finitevolume three-dimensional hyperbolic orbifolds with finite-dimensional unitary representations. Or in other words the Laplacian acting on the Hilbert space of χ −automorphic (χ is a finite-dimensional unitary representation) functions on hyperbolic three-space. We relate the determinant to the Selberg zeta-function using the appropriate version of the Selberg trace formula (proved previously in [Fri05a, Fri05b]). We remark that zeta-regularization of determinants has found application in quantum field theory, in the works of Dowker and Critchley [DC76], Hawking [Haw77], Elizalde et al. [EOR+ 94], and Bytsenko, Cognola and Zerbini [BCZ97]. Main results. Next we define some of the basic notions needed to state our main results. A Kleinian group is a discrete subgroup of PSL(2, C) = SL(2, C)/ ± I. Each element of PSL(2, C) is identified with a Möbius transformation, and has a well-known action on hyperbolic three-space H3 and on its boundary at infinity—the Riemann sphere P1 (see [EGM98, Sect. 1.1]) . A Kleinian group is cofinite iff it has a fundamental domain F ⊂ H3 of finite hyperbolic volume. We use the following coordinate system for hyperbolic three-space, H3 ≡ {(x, y, r ) ∈ R3 | r > 0} ≡ {(z, r ) |z ∈ C, r > 0} ≡ {z + r j ∈ R3 | r > 0}, with the hyperbolic metric ds 2 ≡

d x 2 + dy 2 + dr 2 , r2

and volume form dv ≡

d x d y dz . r3

The Laplace-Beltrami operator is defined by  2  ∂ ∂ ∂2 ∂2 ∆ ≡ −r 2 +r , + + ∂ x 2 ∂ y 2 ∂r 2 ∂r and it acts on the space of smooth functions f : H3 → V, where V is a finite-dimensional complex vector space with inner-product  , V . Suppose that  is a cofinite Kleinian group and χ ∈ Rep(, V ) (Rep(, V ) is the space of finite-dimensional unitary representations of  in V ). Then the Hilbert space of χ −automorphic measurable functions is defined by  H(, χ ) ≡ f : H3 → V | f (γ P) = χ (γ ) f (P) ∀γ ∈ ,  P ∈ H , and  f, f  ≡ 3

F

  f (P), f (P)V dv(P) < ∞ .

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Here F is a fundamental domain for  in H3 , and  , V is the inner product on V . Finally, let ∆ = ∆(, χ ) be the corresponding positive self-adjoint Laplace-Beltrami operator on H(, χ ). Next we  briefly describe the motivation for the functional regularized determinant. Let f (s) = m∈D λ−s m be a sum over the non-zero eigenvalues of ∆. Then formally





  d

−s f (s) =− log(λ)λm =− log(λ) , f (0) = ds s=0 m∈D

and e− f

(0)

m∈D

s=0

=

λm .

λm =0

With this formal calculation in mind, one can think of e− f (0) as the regularized determinant. Now, typically, f (0) does not even converge, but f (s) does converge for Re(s) sufficiently large. Analytic continuation gives a possible value for f (0). The formal argument above works well when the orbifold in question is compact. In the non-compact case we compare ∆ with another self adjoint operator, ∆0 , the self-adjoint extension of the operator   k∞  k∞ k∞     d2 d −r 2 2 + r : C0∞ ([Y, ∞)) → L 2 [Y, ∞), r −3 dr dr dr i=1

i=1

i=1

with respect to Dirichlet boundary conditions ({ f ∈ C0∞ ([Y, ∞)) | f (Y ) = 0}). See §3.3 for the definitions of the notations used above. Define the projection (onto the constant Fourier coefficient) p0 : H(, χ ) →

k∞ 

  L 2 [Y, ∞), r −3 dr

i=1

by  1 P∞ f (x, y, r ) d xd y for r ≥ Y. p0 [ f ](r ) = |P| P Once again, see §3.3 for the definitions of the notations used above. The analogue of f (s) for non-compact spaces, following Müller, is the relative zeta-function  ∞

  1 ζ (s, ∆, ∆0 ) ≡ t s−1 tr e−∆ t − e−∆0 t p0 − dim ker ∆ dt. (s) 0 Here Re(s) > 2. Note that in order for the integral above to converge, we need to know the asymptotics of
 tr e−∆ t − e−∆0 t p0 − dim ker ∆ at both t = 0 and t = ∞. These asymptotics are given in Lemma 4.3.

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Our main article of interest is the regularized characteristic polynomial,
 det ∆ −(1 − s 2 ) , which we call the regularized determinant. For Re(s) > 2 define  ∞ 
1 2 H (w, s) ≡ H (w, s, ∆, ∆0 ) ≡ t w−1 tr e−∆ t − e−∆0 t p0 et (1−s ) dt, (w) 0 and following [Sar87], we define ∂H

det(∆ −(1 − s 2 )) = e− ∂w (0,s) . Our main results are: Theorem. Let  be a cofinite Kleinian group with one cusp at infinity, and let χ ∈ Rep(, V ). Then there exists constants C2 , C3 , D1 , depending on  and χ (they are explicitly determined in §5) such that   log det ∆ −(1 − s 2 ) = log Z (s, , χ ) + s (k(, χ ) log(Y ) + C1 ) +

C2 2 l∞ log (s + 1) + (s) − log s − C3 s 3 − D1 .

[∞ : ∞ ] 2 3

Here Z (s, , χ ) is the Selberg zeta-function (see §5.1), (s) is a meromorphic function (see Eq. 5.3). The constant Y comes about from the decomposition of F = FY ∪ F Y into a compact set FY and a noncompact cusp sector F Y (see §3.1 for more details). The rest of the notation is defined in §5. Corollary. Let  be a cofinite torsion-free Kleinian group with one cusp at infinity, and let χ ∈ Rep(, V ) be a regular character. Then


 3   2 3 vol  \ H det ∆ −(1 − s ) = Z (s, , χ ) exp −s + s L( ∞ , ψ) . 6π The constant L( ∞ , ψ) comes about from regularity at a cusp, and its value is computed using Kronecker’s second limit formula. It can be realized explicitly using the Siegel function g−v,u (τ ) , namely L( , ψ) =

−2π log g−v,u (τ ) . y

See §3.1.1 for more details. Corollary. Let  be a cocompact Kleinian group, χ ∈ Rep(, V ). Then


 3   2 3 vol  \ H det ∆ −(1 − s ) = Z (s, , χ ) exp −s + sC E , 6π where CE =

 tr V χ (R) log N (T0 ) {R}nce

πk 4| E(R) | sin2 ( m(R) )

.

The constant C E is related to the non-cuspidal elliptic elements of . See §3.1.2 for more details.

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2. General Definition of the Relative Zeta-function In this section (following [Mül98]) we state some basic facts concerning Krein’s spectral shift function, and show how they lead to the definition of the general relative zeta-function. Later, in §4, we specialize to ∆ on H(, χ ). For more details on the spectral shift function see [Kre53, BK62, Yaf92]. We first establish some notation. For B, a self-adjoint operator on H, σ (B) and σess (B) are the spectrum and essential spectrum of B respectively. Let A and A0 be bounded self-adjoint operators in H. Suppose that V ≡ A − A0 is of trace-class. Let R0 (z) = (A0 − z)−1 be the resolvent A0 . The spectral shift function of A and A0 , ξ(λ) = ξ(λ; A, A0 ) = π −1 lim arg det (1 + V R0 (λ + i)) , →0

(2.1)

exists for a.e. λ ∈ R, is real-valued, and belongs to L 1 (R). The determinant in (2.1) is the Fredholm determinant. In addition,  tr(A − A0 ) = ξ(λ) dλ, ξ  ≤ A − A0 1 , R

where  · 1 is the trace norm. The theory of the spectral shift function is reminiscent of the Selberg trace formula. Let    ( p)|(1 + | p|) dp < ∞ . G = φ : R → R | φ ∈ L 1 and |φ R

Then for every φ ∈ G, φ(A) − φ(A0 ) is a trace class operator and  tr(φ(A) − φ(A0 )) = φ (λ)ξ(λ) dλ. R

Part 3 of the next lemma will allow us to define the relative zeta function for H, H0 . Lemma 2.1 [Mül98, p. 315]. Let H, H0 be two non-negative self-adjoint operators in H and assume that e−t H − e−t H0 is a trace class operator for t > 0. Then there exists a unique real valued locally integrable function ξ(λ) = ξ(λ; H, H0 ) on R such that for each t > 0, e−tλ ξ(λ) ∈ L 1 (R) and the following conditions hold: ∞ (1) tr(e−t H − e−t H0 ) = −t 0 e−tλ ξ(λ) dλ. (2) For every φ ∈ G, φ(H ) − φ(H0 ) is a trace class operator and  φ (λ)ξ(λ) dλ. tr(φ(H ) − φ(H0 )) = R

(3) In addition, suppose σess (H0 ) ⊂ [c, ∞), where c > 0; then ker H and ker H0 are both finite-dimensional, and there exists c1 > 0 such that tr(e−t H − e−t H0 ) = dim ker H − dim ker H0 + O(e−c1 t ) as t → ∞.

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Let h = dim ker H − dim ker H0 . Then it follows from Lemma 2.1 that for Re(s) > 0, the integral  ∞     t s−1 tr e−t H − e−t H0 − h dt 0

converges absolutely. Definition 2.2 [Mül98, p. 317]. Suppose that σess (H0 ) ⊂ [c, ∞), where c > 0. Then for Re(s) > 0, the relative zeta-function of H and H0 , ζ (s; H, H0 ) is defined by  ∞     1 ζ (s; H, H0 ) = t s−1 tr e−t H − e−t H0 − h dt. (s) 0 3. The Operators ∆, ∆0 and Their Heat Kernels The main goal of this section is to give an explicit formula for tr(e−∆ t − e−∆0 t p0 ). 3.1. Notation. Before we can define ∆0 we must establish some notation (see [Fri05a, Fri05b] for more details). In order to simplify our notation (and to make our paper more readable) we present our results under the assumption: Assumption 3.1. The cofinite Kleinian group  has only one class of cusps at ζ = ∞ ∈ P, and χ ∈ Rep(, V ). The lattice associated with ζ = ∞ is

∞ = Z ⊕ Zτα , Im(τα ) > 0. Let ∞ <  denote the stabilizer subgroup of the cusp at infinity ζ = ∞, ∞ ≡ {γ ∈  | γ (∞) = ∞},

be the maximal torsion-free parabolic subgroup of  . By definition (of and let ∞ ∞

is a free abelian group of rank two. The possible values for the index of a cusp), ∞

] are 1,2,3,4, and 6. See [EGM98]. [∞ : ∞

is canonically isomorphic to a lattice

The subgroup ∞ ∞ = Z ⊕ Zτα . Without loss of generality we can assume that Im(τα ) > 0. Let  be a root of unity of order

]. Then [∞ : ∞    1 b

b ∈ ∞ , (1) ∞ = 0 1   n n     b

b ∈

∞ = , n = 0, . . . [ :  ] /{±I }. (2) ∞ ∞ ∞ 0  −n

Let P ⊂ C be a fundamental domain1 for the action of ∞ on C, and let P be the fundamental parallelogram with base point at the origin for the lattice ∞ . For Y > 0 set F Y ≡ F(Y ) ≡ {z + r j | z ∈ P, r ≥ Y }. 1 The set P is a euclidean polygon.

Regularized Determinants of the Laplacian

665

Then for Y sufficiently large, there exists a compact set FY , disjoint from F Y , so that F ≡ FY ∪ F Y is a fundamental domain for . Define the singular space by V∞ ≡ {v ∈ V | χ (γ )v = v, ∀γ ∈ ∞ },

(3.1)

and the almost singular space



≡ {v ∈ V | χ (γ )v = v, ∀γ ∈ ∞ }. V∞

(3.2)

If dim V∞ > 0, then χ is called singular with index of singularity k(, χ ) ≡ k∞ ≡

. Let P denote the dim V∞ . If dim V∞ = 0, then χ is called regular. Set l∞ ≡ dim V∞ ∞ orthogonal projection P∞ : V → V∞ . k∞ Fix an orthonormal basis {vi }i=1 for V∞ . For P ∈ H3 , Re(s) > 1, and i = 1 . . . k∞ ; we define the Eisenstein series by  E i (P, s) ≡ E(P, s, i, , χ ) ≡ (r (M P))1+s χ (M)∗ vi . M∈∞ \

The series E i (P, s) converges uniformly and absolutely on compact subsets of {Re(s) > 1} × H3 to a χ -automorphic function that satisfies ∆ E( · , s, α, v) = λE( · , s, α, v), λ = 1 − s 2 , and admits a meromorphic continuation to the whole complex plane [Fri05a]. For P = z + r j, P = z + r j ∈ H3 set δ(P, P ) ≡

|z − z |2 + r 2 + r 2 . 2rr

It follows that δ(P, P ) = cosh(d(P, P )), where d denotes the hyperbolic distance in H3 . Next, for k ∈ S ≡ S([1, ∞)) a Schwartz-class2 function, define K (P, Q) by K (P, Q) = k(δ(P, Q)). The function K (P, Q) is called a point-pair invariant. Set  K  (P, Q) ≡ χ (γ )K (P, γ Q). γ ∈

The decay properties of the function k guarantee that the series above converges absolutely and uniformly on compact subsets of H3 × H3 [EGM98, Theorem 6.4.1]. The function K  (P, Q) is the kernel of a bounded operator K : H(, χ ) → H(, χ ). The function k leads to two other useful functions: h, the Selberg–Harish-Chandra transform of k; and g, the Fourier transform of h. Explicitly:       1 π ∞ 1 1 dt 2 s −s t+ (t −t ) t − h(λ) = h(1 − s ) ≡ k , λ = 1 − s 2 , (3.3) s 1 2 t t t 2 The space of smooth functions k : [1, ∞) → C that satisfy lim n (m) (x) = 0 for all n, m ∈ N . x→∞ x k ≥0

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and for r ∈ R set

 1 h(1 + x 2 )e−i xr d x. 2π R For v, w ∈ V let v ⊗ w be the linear operator in V defined by v ⊗ w(x) =< x, w > v. An immediate application of the Spectral Decomposition Theorem ([Fri05a, Fri05b]), and the Selberg–Harish-Chandra Transform yields (see [EGM98, Eq. 6.4.10, p. 278]): g(r ) =

Lemma 3.2. Let k ∈ S and h : C → C be the Selberg–Harish-Chandra Transform of k. Then  K  (P, Q) = h(λm )em (P) ⊗ em (Q) m∈D

 k∞   1  1 + h 1 + x 2 El (P, i x) ⊗ El (Q, i x) dt, |P| R 4π

(3.4)

l=1

where |P| denotes the euclidean area of P ⊂ C. The sum and integrals converge absolutely and uniformly on compact subsets of H3 × H3 . We conclude this section with some notation that will be needed to state our main result. 3.1.1. Regular representations and Siegel’s theta function. Recall that ∞ = Z⊕Zτ ⊂ C with Im(τ ) > 0. It follows that χ restricted to (the abelian group) ∞ diagonalizes into characters3 ψl for l = 1 . . . l∞ , and the identity character for l = l∞ + 1 . . . dim V . For each ψl , u l , vl ∈ R are not both integers, satisfying ψl (1) = e2πiul and ψl (τ ) = e2πivl . We define −2π log g−vl ,ul (τ ) , L( ∞ , ψl ) = y where ga1 ,a2 is the Siegel function, ga1 ,a2 (τ ) =

−qτ(1/2)B2 (a1 ) e2πia2 (a1 −1)/2 (1 − qz )

∞

(1 − qτn qz )(1 − qτn /qz ),

n=1

B2 (X ) =

X2

− X + 1/6, qτ =

e2πiτ ,

qz =

e2πi z ,

and z = a1 τ + a2 .

3.1.2. Non-cuspidal elliptic elements. Let {R}nce be a set of representatives of the non-cuspidal elliptic elements of , the elliptic elements that do not fix a cusp (∞ under Assumption 3.1). Following [EGM98, Definition 5.3.2], the Elliptic number of  is  tr V χ (R) log N (T0 ) . πk 4| E(R) | sin2 ( m(R) ) {R}nce For a fixed representative R, N (T0 ) is the minimal norm of a hyperbolic or loxodromic element of the centralizer C(R). The element R is understood to be a k th power of a primitive non-cuspidal elliptic element R0 ∈ C(R) describing a hyperbolic rotation around 2π the fixed axis of R with minimal rotation angle m(R) . Further, E(R) is the maximal finite subgroup contained in C(R). 3 One-dimensional unitary representations.

Regularized Determinants of the Laplacian

667

3.1.3. Cuspidal elliptic elements. Denote by CE the set of elements of  which are

= {γ ∈  -conjugate to an element of ∞ \ ∞ ∞ | γ is not parabolic nor the identity element}. We fix representatives of conjugacy classes of CE, g1 , . . . , gd 4 that have the form   i i ωi . (3.5) gi = 0 (i )−1 Let C(g) denote the centralizer in  of an element g ∈ CE. In addition, let { pi , ∞} be the set of fixed points in P of the element gi . Since gi is a cuspidal elliptic element it follows that pi is a cusp of  (see [EGM98] p. 52). Hence by Assumption 3.1 there is an element γi ∈  with γi ∞ = pi . Let ci be the lower left-hand (matrix) entry of γi . 3.2. The heat kernel of ∆ as a Poincaré series. The heat kernel for5 ∆ on H3 is a function u : H3 × H3 ×(0, ∞) → R satisfying e−∆ t f (P) =

 H3

u(P, Q, t) f (Q) dv(P),

for all f in the domain of the self-adjoint operator e−∆ t . It is a classical result [Dav89] that   ρ2 ρ u(P, Q, t) = u(ρ, t) = (4π t)−3/2 exp −t − , where ρ = d(P, Q). sinh ρ 4t (3.6) In order to apply the theory of point pair invariants, we need to find a Schwartz-class function kt so that kt (δ(P, Q)) = kt (cosh(d(P, Q)) = u(P, Q, t). Using the formula: cosh−1 (x) = ln(x +



x 2 − 1), for x ≥ 1,

we obtain

2 ⎞  √ 2 − 1) x − ln(x + ln(x + − 1) ⎟ ⎜ kt (x) = exp ⎝ √ ⎠. 3/2 2 (4π t) 4t x −1 e−t

√

Observe6 that as x → 1+ , kt (x) →

⎛

x2

e−t , (4π t)3/2

(3.7)

and that kt ∈ S. We have:

4 There are only finitely many distinct conjugacy classes of elliptic elements in a cofinite Kleinian group. 5 We abuse notation and allow ∆ to represent both the self-adjoint operator on H(, χ ) and the standard differential operator on smooth functions of H3 . 6 The limit follows from applying l’Hôpital’s rule to either (3.6) or (3.7). The fact that the singularity cancels out is one advantage to working the heat kernel instead of the resolvent kernel, which has a singularity and must be iterated.

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Lemma 3.3. Let K  (P, Q, t, χ ) =



χ (γ )kt (δ(P, γ Q)).

γ ∈

Then K  (P, Q, t, χ ) is the heat kernel for ∆ on the Hilbert space H(, χ ). It follows from Eq. 3.3 and Lemma 3.2 that h(x) = e−t x . By definition  2  1 exp(−t) r 2 g(r ) = . e−(1+x )t e−i xr d x = √ exp − 2π R 4t 4π t By applying Lemma 3.2 we obtain the spectral expansion of K  :  K  (P, Q, t, χ ) = e−λm t em (P) ⊗ em (Q) m∈D

+

 k∞   1  1 exp −(1+x 2 )t El (P, i x) ⊗ El (Q, i x) d x. (3.8) |P| R 4π l=1

3.3. The operator ∆0 and its heat kernel. For F ≡ FY ∪ F Y , let ∆0 be the self-adjoint extension of the operator   k∞  k∞ k∞     d2 d −r 2 2 + r : C0∞ ([Y, ∞)) → L 2 [Y, ∞), r −3 dr dr dr i=1

i=1

i=1

with respect to Dirichlet boundary conditions ({ f ∈ C0∞ ([Y, ∞)) | f (Y ) = 0}). Note that ∆0 depends on Y . It is understood that ∆0 acts componentwise with respect to the basis for V∞ fixed in §3.1. The operator ∆0 can be thought of as a Laplacian operator in its own right. In fact ∆0 is closely related to the restriction of ∆ to F Y . Define p0 : H(, χ ) →

k∞ 

  L 2 [Y, ∞), r −3 dr

i=1

by

 1 p0 [ f ](r ) = P∞ f (x, y, r ) d xd y for r ≥ Y. |P| P Lemma 3.4. For all f in the domain of ∆, p0 [∆ f ] = ∆0 p0 [ f ]. Proof. The proof follows from [EGM98, pp. 236–237] and the definition of p0 . For F = FY

∪ FY ,

t > 0, P = z + r j,

P

=

z

+ r j

 

∈ F define



k(P, P , t)     ⎧  2

2 (r/r ) log(rr )−2 log(Y )) ⎪ √ ⎨P∞ rr exp(−t) exp − log 4t for P, P ∈ F Y , −exp − ( 4t |P | 4π t (3.9) ≡ ⎪ ⎩ 0 else. The classical theory of the Heat Equation, for the half line, tells us that k is the heat kernel of ∆0 . In other words:

Regularized Determinants of the Laplacian

669

Lemma 3.5. For all f ∈ H(, χ ), e−∆0 t p0 [ f ] =

 F

k( · , P , t) f (P ) dv(P ).

3.4. The regularized trace. In this section we will prove that the regularized heat kernel e−∆ t − e−∆0 t p0 is of trace-class. Then using ideas from the standard proof of the Selberg trace formula, we will evaluate the trace explicitly. But before we can proceed, we need Theorem 3.6. Its proof is based on the classical Poisson Summation Formula, and it is proved in Appendix A. Theorem 3.6. For P = z + r j, P = z + r j ∈ F Y ,   log2 (r/r )

exp(−t) + O(1). K  (P, P , t, χ ) = P∞ rr exp − √ 4t |P| 4π t Theorem 3.7. The operator e−∆ t − e−∆0 t p0 is of trace-class. Proof. The proof is based on the decay properties of Theorem 3.6 and Eq. 3.9, and a clever trick, using the semi-group properties of the heat kernels, of Deift-Simon. We follow [Mül83, p. 259] and [Par05, Prop. 2.1]. First note that by Theorem 3.6 and Eq. 3.9, e−∆ t − e−∆0 t p0 is Hilbert-Schmidt. Next write e−∆ t − e−∆0 t p0 as

 e−∆ τ e−∆ τ − e−∆0 τ p0 + e−∆ τ − e−∆0 τ p0 e−∆0 τ , (3.10) where τ = t/2. Next choose a function f ∈ C ∞ (F) so that 0 < f ≤ 1, f (P) = 1 if P ∈ FY , F(z + r j) = (Y/r )1/4 if P = z + r j ∈ F Y . Let m f be the multiplication operator by f . Now rewrite (3.10) once again as
−∆ τ 
 −∆0 τ e−∆ τ m f m −1 e − e−∆0 τ p0 + e−∆ τ − e−∆0 τ p0 m −1 . f f mfe The idea is to borrow r −1/4 from e−∆ τ − e−∆0 τ p0 and lend it to e−∆ τ and e
−∆0 τ . It follows from Theorem 3.6 and Eq. 3.9 that each of the operators e−∆ τ m f , m −1 e−∆ τ − f 
 −∆0 τ is Hilbert-Schmidt. Hence e−∆ t − e−∆0 τ p0 , e−∆ τ − e−∆0 τ p0 m −1 f , and m f e −∆ t  e 0 p0 is of trace-class. For more details, see [Mül83, p. 259].  Now we can apply standard Selberg theory to explicitly evaluate the integral trace  (tr V K  (P, P, t, χ ) − tr V k(P, P, t)) dv(P). F

Let S(s) be the scattering matrix of ∆. That is the matrix formed from the constant terms of the Fourier coefficients of the Eisenstein Series. Let φ(s) be the determinant of the scattering matrix (see [Fri05a, Fri05b] for more details).

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J. S. Friedman

Theorem 3.8. 
−∆ t  −∆0 t tr e −e p0 = (tr V K  (P, P, t, χ ) − tr V k(P, P, t)) dv(P) F    φ

 1 −λm t = e − exp −(1 + x 2 )t (i x) d x 4π R φ m∈D

e−t e−t 1 k(, χ ). k(, χ ) log Y + + e−t tr S(0) + √ 4 4 4π t Proof. Since f (P) ≡ tr V K  (P, P, t, χ ) − tr V k(P, P, t) ∈ L 1 (F), we can rewrite the integral above as  lim (tr V K  (P, P, t, χ ) − tr V k(P, P, t)) dv(P) A→∞ F A    = lim tr V K  (P, P, t, χ ) dv(P)− tr V k(P, P, t) dv(P) . (3.11) A→∞

FA

FA

A standard application of the Maaß—Selberg relations (see [Fri05a, EGM98, p. 305, or Ven82, pp. 67–70]) gives us  tr V K  (P, P, t, χ ) dv(P) FA

 e−t = log(A)k(, χ ) √ e−λm t + 4π t m∈D    φ

1 1 − exp −(1 + x 2 )t (i x) d x + e−t tr S(0) + o(1) . 4π R φ 4 A→∞

A straightforward calculation shows that  tr V k(P, P, t) dv(P) FA

   −(log(r ) − log(Y ))2 r 2 e−t 1 − exp dv(P) √ t F Y \F A |P| 4π t  A r 2 e−t d xd ydr = k(, χ ) √ r3 Y P |P| 4π t    A 2 −t r e −(log(r ) − log(Y ))2 d xd ydr − k(, χ ) exp √ t r3 Y P |P| 4π t e−t e−t e−t k(, χ ) + o(1) . = log(A)k(, χ ) √ −√ k(, χ ) log Y − 4 4π t 4π t A→∞ 

= k(, χ )

The term

e−t 4 k(, χ )

is obtained using a simple u−substitution, u=

 

log(r ) − log(Y ) . √ t

Regularized Determinants of the Laplacian

671

4. The Relative Zeta-Function and Relative Determinant Now that we have an explicit formula for the trace 
tr e−∆ t − e−∆0 t p0 (Theorem 3.8), we can proceed and give an explicit evaluation of the relative zetafunction. Since σ (∆0 ) = σess (∆) = [1, ∞) when the representation χ is singular, qχ ≡ dim ker ∆ − dim ker ∆0 = dim ker ∆ . Following Müller we define the relative zeta-function ζ (s, ∆, ∆0 ) by  ∞

  1 t s−1 tr e−∆ t − e−∆0 t p0 − qχ dt. ζ (s, ∆, ∆0 ) = (s) 0

(4.1)

Note that in order for the integral above to converge, we need to know the asymptotics of
 tr e−∆ t − e−∆0 t p0 − qχ at both t = 0 and t = ∞. These asymptotics are given in Lemma 4.3. Since vol(F) < ∞, it follows that [EGM98, Theorem 3.6.4]  dim V if χ is trivial qχ = 0 else. Theorem 4.1. For Re(s) > 2, ζ (s, ∆, ∆0 ) =

 m∈D

λ−s m −

 1 φ

1 (1 + x 2 )−s (i x) d x + (tr S(0) + k(, χ )) 4π R φ 4

k(, χ ) (s − 1/2) log Y. + √ (s) 4π Proof. The proof follows from the standard properties of the Mellin transform, Theorem 3.8, and Lemma 4.3. Note that qχ cancels out with any of the terms coming from the zero eigenvalues of ∆ .   4.1. Asymptotics of the heat kernel. The main tool that allows us to study the asymptotic behavior (near t = 0 and t = ∞) of the regularized heat kernel is the Selberg trace formula for the case of a cofinite Kleinian group with finite-dimensional unitary representations [Fri05a, Fri05b]. Theorem 4.2 (Selberg trace formula). Let  be a cofinite Kleinian group with one cusp at infinity, χ ∈ Rep(, V ), h be a holomorphic function on {s ∈ C | | Im(s)| < 2 + δ} for some δ > 0, satisfying h(1 + z 2 ) = O(1 + |z|2 )3/2− as |z| → ∞, and let  1 g(x) = h(1 + t 2 )e−it x dt. 2π R

672

J. S. Friedman

Then  1 φ

h(1 + t 2 ) (it) dt 4π R φ m∈D
  vol  \ H3 = dimC V h(1 + t 2 )t 2 dt 4π 2 R  tr V χ (R)g(0) log N (T0 )  tr V χ (T )g(log N (T )) + + log N (T0 ) πk |E(T )| |a(T ) − a(T )−1 |2 4| E(R) | sin2 ( m(R) ) {R}nce {T }lox 

h(λm ) −

−

tr(S(0))h(1) 4 ⎛

⎞  ∞ 2g(0) log |c | 1 sinh x i ⎝ + g(x) dx⎠ + 2 |2 |1− 2 |2 |C(gi )| |1 −  |1 − i2 |2 0 i cosh x − 1 + 2 i i=1    η 

l∞ h(1) 1 ∞ 2  + h(1 + t ) (1 + it) dt g(0) + g(0) −γ −

| |∞ : ∞ 4 2 2π R  n  g(0) + L( ∞ , ψl ). (4.2)

|∞ : ∞ | d  tr χ (gi )

l=l∞ +1

Here {λm }m∈D are the eigenvalues of ∆ counted with multiplicity. The summation with respect to {R}nce extends over the finitely many −conjugacy classes of the noncuspidal elliptic elements (elliptic elements that do not fix a cusp) R ∈ , and for such a class N (T0 ) is the minimal norm of a hyperbolic or loxodromic element of the centralizer C(R). The element R is understood to be a k th power of a primitive non-cuspidal elliptic element R0 ∈ C(R) describing a hyperbolic rotation around the 2π fixed axis of R with minimal rotation angle m(R) . Further, E(R) is the maximal finite subgroup contained in C(R). The summation with respect to {T }lox extends over the −conjugacy classes of hyperbolic or loxodromic elements of , T0 denotes a primitive hyperbolic or loxodromic element associated with T . The element T is conjugate in PSL(2, C) to the transformation described by the diagonal matrix with diagonal entries a(T ), a(T )−1 with |a(T )| > 1, and N (T ) = |a(T )|2 . For s ∈ C, S(s) is a k(, χ ) × k(, χ ) matrix-valued meromorphic function, called the scattering matrix of ∆, and φ(s) = det S(s). The elements gi are complete representatives for the conjugacy classes of {γ ∈ ∞ | γ is not parabolic nor the identity element}, |C(gi )| is the order of the centralizer in  of the element gi . The numbers ci ∈ C are constants depending on the gi respectively (see §3.1.3). Finally γ is Euler’s constant, and η∞ is the analogue of Euler’s constant for the lattice ∞ ⊂ R2 . The term L( ∞ , ψl ) is defined in §3.1.1. See [Fri05a, Fri05b, EGM98] for more details. Next, using the Selberg trace formula, we study the regularized heat kernel. We have
 Lemma 4.3. Let θ (t) ≡ tr e−∆ t − e−∆0 t p0 . Then there exists constants a, b, c, d so that √ 3 1 1 θ (t) = at − 2 + b(log t)t − 2 + ct − 2 + d + O( t log t) as t → 0+ , and there exists a positive constant c > 0 so that θ (t) − qχ = O(e−ct ) as t → ∞.

Regularized Determinants of the Laplacian

673

Proof. Our argument is analogous to [Efr88, Efr91, Proposition 1]. The asymptotics as t → ∞ follows immediately from the spectral trace formula (Theorem 3.8). An application of the Selberg trace formula for the pair of functions  2 exp(−t) r , h(z) = exp(−zt), g(r ) = √ exp − 4t 4π t yields (on applying the left side of the Selberg trace formula) the nontrivial terms from the trace of the heat kernel in Theorem 3.8, namely 

 1 2 φ −tλm e − e−t (1+x ) (i x) d x. 4π R φ m∈D

It remains for us to estimate each term on the right as t → 0+ . We start with the loxodromic sum   (log N (T ))2 4t exp(−t)  tr V χ (T ) exp − log N (T0 ). √ |E(T )| |a(T ) − a(T )−1 |2 4π t {T }lox

Since N (T ) > 1 for all loxodromic T , the sum decays to zero as t → 0+ . √ Next note that g(0) = exp(−t) and h(1) = e−t . The finite sum involving the 4π t non-cuspidal elliptic terms is easily estimated, 1 exp(−t)  tr V χ (R) log N (T0 ) = O(t − 2 ), √ πk 2 4π t {R}nce 4| E(R) | sin ( m(R) )

and so are all the other terms with only g(0) or h(1). Next we must estimate  2  exp(−t) ∞ sinh x x exp − d x. √ |1− 2 |2 4t 4π t 0 cosh x − 1 + 2 i x An elementary u−substitution with u = √ shows that the integral above decays to 2 2 + zero (exponentially fast) as t → 0 . The next integral can be calculated explicitly:
 
√ −t  vol  \ H3 vol  \ H3 πe 2 2 dimC V exp(−t (1 + x ))x d x = (dimC V ) . 3 2 2 4π 4π R 2t 2

The last integral  R

e−(1+x

2 )t



(1 + i x) d x 

requires some work. To study the integral near t = 0 we follow [Efr88, Efr91, Proposition 1]. After performing an integration by parts, the integral becomes  ∞ 2 −2tie−t xe−t x log (1 + i x) d x. −∞

674

J. S. Friedman

Using Sterling’s formula √ 1 1 log (1 + i x) = ( + i x) log(1 + i x) − (1 + i x) + log 2π + O( ) as x → ∞, 2 x it follows that for constants β , γ , δ



√ log t γ

2  e−(1+x )t (1 + i x) d x = β √ + √ + δ + O( t log t) as t → 0+ .  t t R See [Efr88, Efr91, Proposition 1] for more details.

 

4.2. The regularized determinant. In this section we define det(∆ −(1 − s 2 )). For Re(s) > 2 define  ∞

 1 2 H (w, s) ≡ H (w, s, ∆, ∆0 ) ≡ t w−1 tr e−∆ t − e−∆0 t p0 et (1−s ) dt. (w) 0 By applying the Mellin Transform to Theorem 3.8 we obtain, for Re(s) > 2,   φ

1 H (w, s) = (λm − (1 − s 2 ))−w − (x 2 + s 2 )−w (i x) d x 4π R φ m∈D

1 k(, χ ) (w − 1/2) log Y. + s −2w (tr S(0) + k(, χ )) + s −(2w−1) √ 4 (w) 4π

In order to define det(∆ −(1 − s 2 )), we will need to know that H (w, s) is regular at w = 0. Lemma 4.4. The following hold: (1) For fixed s > 2, H (w, s) is regular at w = 0. (2) √ 3 1 ∂H (0, s) ∼ a(s 2 − 1) 2 + 2 π b(s 2 − 1) 2 (log(s 2 − 1) + (γ + log(4) − 2)) ∂w √ 1 − 2 π c(s 2 − 1) 2 − d log(s 2 − 1) as s → ∞, where γ is Euler’s constant, and the constants a, b, c, d are from Lemma 4.3. Proof. The proof is a standard exercise using the Mellin transform, Lemma 4.3, and the following formulas [GR65]:  ∞ 1 1 1 3 2 (s 2 − 1)w− (w − ),  = 0, , , t − et (1−s ) t w−1 dt = (w) 0 (w) 2 2    ∞ 1 (w− 2 ) 2 1 1 1 log t t (1−s 2 ) w−1 −w 2 2 (s −1) (w − )−log(s − 1) . t dt = √ e (w) 0 (w) 2 t Here (z) is the logarithmic derivative of (z). Regularity follows from the fact that 1  (w) vanishes at w = 0. See [Efr88, Efr91, Prop 2 and 3] for more details.  Next, for Re(s) > 2, define the regularized determinant by ∂H

det(∆ −(1 − s 2 )) = e− ∂w (0,s) .

(4.3)

Our main result, Theorem 5.2, will give the meromorphic continuation to Re(s) ≤ 2.

Regularized Determinants of the Laplacian

675

5. Selberg’s Zeta-Function and the Regularized Determinant 5.1. The definition of the Selberg zeta-function. In this section we define the Selberg zeta-function for cofinite Kleinian groups with finite-dimensional unitary representations. For more details see [Fri05b]. Suppose T ∈  is loxodromic (we consider hyperbolic elements as loxodromic elements). Then T is conjugate in PSL(2, C) to a unique element of the form  D(T ) =

a(T ) 0 0 a(T )−1



such that a(T ) ∈ C has |a(T )| > 1. Let N (T ) denote the norm of T , defined by N (T ) ≡ |a(T )|2 , and let C(T ) denote the centralizer of T in . There exists a (primitive) loxodromic element T0 , and a finite cyclic elliptic subgroup E(T ) of order m(T ), generated by an element E T such that C(T ) = T0  × E(T ) . Here T0  = {T0n | n ∈ Z }. Next, let t1 , . . . , tn , and t 1 , . . . , t n denote the eigenvalues of χ (T0 ) and χ (E T ) respectively. The elliptic element E T is conjugate in PSL(2, C) to an element of the form 

0 ζ (T0 ) 0 ζ (T0 )−1

 ,

where here ζ (T0 ) is a primitive 2m(T )th root of unity. For Re(s) > 1 the Selberg zeta-function Z (s, , χ ) is defined by

Z (s, , χ ) ≡

dim

V

{T0 }∈R

j=1

l,k≥0 c(T, j,l,k)=1

  1 − t j a(T0 )−2k a(T0 )−2l N (T0 )−s−1 .

Here the product with respect to T0 extends over a maximal reduced system R of -conjugacy classes of primitive loxodromic elements of . The system R is called reduced if no two of its elements have representatives with the same centralizer.7 The function c(T, j, l, k) is defined by c(T, j, l, k) = t j ζ (T0 )2l ζ (T0 )−2k . 7 See [EGM98] Sect. 5.4 for more details.

676

J. S. Friedman

5.2. The relationship between the Selberg zeta-function and the regularized determinant. One way to study the Selberg zeta-function is to apply the Selberg trace formula to the pair of functions, 1 1 − 2 and s2 + w − 1 B +w−1 1 −B|x| 1 e , g(x) = e−s|x| − 2s 2B

h(w) =

where 1 < Re(s) < Re(B). Let Z (s) ≡ Z (s, , χ ) be the Selberg zeta-function under Assumption 3.1. We have [Fri05b]: Lemma 5.1. 1 Z

1 Z

(s) − (B) 2s Z 2B Z  1 tr(χ (T )) log N (T0 ) = N (T )−s 2s m(T )|a(T ) − a(T )−1 |2 {T } lox

tr(χ (T )) log N (T0 ) 1  − N (T )−B 2B m(T )|a(T ) − a(T )−1 |2 {T } lox 

    1 1 φ 1 1 1 − − (i x) d x = − λn − (1 − s 2 ) λn −(1 − B 2 ) 4π R s 2 + x 2 B 2 + x 2 φ n∈D 

  1  tr S(0) tr S(0) l∞ 1 (1 + i x) d x + + − −

] 2 + x2 2 + x2 2π [∞ : ∞ s B  4s 2 4B 2 R l∞ l∞ − +

2

]B 2 4[∞ : ∞ ]s 4[∞ : ∞   ∞  −sx l  e e−Bx sinh x tr χ (gi ) − − dx 2 |2 |1− 2 |2 2s 2B |C(g )||1 −  0 i i cosh x − 1 + 2 i i=1 
   1 tr V χ (R) log N (T0 ) vol  \ H3 dim V 1 − (s − B) − + πk 2s 2B 4π 4| E(R) | sin2 ( m(R) ) 

{R}nce

 l

2 tr χ (gi ) log |ci | |C(gi )||1 − i2 |2 i=1 ⎞ ⎛   n  η  1 1 1 ⎝l∞ ∞ − γ + − − L( ∞ , ψl )⎠ .

] 2s 2B [∞ : ∞ 2

−

1 1 − 2s 2B

(5.1)

l=l∞ +1

Recall that H (w, s) =

 m∈D

((λm − 1) + s 2 )−w −

 φ

1 (x 2 + s 2 )−w (i x) d x 4π R φ

1 k(, χ ) (w − 1/2) + s −2w (tr S(0) + k(, χ )) + s (1−2w) √ log Y. 4 (w) 4π

Regularized Determinants of the Laplacian

677

Applying the following elementary equations:      1 d ∂ ∂ d 1 d −2s d 2 −w 2 −w =− = , − (u + s ) (u + s ) 2 2 w=0 ds 2s ds ∂w ∂w ds 2s ds w=0 (u + s ) √    d 1 d ∂ (1−2w) (w − 1/2) π = − − s , 2 ds 2s ds ∂w (w) s w=0 to H (w, s), we see that    1 d ∂ d H (w, s)|w=0 − ds 2s ds ∂w    1 d ∂ d −2s =− = H (w, s) ∂w ds 2s ds ((λ − 1) + s 2 )2 m w=0 m∈D  k(, χ ) −2s φ

1 (i x) d x − − log Y. (5.2) 4π R (x 2 + s 2 )2 φ 2s 2 Caution. Differentiation through the sum and integral is justified by regularity at w = 0 (Lemma 4.4). Instead of differentiating first with respect to w at w = 0, we switch the order of differentiation, differentiate with respect to s, and restrict w so that w > 2 (where the sum and integral converge uniformly). Finally we differentiate with respect to w, and using analytic continuation (and uniqueness of analytic continuation), we obtain (5.2). For Re(s) > 0 set ⎞ ⎛  ∞ l  1 tr χ (gi ) sinh x ⎠ d x, for i = 1, ⎝ (s) = − e−sx 2 |2 |1−i2 |2 x |C(g )||1− 0 i i cosh x − 1+ 2 i=1 (5.3) (see §3.1.3 for the definition of i and gi ). Using the equation  1 

1 

1 (1 + i x) d x = (1 + s), 2 2 π R (s + x )  s  it follows that   1 Z

d (s) ds 2s Z   −2s −2s φ

1 (i x) d x = − (λn − 1 + s 2 )2 4π R (s 2 + x 2 )2 φ n∈D   l∞ l∞ 

1 d tr S(0) − (s + 1) − +

] 

] ds 2s[∞ : ∞ 2s 3 [∞ : ∞ 1  tr V χ (R) log N (T0 ) d 1 d ( (s)) + 2 − πk ds 2s ds 2s 4| E(R) | sin2 ( m(R) ) {R}nce ⎞⎤ ⎡ ⎛ l n   η 1 1 ⎣ 2 tr χ (gi ) log |ci | ∞ ⎝l∞ −γ + + L( ∞ , ψl )⎠⎦ + 2 2 |2

] 2s [ :  2 |C(g )||1− ∞ i ∞ i i=1 l=l∞ +1
 3 vol  \ H dim V + . (5.4) 4π

678

J. S. Friedman

Simplifying, and recalling the definition of det(∆ −(1 − s 2 )) we arrive at     d 1 Z

d 1 d  (s) = log(det(∆ −(1 − s 2 )) ds 2s Z ds 2s ds   l∞ 

d 1 k(, χ ) (s + 1) log Y − +

]  2s 2 ds 2s [∞ : ∞   1 d d 1 1 − (s) + 2 C1 − 3 C2 + C3 , ds 2s ds 2s 2s where C1 , C2 , C3 are easily read off from (5.4). Next, integrating twice, we obtain:   l∞ log (s + 1) log Z (s) = log det ∆ −(1 − s 2 ) − (s)k(, χ ) log Y −

] [∞ : ∞ 2 C2 log s + C3 s 3 + D1 + s 2 D2 , − (s) − sC1 + 2 3 where D1 , D2 are constants of integration. They can be determined by letting s → ∞, and applying
Lemma 4.4.  More specifically, Lemma 4.4 tells us the asymptotic growth of log det ∆ −(1 − s 2 ) as s → ∞. Noting that lim log Z (s) = 0,

s→∞

lim (s) = 0,

s→∞

and applying Sterling’s formula we obtain: D1 = γ + log(4) − 2 +

√ l∞ log 2π ,

] [∞ : ∞

D2 = 0.

We have proved Theorem 5.2. For Re(s) > 2,   log det ∆ −(1 − s 2 ) = log Z (s, , χ ) + s (k(, χ ) log(Y ) + C1 ) +

C2 2 l∞ log (s + 1)+(s)− log s − C3 s 3 − D1 ,

[∞ : ∞ ] 2 3

where C1 =

 tr V χ (R) log N (T0 ) {R}nce

πk 4| E(R) | sin2 ( m(R) )

⎞⎤ ⎡ ⎛ l n   η  2 tr χ (g ) log |c | 1 i i ⎝l∞ ∞ −γ + + L( ∞ , ψl )⎠⎦ , +⎣ 2 |2 [ :  ] 2 |C(g )||1 −  ∞ i ∞ i i=1 l=l∞ +1   l∞ , C2 = tr S(0) −

] [∞ : ∞ 
vol  \ H3 dim V C3 = , 4π and √ l∞ D1 = γ + log(4) − 2 + log 2π.

[∞ : ∞ ]

Regularized Determinants of the Laplacian

679

Corollary 5.3. Let  be torsion-free with one cusp at ∞, and let χ be a regular character (a one-dimensional unitary representation). Then


3   2 3 vol  \ H + s L( ∞ , ψ) . det ∆ −(1 − s ) = Z (s, , χ ) exp −s 6π Corollary 5.4. Let  be cocompact, and let χ be a regular character. Then


3   2 3 vol  \ H + sC , det ∆ −(1 − s ) = Z (s, , χ ) exp −s 6π where C=

 tr V χ (R) log N (T0 ) {R}nce

πk 4| E(R) | sin2 ( m(R) )

.

Appendix A. Proof of Theorem 3.6 In this section we prove Theorem 3.6, that is for P = z + r j, P = z + r j ∈ F Y , we show that   exp(−t) log2 (r/r ) K  (P, P , t, χ ) = P∞ rr

+ O(1). exp − √ 4t |P| 4π t As usual we are under Assumption 3.1. The first step is to split up K  (P, P , t, χ ) as  χ (γ )kt (δ(P, γ P )) K  (P, P , t, χ ) = γ ∈



=

χ (γ )kt (δ(P, γ P )) +

γ ∈∞



χ (γ )kt (δ(P, γ P )).

γ ∈\∞

From [EGM98, Eq. 4.5.9 or Lemma 6.4.2], it follows that  |kt (δ(P, γ P ))| = O(1) as r → ∞. γ ∈\∞

Hence



χ (γ )kt (δ(P, γ P )) = O(1) as r → ∞.

γ ∈\∞

It remains to estimate f ∞ (P, P ) ≡



χ (γ )kt (δ(P, γ P )).

γ ∈∞

The subgroup ∞ is not an abelian group. So in general we can not diagonalize the unitary representation, χ restricted to ∞ , into unitary characters. However, we have the following lemma which is almost as good as diagonalizing χ (see [Fri05b, Lemma 2.4]).

680

J. S. Friedman

Lemma A.1. Let ∞ be the one cusp of  (Assumption 3.1). Then there exist E, R, S ∈ ∞ with the following properties: (1) ∞ = {E k R i S j | 0 ≤ k < m, i, j ∈ Z}. Here R, S are parabolic elements with R(P) = P + 1 and S(P) = P + τ (here ∞ = Z ⊕ Zτ ) for all P ∈ H3 , and E is elliptic of order m.

= {R i S j | i, j ∈ Z}. (2) ∞ (3) The elements R and S commute, but the group ∞ is not abelian when m > 1.

onto itself. Furthermore, there exists a (4) If in addition, m > 1, then χ (E) maps V∞

basis of V∞ so that χ (E)|V∞ is diagonal. The notation used in the lemma above is explained in §3.1. Next we split f ∞ into two sums,   f ∞ (P, P ) = P∞ χ (γ )kt (δ(P, γ P )) + (I V − P∞ ) χ (γ )kt (δ(P, γ P )). γ ∈∞

γ ∈∞

Lemma A.2 (Poisson Summation Formula). Let f : R2 → C be a Schwartz-class function, and let be a two-dimensional lattice in R2 = C. Then 

f (ω) =

ω∈

1   f (ω). | | 0 ω∈

Here 0 is the dual lattice to , and  f is the Fourier transform of f ,   f (z) = f (u)e−2πiu,z du, R2

where ·, · is the standard real inner product on C = R2 . Lemma A.3.  γ ∈∞

  exp(−t) log2 (r/r ) + O(1). P∞ χ (γ )kt (δ(P, γ P )) = P∞ rr exp − √ 4t |P| 4π t



Proof. It follows from the definition of singularity that for γ ∈ ∞ , P∞ χ (γ ) = P∞ . Thus   P∞ χ (γ )kt (δ(P, γ P )) = P∞ kt (δ(P, γ P )). γ ∈∞

γ ∈∞

Using Part 1 of Lemma A.1, we can rewrite the above sum as P∞

m  

kt (δ(P, E k R i S j P )) = P∞

k=0 i, j∈Z2

m  

kt (δ(E −k P, R i S j P )).

k=0 i, j∈Z2

The last equality follows because δ is a point-pair invariant. Once again applying Lemma A.1, and the definition of δ we can write the above sum as P∞

m   k=0 ω∈ ∞

f k (ω),

Regularized Determinants of the Laplacian

where

 f k (u) = kt

681

|z − z(E k P ) + u|2 + r 2 + r 2 2rr

 .

We have used8 the fact that r (E k P) = r (P) = r . Next we apply the Poisson summation to obtain   f k (ω) = | ∞ |−1 fˆk (0) + | ∞ |−1 fˆk (ω). ω∈ ∞

ω∈ ∞ ω=0

A straightforward computation [EGM98, Lemma 3.5.5] shows9 that   r  fˆk (0) = rr g log , r where  2 exp(−t) x . g(x) = √ exp − 4t 4π t Noting that fˆk (0) is independent of k, and that

] |E| [∞ : ∞ 1 m = = = , | ∞ | | ∞ | | ∞ | |P|

we recover the leading term of the lemma. To conclude we show that  fˆk (ω) = O(1). ω∈ ∞ ω=0

Since f k is smooth, fˆk is of rapid decay, hence [EGM98, Lemma 6.4.2] fˆk (v) = O((rr )−N |v|−2N ) for any N > 0. The lemma now follows.

(A.1)

 

Equation A.1 will be used to show that  (I V − P∞ ) χ (γ )kt (δ(P, γ P )) = O(1) as r → ∞. γ ∈∞

In order to proceed, we need to understand I V − P∞ as a projection operator on the ⊥ . First, decompose V into subspace V∞

b ⊕ V∞ . V = V∞ ⊕ V∞

By Lemma A.1 the unitary representation χ restricted to ∞ can be diagonalized in a

⊕ V b . Hence we block matrix form with respect to the decomposition V = V∞ ⊕ V∞ ∞

and P can write I V − P∞ = Pa + Pb , where Pa is the orthogonal projection onto V∞ b b is the orthogonal projection onto V∞ . 8 We abuse notation here and let r, z represent both coordinates in H3 and coordinate functions. 9 [EGM98, Lemma 6.4.2] seems to have an extra factor of “2.”

682

J. S. Friedman

Lemma A.4.



(I V − P∞ ) χ (γ )kt (δ(P, γ P )) = O(1).

γ ∈∞

Proof. We first show that 

Pa χ (γ )kt (δ(P, γ P )) = O(1).

γ ∈∞

The proof is almost identical to the proof of Lemma A.3 except that the term corresponding to ω = 0 is zero. Indeed, Pa

m  

χ (E k R i S j )kt (δ(P, E k R i S j P ))

k=0 i, j∈Z2

=

m 



χ (E)

k

Pa



kt (δ(P, E k R i S j P )),

i, j∈Z2

k=0

where χ (E) is a diagonal matrix with each element on the diagonal a finite root of unity not equal to one. Since the order of each root of unity divides m we must have m 

χ (E)k = 0.

k=0

Hence the “constant term” (the term corresponding to ω = 0 cancels out). Next, to show that  Pb χ (γ )kt (δ(P, γ P )) γ ∈∞

is bounded it suffices to estimate the lattice sum  kt (δ(P, R l S j P )). Pb

(A.2)

l, j∈Z2

) restricted to V b . Hence we can Since R and S commute, we can diagonalize χ (∞ ∞ assume that χ is a lattice character of the form

χ (R l S n ) = exp(2πi(lθ R + nθ S )). Now we can rewrite (A.2) as  exp(2πi(lθ R + nθ S ))kt (δ(z + r j, l + nτ (z + r j))). l,n∈Z2

By unraveling the definition of Pb , it follows that at least one of θ R , θ S is not an integer. By applying the Poison summation formula to the function f 1 (w, v) = exp(2πi(wθ R + vθ S ))kt (δ(z + r j, w + vτ (z + r j))),

Regularized Determinants of the Laplacian

we obtain



683

f 1 (l, n) =

l,n∈Z2



 f 1 (l, n).

l,n∈Z2

However the exponential factor exp(2πi(lθ R + nθ S )) shifts the Fourier transform of the function f 2 (w, v) = kt (δ(z + r j, w + vτ (z + r j))), and wipes out the unbounded “constant term”  f 2 (0, 0). In other words, if we applied the Poisson summation formula to f 2 , we would see, as we did with f k , that    f 2 (l, n) =  f 2 (0, 0) + f 2 (l, n). l,n∈Z2

l,n∈Z2 (l,n)=(0,0)

The latter sum decays, while the first term would not. The effect of multiplying exp(2πi(wθ R + vθ S )) is to shift the sum away from the integers. That is    f 2 (l + α, n + β), f 1 (l, n) = l,n∈Z2

l,n∈Z2

where (0, 0) ∈ / Z2 + (α, β). Hence we can apply (A.1) to conclude the lemma.

 

Acknowledgements. I would like to thank Professor Leon Takhtajan for originally suggesting this problem to me, for reading over this paper, and for useful discussions. I would also like to thank Professor Werner Müller for answering some questions of mine related to the writing of this paper, and the anonymous referee, for pointing out some important errors, and for suggesting very useful comments.

References [BCZ97]

Bytsenko, A.A., Cognola, G., Zerbini, S.: Determinant of the laplacian on a non-compact three-dimensional hyperbolic manifold with finite volume. J. Phys. A 30(10), 3543–3552 (1997) [BJP] Borthwick, D., Judge, C., Perry, P.: Determinants of laplacians and isopolar metrics on surfaces of infinite area. Duke Math. J. 118, 69–102 (2003) [BK62] Birman, M.Š., Kre˘ın, M.G.: On the theory of wave operators and scattering operators. Dokl. Akad. Nauk SSSR 144, 475–478 (1962) [Dav89] Davies, E.B.: Heat kernels and spectral theory. Cambridge Tracts in Mathematics, Vol. 92. Cambridge: Cambridge University Press, 1989 [DC76] Dowker, J.S., Critchley, R.: Scalar effective lagrangian in de sitter space. Phys. Rev. D (3) 13(2), 224–234 (1976) [DP86] D’Hoker, E., Phong, D.H.: On determinants of laplacians on riemann surfaces. Commun. Math. Phys. 104(4), 537–545 (1986) [Efr88] Efrat, I.: Determinants of laplacians on surfaces of finite volume. Commun. Math. Phys. 119(3), 443–451 (1988) [Efr91] Efrat, I.: Erratum: Determinants of Laplacians on surfaces of finite volume [Commun. Math. Phys. 119 (3), 443–451 (1988)]. Commun. Math. Phys. 138(3):607 (1991) [EGM98] Elstrodt, J., Grunewald, F., Mennicke, J.: Groups acting on hyperbolic space. Springer Monographs in Mathematics, Berlin: Springer-Verlag, 1998 [EOR+ 94] Elizalde, E., Odintsov, S.D., Romeo, A., Bytsenko, A.A., Zerbini, S.: Zeta regularization techniques with applications. River Edge, NJ: World Scientific Publishing Co. Inc., 1994 [Fri05a] Friedman, J.S.: The Selberg trace formula and Selberg zeta-function for cofinite Kleinian groups with finite-dimensional unitary representations. Ph.D. thesis, Stony Brook University, 2005, available at http://arxiv.org/abs/math.NT/0612807, 2006

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J. S. Friedman

Friedman, J.S.: The selberg trace formula and selberg zeta-function for cofinite kleinian groups with finite-dimensional unitary representations. Math. Z. 250(4), 939–965 (2005) Gradshteyn, I.S., Ryzhik, I.M.: Table of integrals, series, and products. Fourth edition prepared by Ju. V. Geronimus and M. Ju. Ce˘ıtlin. Translated from the Russian by Scripta Technica, Inc. Translation edited by Alan Jeffrey, New York: Academic Press, 1965 Hawking, S.W.: Zeta function regularization of path integrals in curved spacetime. Commun. Math. Phys. 55(2), 133–148 (1977) Kre˘ın, M.G.: On the trace formula in perturbation theory. Mat. Sbornik N.S. 33(75), 597–626 (1953) Müller, W.: Spectral theory for riemannian manifolds with cusps and a related trace formula. Math. Nachr. 111, 197–288 (1983) Müller, W.: Manifolds with cusps of rank one, Lecture Notes in Mathematics, Vol. 1244. Berlin: Springer-Verlag, 1987 Müller, W.: Spectral geometry and scattering theory for certain complete surfaces of finite volume. Invent. Math. 109(2), 265–305 (1992) Müller, W.: Relative zeta functions, relative determinants and scattering theory. Commun. Math. Phys. 192(2), 309–347 (1998) Park, J.: Eta invariants and regularized determinants for odd dimensional hyperbolic manifolds with cusps. Amer. J. Math. 127(3), 493–534 (2005) Sarnak, P.: Determinants of laplacians. Commun. Math. Phys. 110(1), 113–120 (1987) Venkov, A.B.: Spectral theory of automorphic functions. Proc. Steklov Inst. Math. (1982), no. 4(153), ix+163 pp. (1983), a translation of Trudy Mat. Inst. Steklov. 153 (1981) Venkov, A.B., Kalinin, V.L., Faddeev, L.D.: A nonarithmetic derivation of the selberg trace formula. Zap. Nauˇcn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 37, 5–42 (1973) Yafaev, D.R.: Mathematical scattering theory. Translations of Mathematical Monographs, Vol. 105. Amer. Math. Soc. 1992 Translated from the Russian by J. R. Schulenberger

Communicated by P. Sarnak

Commun. Math. Phys. 275, 685–708 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0296-1

Communications in

Mathematical Physics

Hidden Symmetries and Integrable Hierarchy of the N = 4 Supersymmetric Yang-Mills Equations Alexander D. Popov
, Martin Wolf

Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany. E-mail: [email protected]; [email protected] Received: 7 September 2006 / Accepted: 21 February 2007 Published online: 15 August 2007 – © Springer-Verlag 2007

Abstract: We describe an infinite-dimensional algebra of hidden symmetries of N = 4 supersymmetric Yang-Mills (SYM) theory. Our derivation is based on a generalization of the supertwistor correspondence. Using the latter, we construct an infinite sequence of flows on the solution space of the N = 4 SYM equations. The dependence of the SYM fields on the parameters along the flows can be recovered by solving the equations of the hierarchy. We embed the N = 4 SYM equations in the infinite system of the hierarchy equations and show that this SYM hierarchy is associated with an infinite set of graded symmetries recursively generated from supertranslations. Presumably, the existence of such nonlocal symmetries underlies the observed integrable structures in quantum N = 4 SYM theory. 1. Introduction Let us consider an open subset P 3 := P 3 \ P 1 of the three-dimensional complex projective space P 3 together with the following double fibration: 4

P3

×



P1 @ R @

(1.1) 4

Here, 4 is complexified Minkowski space and P 3 its associated twistor space [1]. It can be shown that P 3 is a rank 2 holomorphic vector bundle P 3 = O(1) ⊕ O(1) over the Riemann sphere P 1 , where O(n) denotes the holomorphic line bundle O(n) → P 1 parametrized by the first Chern number n ∈ . The importance of the diagram (1.1) lies





On leave from Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Russia.



Address after October 1st, 2006: Theoretical Physics Group, The Blackett Laboratory, Imperial College

London, Prince Consort Road, London SW7 2BW, United Kingdom.

686

A. D. Popov, M. Wolf

in the fact that there is a correspondence between holomorphic vector bundles over P 3 obeying certain triviality conditions and holomorphic vector bundles over 4 equipped with a connection satisfying the self-dual Yang-Mills (SDYM) equations on 4 [2]. This map between vector bundles has been termed the Penrose-Ward transform [3]. Substituting the twistor space P 3 by Pn3 := O(n) ⊕ O(n) (for n > 1) which has the same dimension but different topology, one obtains the double fibration 2(n+1)

Pn3



× P1 @ R @

(1.2)

2(n+1)

and the above-mentioned Penrose-Ward transform is generalized to a correspondence between holomorphic vector bundles over Pn3 and solutions to a system of equations (called the SDYM hierarchy truncated up to level n [3]) for Yang-Mills-Higgs fields on the space 2(n+1) ∼ (1.3) = 4 × 2(n−1) . Here, 4 is interpreted as (complexified) space-time and 2(n−1) as a space of “higher times” parametrizing solutions to the SDYM equations which in turn appear as a subset of the truncated SDYM hierarchy [3–8]. Letting n tend to infinity, one obtains the full SDYM hierarchy which is associated with an affine extension of translation symmetries [3, 9–12]. The above construction can be generalized to N -extended supersymmetric SDYM theory [13] by substituting Pn3 by a generalized supertwistor space [14], Pn3|N := O(n) ⊗

2

⊕ O(n) ⊗

N

,

(1.4)

associated with the superspace 2(n+1)|2N extending the space (1.3). The operator  inverts the Grassmann parity of the fibre coordinates. In a seminal recent paper [15], 3|4 Witten observed that the supertwistor space P 3|4 := P1 is a Calabi-Yau supermanifold and showed that B-type open topological string theory with the supertwistor space as target space (twistor string theory1 ) is equivalent to holomorphic Chern-Simons (hCS) theory2 on the same space. This theory is in turn equivalent to N = 4 SDYM theory in four dimensions. Generalizing an earlier construction [17], it was also shown in [15] that it is possible to recover perturbative N = 4 supersymmetric Yang-Mills (SYM) theory in terms of integrals over moduli spaces of algebraic curves (D-instantons) in the supertwistor space P 3|4 . For an account of progress made in this area, see e.g. [18, 19] and references therein. For other aspects of twistor string theories discussed lately see e.g. [20–29]. A twistor construction was developed not only for (supersymmetric) SDYM theory but also for full N -extended SYM theory [30–35] with 0 ≤ N ≤ 4 (for recent reviews see [21, 29]). Recall that the twistor description of N = 4 SYM theory is based on the N = 3 superspace formulation of the latter [30]. For that one considers the supertwistor 3|3 space P 3|3 as in (1.4) with n = 1, N = 3 and the dual supertwistor space P∗ = 3|3 1|3 P∗ \ P∗ = O(1) ⊗ 2∗ ⊕ O(1) ⊗ 3∗ which is fibred over P∗1 . Let πα˙ ˙ 2˙ and ρα with α = 1, 2 be homogeneous coordinates on P 1 and P∗1 , with α˙ = 1, 1 For other variants of twistor string models see [16]. 2 Holomorphic Chern-Simons theory describes (inequivalent) holomorphic structures on a vector bundle over a given complex (super)manifold, which in the case at hand is P 3|4 .

Hidden Symmetries and Integrable Hierarchy of the N = 4 Supersymmetric Yang-Mills Equations

687

respectively. Then the superambitwistor space, denoted by L5|6 , is defined as a quadric surface z α ρα − w α˙ πα˙ + 2θ i ηi = 0 (1.5) in P 3|3 × P∗ [30]. Here, (z α , πα˙ , ηi ) and (w α˙ , ρα , θ i ) are homogeneous coordinates 3|3 on P 3|3 and P∗ , respectively, and ηi , θ i are Grassmann odd variables with i = 1, 2, 3. Interestingly, L5|6 is a Calabi-Yau supermanifold and it is possible to formulate twistor string theory on it [15]. Note that if pr1 and pr2 are the two projections from P 1 × P∗1 onto P 1 and P∗1 , then the space L5|6 can be defined as a holomorphic vector bundle over P 1 × P∗1 in terms of a short exact sequence of vector bundles 3|3

0 → L5|6 → pr ∗1 P 3|3 ⊕ pr ∗2 P∗

3|3

κ

→ O(1, 1) → 0

(1.6)

generalizing the analogous sequence for the purely bosonic case [36, 37]. The mapping κ is given by κ(z α , πα˙ , ηi , w α˙ , ρα , θ i ) := z α ρα −w α˙ πα˙ +2θ i ηi and O(1, 1) := pr ∗1 O(1)⊗ pr ∗2 O(1). The corresponding long exact cohomology sequence gives H 1 ( P 1 × P∗1 , L5|6 ) = 0 and H 0 ( P 1 × P∗1 , L5|6 ) = 4|12 . Hence, by virtue of the Kodaira theorem [38] we obtain the double fibration F 6|12 @ R @

L5|6

(1.7) 4|12

where F 6|12 := 4|12 × P 1 × P∗1 is called correspondence space and 4|12 is complexified N = 3 Minkowski superspace appearing as the (4|12)-dimensional moduli space of global holomorphic sections of the bundle L5|6 → P 1 × P∗1 [30, 32, 34, 39]. Furthermore, there is a one-to-one correspondence between gauge equivalence classes of solutions to the N = 3 SYM equations in four dimensions and equivalence classes of holomorphic vector bundles3 E over the quadric L5|6 such that the vector bundles E are holomorphically trivial on each submanifold Yx˜ ∼ = P 1 × P∗1 → L5|6 with 4|12 x˜ ∈ [30, 32, 34]. Vector bundles E having such properties are sometimes called 4|12 -trivial [32]. 5|6 In this paper, we introduce a generalized superambitwistor space Lm,n via the short exact sequence κm,n

6|6 0 → L5|6 m,n → Cm,n → O(m, n) → 0,

(1.8)

where the holomorphic vector bundles 6|6 := [O(m, n −1)⊕O(m −1, n)]⊗ Cm,n

and

2

⊕[O(m, 0)⊕O(0, n)]⊗

O(m, n) := pr ∗1 O(m) ⊗ pr ∗2 O(n)

are fibred over

P1

×

P∗1 .

3

(1.9) (1.10)

Here, the map κm,n is defined as

κm,n : (z α , w α˙ , πα˙ , ρα , ηi , θ i ) → z α ρα − w α˙ πα˙ + 2θ i ηi ,

(1.11)

3 In the Dolbeault picture, these classes are described by gauge equivalence classes of solutions to the equations of motion of hCS theory on L5|6 .

688

A. D. Popov, M. Wolf

where (z α , w α˙ , πα˙ , ρα , ηi , θ i ) are homogeneous coordinates on Cm,n . Clearly, for m = 5|6 3|3 6|6 n = 1 the sequence (1.8) reduces to (1.6), i.e. L5|6 ≡ L1,1 and pr ∗1 P 3|3 ⊕pr ∗2 P∗ ≡ C1,1 . One of the goals of our paper is to establish a supertwistor correspondence involving 5|6 the space Lm,n . We first discuss its geometry and the associated double fibration 6|6

F M+2|N π2 @ π1 R @

5|6

Lm,n with F M+2|N := M|N

:=

M|N

×

P1 ×

(1.12) M|N

P∗1 and

3mn+m+n−1|3(m+n+2)

∼ =

4|12

×

3mn+m+n−5|3(m+n−2)

(1.13)

containing 4|12 as a subspace. Afterwards, we will describe a one-to-one correspondence between the moduli space of solutions to SYM-type equations on the space (1.13) 5|6 and the moduli space of M|N -trivial holomorphic vector bundles over Lm,n . We shall M|N call the obtained Yang-Mills-Higgs type equations on the SYM hierarchy truncated up to level (m, n). In addition, we show that the N = 3 SYM equations are contained in this system of differential equations. The full SYM hierarchy is introduced as an asymptotic limit n → ∞ after putting m = n. Further, we show that the SYM hierarchy is associated with an infinite-dimensional algebra, denoted by T 4|12 [λ, µ], with generators ˙

µa λb Pα β˙ ,

µa Q iα

and

˙

λb Q iβ˙

(1.14)



for a, b˙ ∈ . Here, λ and µ are local coordinates on P 1 × P∗1 and Pα β˙ , Q iα and Q iα˙ are the generators of the algebra T 4|12 of supertranslations on 4|12 . We have
 ˙ j ˙ j µa Q iα , λb Q β˙ = −2δi µa λb Pα β˙

(1.15)

as the only nonvanishing (anti)commutation relations of T 4|12 [λ, µ]. For a Minkowski metric on space-time and proper reality conditions on the extra coordinates, we will obtain a corresponding real form of the above algebra. Notice that T 4|12 [λ, µ] is a subalgebra of an algebra T 4|12 [λ, λ−1 , µ, µ−1 ] with the same generators (1.14) but for a, b˙ ∈ . In addition, we demonstrate that T 4|12 [λ, µ] generates hidden symmetries of the N = 3 SYM equations. In this respect, we again emphasize that the N = 3 and N = 4 SYM theories are actually equivalent, so we will not make any distinction between these theories and also refer to them interchangeably. The symmetries under consideration are in fact point symmetries of the SYM hierarchy and the dependence of the space-time components of the SYM fields on the moduli parametrizing higher flows can be recovered by solving a part of the infinite set of equations of the SYM hierarchy. In other words, the generators of the algebra T 4|12 [λ, µ] are realized as (super)derivatives along “higher times” and they generate tangent vectors to the infinite-dimensional space of solutions to the N = 4 SYM equations. The existence of such symmetries could play an important role in quantum integrability of N = 4 SYM theory.




Hidden Symmetries and Integrable Hierarchy of the N = 4 Supersymmetric Yang-Mills Equations

689

2. Generalized Superambitwistor Space 5|6

Moduli space. To clarify the geometry of Lm,n , we use the exact sequence (1.8). This sequence induces a long exact cohomology sequence κm,n

0 6|6 0 0 → H 0 (Y, L5|6 m,n ) → H (Y, Cm,n ) → H (Y, O(m, n)) → 1 6|6 1 → H 1 (Y, L5|6 m,n ) → H (Y, Cm,n ) → H (Y, O(m, n)) → · · · ,

(2.1)

where Y := P 1 × P∗1 and the map κm,n is defined by the formula (1.11).4 Using 6|6 Künneth’s formula, we have H 1 (Y, Cm,n ) = 0 = H 1 (Y, O(m, n)) for m, n ≥ 1. Furthermore, the map κm,n is surjective. Therefore, the sequence (2.1) yields a short exact sequence 0 6|6 0 0 → H 0 (Y, L5|6 m,n ) → H (Y, Cm,n ) → H (Y, O(m, n)) → 0.

(2.2)

In addition, we obtain 6|6 ) = H 0 (Y, Cm,n

4mn+2m+2n|3(m+n+2)

,

H 0 (Y, O(m, n)) =

(m+1)(n+1)

(2.3)

and from (2.2) and (2.3) we find H 0 (Y, L5|6 m,n ) =

with M := 3mn + m + n − 1 and N := 3(m + n + 2). (2.4) Thus, we conclude that the moduli space of global holomorphic sections of the fibration M|N

L5|6 m,n → Y is the (M|N )-dimensional superspace Decomposition of

M|N .

M|N

(2.5)

with M, N given in (2.4).

Let us consider the bosonic part of the sequence (1.8),

0 → L5|0 m,n → [O(m, n − 1) ⊕ O(m − 1, n)] ⊗

2

→ O(m, n) → 0.

(2.6) 5|0

Upon dualizing it and using the Euler sequence (see e.g. [40]), we conclude that Lm,n can be identified with the vector bundle Jet1 O(m, n) of first order jets of O(m, n). Moreover, by recalling the exact sequence (see e.g. [32]) 0 → T ∗ X ⊗ E → Jet1 E → E → 0,

(2.7)

where E is a vector bundle over a manifold X , we obtain 0 → O(m − 2, n) ⊕ O(m, n − 2) → L5|0 m,n → O(m, n) → 0,

(2.8)

∼ where we have substituted E = O(m, n) and T ∗ X = T ∗ Y = T ∗ ( P 1 × P∗1 ) = O(−2, 0) ⊕ O(0, −2) into (2.7). It is not difficult to show that the long exact cohomology sequence arising from the short exact sequence (2.8) reduces to 0 0 → H 0 (Y, O(m −2, n)⊕O(m, n −2)) → H 0 (Y, L5|0 m,n ) → H (Y, O(m, n)) → 0. (2.9) 4 By a slight abuse of notation, we use the same symbol κ m,n for both the map between vector bundles and the map between cohomology groups.

690

A. D. Popov, M. Wolf 5|0

As H 0 (Y, Lm,n ) = M|0

M|0 ,

∼ =

(2.9) implies the decomposition (m+1)(n+1)|0

(m+1)(n−1)|0

×

×

(m−1)(n+1)|0

(2.10)

of the bosonic part M|0 ⊂ M|N of the moduli space (2.4) of global holomorphic sections of the bundle (2.5). Using this natural decomposition, we choose the following bosonic coordinates on M|N : x α1 ···αn α˙ 1 ···α˙ m ,

t α1 ···αn−2 α˙ 1 ···α˙ m

and

s α1 ···αn α˙ 1 ···α˙ m−2 .

(2.11)

As fermionic coordinates, we may take ηiα˙ 1 ···α˙ m

and

θ iα1 ···αn .

(2.12)

Note that all of these coordinates are totally symmetric in their spinorial indices. 5|6

6|6

Geometry of the space Lm,n . As homogeneous coordinates on Cm,n we take (z α , w α˙ , πα˙ , ρα , ηi , θ i ) ∈ 8|6 subject to the identification (z α , w α˙ , πα˙ , ρα , ηi , θ i ) ∼ (t1m t2n−1 z α , t1m−1 t2n w α˙ , t1 πα˙ , t2 ρα , t1m ηi , t2n θ i )

(2.13)

for any pair (t1 , t2 ) ∈ ∗ × ∗ such that (πα˙ )t = (0, 0) and (ρα )t = (0, 0). Upon imposing the quadric constraint z α ρα − w α˙ πα˙ + 2θ i ηi = 0

(2.14) 5|6 Lm,n .

on these coordinates, we may also use them as coordinates on Note that this constraint can be solved on each of the four patches W p , with p, q, . . . = 1, . . . , 4, 5|6 6|6 covering Lm,n . Recall that Cm,n is a holomorphic vector bundle (1.9) over Y = P 1 × 1 P∗ and as such it can be covered by four patches U p = 4|6 × W p , where {W p } is the 5|6 5|6 6|6 standard acyclic covering of Y . We may choose W p = U p ∩ Lm,n since Lm,n ⊂ Cm,n . 6|6 In terms of the homogeneous coordinates on Cm,n , global holomorphic sections of the bundle (2.5) have the form z α1 = [x αR1 ···αn α˙ 1 ···α˙ m + α1 (α2 t α3 ···αn )α˙ 1 ···α˙ m ]ρα2 · · · ραn πα˙ 1 · · · πα˙ m , wα˙ 1 = [x Lα1 ···αn α˙ 1 ···α˙ m − s α1 ···αn (α˙ 3 ···α˙ m α˙ 2 )α˙ 1 ]ρα1 · · · ραn πα˙ 2 · · · πα˙ m , ηi =

ηiα˙ 1 ···α˙ m πα˙ 1

· · · πα˙ m ,

(2.15)

θ i = θ iα1 ···αn ρα1 · · · ραn .

˙ 1 ···α˙ m 1 ···αn α Furthermore, the moduli x αR,L appearing in (2.15) are not independent but satisfy

x αR1 ···αn α˙ 1 ···α˙ m − x Lα1 ···αn α˙ 1 ···α˙ m + 2θ iα1 ···αn ηiα˙ 1 ···α˙ m = 0

(2.16)

as a result of the quadric constraint (2.14). These equations can be solved in terms of the coordinates on M|N as ˙ 1 ···α˙ m α1 ···αn α˙ 1 ···α˙ m iα1 ···αn α˙ 1 ···α˙ m 1 ···αn α = x α1 ···αn α˙ 1 ···α˙ m ∓ γ R,L θ ηi , x αR,L

(2.17)

α1 ···αn α˙ 1 ···α˙ m obey where the constant factors γ R,L

γ Rα1 ···αn α˙ 1 ···α˙ m + γ Lα1 ···αn α˙ 1 ···α˙ m = 2. Below, we shall choose them appropriately.

(2.18)

Hidden Symmetries and Integrable Hierarchy of the N = 4 Supersymmetric Yang-Mills Equations

691 5|6

Double fibration. Recall that the sections (2.15) describe embeddings Yx˜ → Lm,n , where x˜ collectively denotes all the coordinates (x, t, s, η, θ ) ∈ M|N . On the other 5|6 hand, for each point  = (z α , w α˙ , πα˙ , ρα , ηi , θ i ) ∈ Lm,n , the incidence relations (2.15) 5|6 define a subspace M−3|N −6 in M|N . The correspondence between subspaces in Lm,n M|N and can be described by the double fibration (1.12) mentioned in the previous section. There, F M+2|N = M|N × P 1 × P∗1 and the projections π1 and π2 are defined by the formulae π1 : (x, t, s, πα˙ , ρα , ηiα˙ 1 ···α˙ m , θ iα1 ···αn ) → (x, t, s, ηiα˙ 1 ···α˙ m , θ iα1 ···αn ), π2 : (x, t, s, πα˙ , ρα , ηiα˙ 1 ···α˙ m , θ iα1 ···αn ) → (z α , w α˙ , πα˙ , ρα , ηi , θ i ),

(2.19)

where z α , w α˙ , ηi and θ i are given in (2.15). So, the diagram (1.12) describes the oneto-one correspondences 5|6

{submanifolds Yx˜ in Lm,n } 5|6 {points  in Lm,n }

←→ ←→

{points x˜ in M|N }, M−3|N −6 {subspaces  in

M|N },

(2.20) ∼ Yx˜ → L5|6 where a fixed point x˜ ∈ M|N corresponds to a submanifold π2 (π1−1 (x)) ˜ = m,n 5|6 and, conversely, a fixed point  ∈ Lm,n corresponds to a codimension 3|6 subspace M−3|N −6 π1 (π2−1 ()) ∼ → M|N . Note that the correspondence space F M+2|N can =  ˆ p := M|N × W p with local coordinates (2.11), (2.12) be covered by four patches W M|N and (λ( p) , µ( p) ) on W p , where {W p } forms the four-set acyclic covering of on P 1 × P∗1 . Parametrization of M|N . From now on, we specialize to the case when m = n and 5|6 5|6 denote Ln := Ln,n . The discussion for m = n can be given in an analogous way as for m = n. Furthermore, we switch from the spinorial notation used above to a polynomial one. This will allow us to write many formulae more concisely and to also take the limit n → ∞ in the field equations. So, let us now consider the double fibration π2

5|6 Ln

F M+2|N @ π1 R @

(2.21) M|N

with M = 3n 2 + 2n − 1 and N = 6(n + 1). In particular, we parametrize coordinates ˙ ˙ ˙ ˙ (x A B , t a B , s Ab , ηiA , θ i A ),

M|N

by the (2.22)

˙ B, ˙ . . . = 1, ˙ . . . , n˙ + 1˙ and a, b, . . . = 1, . . . , n − 1, where A, B, . . . = 1, . . . , n + 1, A, 6|6 6|6 ˙ ˙ ˙ a, ˙ b, . . . = 1, . . . , n˙ − 1. On the patch U1 of Cn := Cn,n defined by the conditions π1˙ = 0 and ρ1 = 0, we may choose the following (local) coordinates: α := z (1)

zα π1˙n ρ1n−1

,

3 z (1) :=

(1) ηi

π2˙ , π1˙

ηi := n , π1˙

α˙ w(1) :=

i θ(1)

w α˙ π1˙n−1 ρ1n

θi := n . ρ1

,

˙

3 w(1) :=

ρ2 , ρ1 (2.23)

692

A. D. Popov, M. Wolf

3 = λ Note that by virtue of the projection π2 appearing in (2.21) we can set z (1) (1) and ˙

3 = µ . Locally, the sections (2.15) over W ⊂ Y then take the form5 w(1) 1 (1)

z1 =

n+ ˙ 1˙ 

 ˙ x 1RB

˙ 1˙ B=  n+ ˙ 1˙

z

=

2

˙

w1 =

n+1  A=1

˙

w2 =

a=1



˙ 1˙ B=

n+1 

1 2

n+ ˙ 1˙ 

n−1 



B˙ (x a+1 R

−t

a B˙

)µ

a−1

+

B˙ n−1 x n+1 µ R

a=1

⎛

⎝x LA1˙ +

1 2

⎛

n− ˙ 1˙ 

˙

˙

λ B−1 ,

⎞ ˙ ˙

˙

˙

(x LAb+1 + s Ab )λb ⎠ µ A−1 ,

(2.24)

˙ 1˙ b=

⎞ n− ˙ 1˙  ˙ ˙ ˙ ˙ ˙ ˙ ˙ 1˙ n− ⎝1 (x LAb+1 − s Ab )λb−1 + x LAn+ λ ˙ 1 ⎠ µ A−1 , 2 ˙ 1˙ b=

A=1

ηi =

 n−1  ˙ ˙ ˙ ˙ B + 21 (x a+1 + t a B )µa λ B−1 , R

˙

˙ ˙

ηiA λ A−1

n+1 

θi =

and

˙ 1˙ A=

θ i A µ A−1 .

A=1

Similar expressions can also be written for the other patches W p with p = 2, 3, 4. However, for illustrating purposes, we will mostly write formulae only for the patches ˆ 1 . As before, (2.14) implies W1 , W1 and W ˙

˙

˙

x RA B − x LA B + 2θ i A ηiB = 0,

(2.25)

which can be solved by putting ˙

˙

˙

˙

AB AB i A B x R,L = x A B ∓ γ R,L θ ηi ˙

(2.26)

˙

with γ RA B + γ LA B = 2. We shall specify these factors in a moment. Among the coordinates (2.22) on M|N we will single out those which correspond to coordinates on complexified N = 3 Minkowski superspace 4|12 entering in the decomposition M|N

=

3n 2 +2n−1|6(n+1)

∼ =

4|12

×

3n 2 +2n−5|6(n−1)

.

(2.27)

The remaining coordinates are then interpreted as additional parameters (moduli) or “higher times” from the viewpoint of N = 3 SYM theory. The decomposition (2.27) ˙ = (α, ˙ with means the decomposition of the indices (A) = (α, a + 2) and ( A) ˙ a˙ + 2) ˙ ˙ ˙ ˙ α = 1, 2, a = 1, . . . , n − 1 and α˙ = 1, 2, a˙ = 1, . . . , n˙ − 1. Using this, we denote the coordinates on 4|12 by ˙ (x α β , ηiα˙ , θ iα ) (2.28) and those on

3n 2 +2n−5|6(n−1) ˙ ˙

˙

by ˙ ˙

˙

˙

˙

˙ ˙

˙

˙ 2 ia+2 (x α b+2 , x a+2 β , x a+2 b+2 , t α b , t a+2 b , s a β , s a b+2 , ηia+ ,θ ). 5 We suppress (in most cases) the patch indices.

(2.29)

Hidden Symmetries and Integrable Hierarchy of the N = 4 Supersymmetric Yang-Mills Equations

693

Integrable distribution T . Below, we shall establish a correspondence between holo5|6 morphic vector bundles6 over Ln and solutions to the equations of the SYM hierarchy truncated up to level n. Since the lowest level flows of the hierarchy will – by construction – correspond to supertranslations on 4|12 , the N = 3 SYM equations are embedded into the SYM hierarchy. Moreover, the set of vector fields spanning the tangent spaces to the leaves of the fibration F M+2|N → L5|6 (2.30) n must contain the vector fields λµ

∂ ∂ x 11˙

−µ

∂ ∂ x 12˙

with

−λ

∂ ∂ x 21˙

+

∂

, µDi1 − Di2 and λD1i˙ − D2i˙

∂ x 22˙

∂ β˙ ∂ + ηi iα ∂θ ∂ x α β˙

Diα :=

Dαi˙ :=

and

∂ ∂ + θ iβ β α˙ α ˙ ∂x ∂ηi

(2.31)

(2.32)

entering into the linear system for the N = 3 SYM equations. To implement this, we A B˙ such that need to choose the factors γ R,L α β˙

˙

β˙

˙ ˙

˙ ˙

x Lα b+2 = x α b+2 + 2θ iα ηib+2 ,

x R,L = x α β ∓ θ iα ηi , x αR b+2 = x α b+2 , a+2 β˙

xR

˙ ˙

˙ ˙

β˙

˙

a+2 β˙

= x a+2 β − 2θ ia+2 ηi ,

˙ ˙

˙ ˙

xL

˙ ˙

˙

= x a+2 β ,

(2.33)

˙ ˙

b+2 = x a+2 b+2 ∓ θ ia+2 ηib+2 . x a+2 R,L

In the sequel, we denote the relative holomorphic tangent bundle of the fibration (2.30) by T .7 It is defined by the short exact sequence 0 → T → T F M+2|N → π2∗ T L5|6 → 0. n

(2.34)

In other words, by dualizing this sequence, we obtain the sheaf 1 (F M+2|N )/π2∗ 1 (Ln ) of relative differential one-forms on F M+2|N . Short calculations reveal that for n > 1 the bosonic part of the distribution T is spanned by the vector fields 5|6

D A B˙ = λµ

Dn+1 A˙ D An+ ˙ 1˙ Ta B˙

∂ ∂ x A B˙

−µ

∂ ˙ 1˙ ∂ x A B+

−λ

∂ ∂ x A+1 B˙

+

∂ ˙ 1˙ ∂ x A+1 B+

∂ ˙ ˙ = µα λβ =: µα λβ Dα β˙ A B˙ , (2.35a) ˙ β− ˙ 1˙ A+α−1 B+ ∂ x 



∂ ∂ ∂ n˙ 1˙ ∂ − =: λα˙ Dn+1 A˙ α˙ , (2.35b) =λ − δ + δ ˙ 1˙ A˙ A˙ ˙ 1˙ ∂s 1 n− ∂s 11˙ ∂ x 1 A˙ ∂ x 1 A+ 



∂ ∂ ∂ n 1 ∂ − =: µα Dα An+ =µ − δ + δ A 11˙ A n−1 1˙ ˙ 1˙ , (2.35c) ∂ x A1˙ ∂t ∂ x A+1 1˙ ∂t ∂ ∂ ∂ ˙ ˙ =λ − = λβ =: λβ Da B˙ β˙ for a ≤ n − 1, ˙ ˙ 1˙ ˙ β− ˙ 1˙ a B a B+ a B+ ∂t ∂t ∂t (2.35d)

6 In the Dolbeault picture, they can be described by hCS theory on L5|6 . n 7 We shall use the same letter T for both the bundle and the distribution generated by its sections.

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Ta n+ ˙ 1˙ = µ

∂

−

∂t a 1˙

∂

= µα

∂t a+1 1˙

∂ ∂t a+α−1 1˙

=: µα Dαa n+ ˙ 1˙ for a ≤ n − 2, (2.35e)

S Ab˙ = µ

∂

−

∂s Ab˙

∂ ∂s A+1 b˙

= µα

∂ ∂s A+α−1 b˙

˙ =: µα Dα Ab˙ for b˙ ≤ n˙ − 1,

(2.35f) ∂ ∂ ∂ α˙ α˙ ˙ Sn+1a˙ = λ 1 a˙ − = λ =: λ Dn+1a˙ α˙ for a˙ ≤ n˙ − 2, ˙ 1˙ ˙ α− ˙ 1˙ ∂s ∂s 1 a+ ∂s 1 a+ (2.35g) ˙ B, ˙ . . . ≤ n˙ and where A, B, . . . ≤ n and A,



 λ µ (λα˙ ) := and (µα ) := . (2.36) −1 −1 For n = 1, the bosonic part of T is spanned by the vector field D11˙ =: D from (2.35a). In addition, the fermionic part of the distribution T is spanned by the vector fields Vi A = µDi A − Di A+1 =: µα Diα A

α˙ i V Ai˙ = λD iA˙ − D iA+ ˙ 1˙ =: λ Dα˙ A˙ ,

and

(2.37) where Diα and

Dαi˙

were already given in (2.32) and

Dia+2 = i Da+ = ˙ 2˙

∂

β˙

∂θ ia+2 ∂

˙

˙ 2 ∂ηia+

+ 2ηi

+ 2θ iβ

∂ ∂ x a+2 β˙ ∂

˙ 2˙ ∂ x β a+

˙ ˙

+ ηib+2 +θ

∂

˙ 2˙ ∂ x a+2 b+ ∂ ib+2

, (2.38)

, ˙

˙ 2 ∂ x b+2 a+

ˆ 1, ˙ Besides the expressions (2.35) and (2.37) on W for a, b ≤ n − 1 and a, ˙ b˙ ≤ n˙ − 1. ˆ 2, W ˆ3 one can easily write down the basis vector fields of T on the remaining patches W M+2|N ˆ ˆ and W4 covering together with W1 the correspondence space F . The only nonvanishing (anti)commutators among the above vector fields are j j Vi A , VB˙ = 2δi D A B˙ . (2.39) Hence, the distribution T is integrable. To homogenize notation, we define ˙

D I  := µα λβ Dα β˙ I  ,

D I˙ := µα Dα I˙

and

D I := λα˙ D I α˙

(2.40)

with {D I  } = {D A B˙ } {D I˙ } = {S Ab˙ , Ta n+ ˙ 1˙ , D An+ ˙ 1˙ } {D I } = {Ta B˙ , Sn+1a˙ , Dn+1 A˙ }

and and and

{Dα β˙ I  } = {Dα β˙ A B˙ }, {Dα I˙ } = {Dα Ab˙ , Dαa n+ ˙ 1˙ , Dα An+ ˙ 1˙ }, (2.41) {D I α˙ } = {Da B˙ α˙ , Dn+1a˙ α˙ , Dn+1 A˙ α˙ }.

Also, from (2.37) we see that Diα A = Di A+α−1

and

Dαi˙ A˙ = D iA+ , ˙ α− ˙ 1˙

(2.42)

i.e. the derivatives Di1A and Di2 A , D1i˙ A˙ and D2i˙ A˙ are not linearly independent. Similarly, (2.35), (2.40) and (2.41) imply that for n > 1 the derivatives Dα β˙ I  , Dα I˙ and D I α˙ are not independent. For n = 1, there is only one bosonic and six fermionic vector fields given in (2.31).

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695

N = 3 SYM equations. From the above formulae we see that D11˙ , Vi1 and V1˙i coincide with the vector fields (2.31) entering into the linear system [30] ˙

˙

µα λβ (∂α β˙ + Aα β˙ )ψ1 =: µα λβ ∇α β˙ ψ1 = 0, µα (Diα + Aiα )ψ1 =: µα ∇iα ψ1 = 0, λα˙ (Dαi˙ + Aiα˙ )ψ1 =: λα˙ ∇αi˙ ψ1 = 0,

(2.43)

˙

ˆ 1 depending holomorwhere ∂α β˙ := ∂/∂ x α β and ψ1 is a matrix-valued function on W phically on λ and µ. The compatibility conditions of (2.43) are the constraint equations

j j j = 0 and ∇iα , ∇β˙ − 2δi ∇α β˙ = 0 (2.44) ∇i(α , ∇ jβ) = 0, ∇(iα˙ , ∇β) ˙ of N = 3 SYM theory. Recall that these equations are equivalent to the field equations of N = 3 SYM theory [30, 41]. Therefore, we will not make any distinction between the constraint and the field equations. 3. General Form of the SYM Hierarchy Relative connection AT . Let X be a smooth (super)manifold and T an integrable distribution on X . For any smooth function f on X , let dT f be the restriction of d f to T , i.e. dT is the composition d

C ∞ (X ) → 1 (X ) → (X, T ∗ ),

(3.1)

where 1 (X ) := (X, T ∗ X ) and T ∗ denotes the sheaf of (smooth) differential oneforms dual to T . The operator dT can be extended to act on relative differential k-forms from the space kT (X ) := (X, k T ∗ ), dT : kT (X ) → k+1 T (X )

and

k ≥ 0.

(3.2)

Let Eˆ be a smooth complex vector bundle over X . A covariant differential (or connection) on Eˆ along the distribution T – a T -connection [42] – is a -linear mapping ˆ → (X, T ∗ ⊗ E) ˆ DT : (X, E)

(3.3)

satisfying the Leibniz formula DT ( f σ ) = f DT σ + dT f ⊗ σ,

(3.4)

ˆ is a local section of Eˆ and f is a local smooth function. This where σ ∈ (X, E) T -connection extends to a map ˆ → k+1 (X, E), ˆ DT : kT (X, E) T

(3.5)

ˆ := (X, k T ∗ ⊗ E). ˆ Locally, DT has the form where kT (X, E) DT = dT + AT ,

(3.6)

ˆ where the standard End E-valued T -connection one-form AT has components only along the distribution T . As usual, DT2 naturally induces the curvature FT = dT AT + ˆ of AT . We say that DT (or AT ) is flat, if FT = 0. AT ∧ AT ∈ (X, 2 T ∗ ⊗ End E) ˆ DT ) is called a T -flat vector bundle [42]. For a flat DT , the pair (E,

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A. D. Popov, M. Wolf

Linear system on F M+2|N . Consider X = F M+2|N and the integrable distribution T which is locally spanned by the vector fields (2.35) and (2.37). We have

(3.7) T = span D I  , D I˙ , D I , Vi A , V Ai˙ in the shorthand notation introduced in (2.37), (2.40) and (2.41). Suppose we are given a topologically trivial and M|N -trivial rank r holomorphic vector bundle E over the 5|6 generalized superambitwistor space Ln from the double fibration (2.21). In particular, 5|6 this means that E is holomorphically trivial on any submanifold Yx˜ → Ln with x˜ ∈ M|N . Let us consider the pulled-back bundle E˜ := π ∗ E over the supermanifold F M+2|N 2 ˆ p } as defined in Sect. 2. By definition, the pull-backs f˜ := π ∗ f of with a covering {W 2 the transition functions f = { f pq } of E to E˜ = π2∗ E must be constant along the fibres of π2 , i.e.8 dT f˜pq = 0, (3.8) where the relative differential dT (exterior differential along the fibres of (2.30)) is defined by (3.1), (3.2) and (3.7). 5|6 Note that M|N -triviality of the vector bundle E → Ln is equivalent to the triviality of the vector bundle E˜ → F M+2|N along the fibres Yx˜ of the projection π1 from (2.21). Hence, there exists a trivialization {ψ p } of E˜ such that f˜pq = ψ p−1 ψq

on

ˆ p ∩W ˆ q = ∅, W

(3.9)

where the ψ p s are holomorphic in λ( p) and µ( p) . From (3.8) and (3.9), it follows that ψ p dT ψ p−1 = ψq dT ψq−1

(3.10)

ˆ p ∩W ˆ q and therefore on any W AT |Wˆ

p

:= ψ p dT ψ p−1

(3.11)

is a globally defined flat T -connection on a trivial vector bundle Eˆ → F M+2|N which is ˜ In fact, we have an equivalence topologically equivalent to E. ˆ fˆ = { ˜ f˜ = { f˜pq }, dT ) ∼ (E, (E,

r }, DT

= dT + AT ),

(3.12)

ˇ which is the equivalence of the Cech and Dolbeault descriptions of T -flat vector bundles. By an extension of Liouville’s theorem to Y → F M+2|N , it follows that on patch ˆ 1 the components of AT are given by the formulae W ˙

D I 
AT := ψ1 D I  ψ1−1 = µα λβ Aα β˙ I  , D I˙
AT := ψ1 D I˙ ψ1−1 = µα Aα I˙ ,

D I
AT := ψ1 D I ψ1−1 = λα˙ A I α˙ , Vi A
AT := V Ai˙
AT

:=

ψ1 Vi A ψ1−1 ψ1 V Ai˙ ψ1−1

(3.13)

α

= µ Aiα A , = λα˙ Aiα˙ A˙ ,

8 Recall that a (local) function on the correspondence space F M+2|N descends to generalized superambit5|6 wistor space Ln if and only if it lies in the kernel of dT .

Hidden Symmetries and Integrable Hierarchy of the N = 4 Supersymmetric Yang-Mills Equations

697

where the fields {A} = {Aα β˙ I  , Aα I˙ , . . .} do not depend on λ and µ. Similar formulae can be written down for the other patches. Equations (3.13) can be rewritten as the following system of linear differential equations: ˙

˙

µα λβ (Dα β˙ I  + Aα β˙ I  )ψ1 =: µα λβ ∇α β˙ I  ψ1 = 0, µα (Dα I˙ + Aα I˙ )ψ1 =: µα ∇α I˙ ψ1 = 0,

λα˙ (D I α˙ + A I α˙ )ψ1 =: λα˙ ∇ I α˙ ψ1 = 0, µ (Diα A + Aiα A )ψ1 =: µα ∇iα A ψ1 = 0, λα˙ (Dαi˙ A˙ + Aiα˙ A˙ )ψ1 =: λα˙ ∇αi˙ A˙ ψ1 = 0.

(3.14)

α

It is not too difficult to see that these equations are invariant under the gauge transformations and A → g −1 Ag + g −1 Dg, (3.15) ψ1 → g −1 ψ1 where g is any smooth G L(r, )-valued function on invariant under the equivalence map ψ1 → ψ1 h 1

and

M|N .

In addition, (3.14) is also

A → A,

(3.16)

ˆ 1 ⊂ F M+2|N . Similar where h 1 is any holomorphic G L(r, )-valued function on W ˆ p for p = 2, 3, 4. Note invariance transformations hold for the remaining patches W that the transition functions (3.9) are invariant under the transformations (3.15) and furthermore are mapped to equivalent ones for (3.16). SYM(n) equations. The compatibility conditions of Eq. (3.14) are given by       ∇(α(α˙ I  , ∇β)β)J ∇(α α˙ I  , ∇β) J˙ = 0, ∇α(α˙ I  , ∇ J β) ˙  = 0, ˙     i ∇(α α˙ I  , ∇iβ)B = 0, ∇α(α˙ I  , ∇β) ˙ B˙       ∇(α I˙ , ∇β) J˙ = 0, ∇α I˙ , ∇ J β˙ = 0, ∇ I (α˙ , ∇ J β) ˙         i ∇(α I˙ , ∇iβ)B = 0, ∇α I˙ , ∇βi˙ B˙ = 0, ∇ I α˙ , ∇iβ B = 0, ∇ I (α˙ , ∇β) ˙ B˙

j j j ∇i(α A , ∇ jβ)B = 0, ∇(iα˙ A˙ , ∇β) = 0, ∇iα A , ∇β˙ B˙ − 2δi ∇α β˙ A B˙ ˙ B˙

= 0, = 0, = 0, = 0, = 0, (3.17)

where parentheses mean normalized symmetrization of the spinorial indices. We shall call the finite system (3.17) of nonlinear differential equations the SYM hierarchy truncated up to level n and refer to the field equations (3.17) as the SYM(n) equations. The full SYM hierarchy is obtained by taking the asymptotic limit n → ∞.9 We emphasize that because of Bianchi identities, (3.17) is not the “minimal” set of equations since some equations of (3.17) are implied by others. As our subsequent discussion will not be affected by this issue, we can leave it aside in the remainder of this work. It is not difficult to see that the constraint equations (2.44) of N = 3 SYM theory are embedded into the SYM(n) equations. Indeed, we find them ˙

9 One can also allow the indices to run over all negative values with corresponding coordinates x A B (hence, A, B˙ ∈ ), etc. parametrizing “negative flows”.

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A. D. Popov, M. Wolf

∇i(α1 , ∇ jβ)1

= 0,

i j j j ∇(α˙ 1˙ , ∇β) ∇iα1 , ∇β˙ 1˙ − 2δi ∇α β1 ˙ 1˙ = 0, (3.18) ˙ 1˙ = 0,

as part of (3.17) since ∇iα1 ≡ ∇iα , ∇αi˙ 1˙ ≡ ∇αi˙ and ∇α β1 ˙ 1˙ ≡ ∇α β˙ . Now one could continue to work with (3.17) by imposing a sort of transversal gauge [41] (or follow a more general approach [43]) to extract the field content, the superfield expansions, etc. of the SYM(n) equations. However, already in describing truncated hierarchies for N -extended self-dual SYM theories, this procedure was rather involving [14]. That is why we leave this discussion to future work. In the next section, we will write the SYM(n) equations in light-cone gauge in which they take a simpler form. Reality conditions. Up to now, we have only worked in a complex setting. However, it is well known that for N = 4 SYM theory described in terms of supertwistors, one can obtain real SYM fields on Minkowski space 1,3 with metric g = diag(−1, +1, +1, +1) 3|3 by choosing appropriate real structures on 4|12 , P 3|3 × P∗ and on the quadric L5|6 (see e.g. [32]). In fact, by introducing an antiholomorphic involution on L5|6 , one obtains an induced involution on 4|12 such that its fixed point set is identified with Minkowski superspace. In the present case of generalized superambitwistor space, we may proceed similarly. 5|6 5|6 In particular, we consider the antiholomorphic involution τ M : Ln → Ln which is 6|6 defined by its action on the homogeneous coordinates of Cn as the map



τ M : (z α , w α˙ , πα˙ , ρα , ηi , θ i ) → (−w α˙ , −z α , ρα , πα˙ , θ i , ηi ),

(3.19)

where bar denotes complex conjugation. Inserting (2.24) into (3.19), we find that τ M acts on M|N as10 ˙

τ M (x A B ) = −x B A˙ ,

˙

τ M (t a B ) = −s B a˙

Clearly, the fixed point set is a real slice

 M|N

∼ =

4|12 ×  M−4|N −12

 M|N ⊂

˙

τ M (ηiA ) = θ i A .

and M|N ,

(3.20)

with

for M = 3n 2 +2n −1 and N = 6(n +1). (3.21) ˙

If we choose the parametrization of x α β according to 

0 x + x 3 x 1 − ix 2 ˙ (x α β ) := −i 1 x + ix 2 x 0 − x 3



(3.22)



with (x 0 , x 1 , x 2 , x 3 ) ∈ 4 , the factor 4|12 appearing in the decomposition (3.21) can be identified with N = 3 Minkowski superspace. The extension of τ M to matrix-valued holomorphic functions h is given by τ M (h(· · · )) := [h(τ M (· · · ))] t ,

(3.23)

where “t” means matrix transposition. With this rule we can extend the involution τ M to the holomorphic vector bundles E and E˜ = π2∗ E. In particular, for the transition functions f˜ = { f˜pq } of E˜ the reality condition τ M ( f˜) = f˜ yields † = f˜31 , f˜12

† f˜14 = f˜41

and

† f˜23 = f˜23 .

10 By a slight abuse of notation, we use the symbol τ for maps defined on different spaces. M

(3.24)

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699

† Here, we have used the shorthand notation f˜pq := [ f˜pq (τ M (· · · ))]† . Equations (3.24) can be satisfied by imposing the conditions

ψ1† = ψ1−1 ,

ψ2† = ψ3−1

and

ψ4† = ψ4−1

(3.25)

which yield the relations (Aα β˙ A B˙ )† = Aβ α˙ B A˙ , (Aα I˙ )† = A I α˙ and (Aiα˙ A˙ )† = Aiα A .

(3.26)

Therefore, the gauge group is reduced from G L(r, ) to the unitary group U (r ). If one in addition assumes that det( f˜pq ) = 1, then one obtains SU (r ). To sum up, we have shown that there is a one-to-one correspondence between M|N trivial τ M -invariant holomorphic vector bundles E over the generalized superambitwistor 5|6 space Ln and gauge equivalence classes of solutions to the equations of the truncated SYM(n) hierarchy which include the field equations of N = 3 SYM theory as a subset.



4. SYM Hierarchy in Light-Cone Gauge ˆ1 Extended linear system. For the sake of concreteness, we again work on the patch W of F M+2|N . We start by reconsidering the vector fields (2.35) and (2.37) and the resulting linear system (3.14). Note that the vector fields (2.35a) are a linear combination of the vector fields



 ∂ ∂ ∂ n˙ 1˙ ∂ − δ + δ X A B˙ = λ − with A ≤ n+1, B˙ ≤ n˙ ˙ 1˙ B˙ B˙ ˙ 1˙ ∂s An− ∂s A1˙ ∂ x A B˙ ∂ x A B+ (4.1a) or 



∂ ∂ ∂ n 1 ∂ − with A ≤ n, B˙ ≤ n˙ + 1˙ − δA + δA Y A B˙ = µ ∂ x A B˙ ∂t n−1 B˙ ∂ x A+1 B˙ ∂t 1 B˙ (4.1b) together with the remaining vector fields of (2.35). In particular, we have ˙

D A B˙ = µX A B˙ − X A+1 B˙ + δ 1B˙ S A1˙ + λδ nB˙˙ S An− ˙ 1˙ =

1 λY A B˙ − Y A B+ ˙ 1˙ + δ A T1 B˙

+ µδ nA Tn−1 B˙

(4.2a) with

A ≤ n, B˙ ≤ n. ˙

(4.2b)

In other words, both sets (4.1a) and (4.1b) of vector fields belong to the distribution T 5|6 tangent to the fibres of the projection π2 : F M+2|N → Ln . Together with vector fields (2.35d)–(2.35g) they form an “overcomplete basis” for the distribution T . However, they all do annihilate the transition functions of the holomorphic vector bundle E˜ → F M+2|N and their linear span forms T . Therefore, one can use them for introducing an extended linear system11 which eventually yields a rather homogeneous form of the equations of the SYM hierarchy when written in light-cone gauge. Instead of (2.35a)–(2.35c) for n > 1, we now take the vector fields (4.1) and combine them together with the remaining bosonic vector fields (2.35d)–(2.35g) into the following expressions: Dϒ˙ = µα Dα ϒ˙ and Dϒ = λα˙ Dϒ α˙ , (4.3a) 11 Of course, one now has to deal with additional constraints among the obtained superfields caused by the linear dependence of the used vector fields. However, this fact does not affect our subsquent discussion.

700

A. D. Popov, M. Wolf

where {Dϒ˙ } = {Y A B˙ , S Ab˙ , Ta n+ ˙ 1˙ } {Dϒ } = {X A B˙ , Ta B˙ , Sn+1a˙ }

{Dα ϒ˙ } = {Dα A B˙ , Dα Ab˙ , Dαa n+ ˙ 1˙ }, (4.3b) {Dϒ α˙ } = {D A B˙ α˙ , Da B˙ α˙ , Dn+1a˙ α˙ }. (4.3c)

and and

So, the abstract indices ϒ and ϒ˙ run over the appropriate index set which is easily extracted from (4.3). Note that the vector fields (4.3b) and (4.3c) are related by the involution τ M . Now, instead of the linear system (3.14), we find the following extended linear system: µα (Dα ϒ˙ + Aα ϒ˙ )ψ1 = 0,

λα˙ (Dϒ α˙ + Aϒ α˙ )ψ1 = 0, µα (Diα A + Aiα A )ψ1 = 0, λα˙ (Dαi˙ A˙ + Aiα˙ A˙ )ψ1 = 0.

(4.4)

Light-cone gauge. Let us now fix a gauge by imposing the condition ψ1 (λ = 0 = µ) = 1. Such a gauge can always be obtained from the general (λ, µ)-expansion, ψ1 =

∞ 

(k,l)

λk µl ψ1

,

(4.5)

k,l=0 (0,0) by performing the gauge transformation12 ψ1 → ψ˜ 1 = (ψ1 )−1 ψ1 so that ψ˜ 1 (λ = 0 = µ) = 1. Hence, from the very beginning we can assume that ψ1(0,0) = 1 in (4.5). Then we have (4.6) ψ1 = 1 + µ + λ + λµ + · · · .

Imposing the reality condition (3.25) on ψ1 , we obtain  = − †

and

 † = − +  + .

(4.7)

Next, we substitute the expansion (4.6) into the linear system (4.4) and find that all the fields Aα ϒ˙ , Aϒ α˙ , Aiα A and Aiα˙ A˙ are expressed in terms of the prepotentials ,  and  as A1ϒ˙ = D2ϒ˙ , Ai1A = Di2 A ,

A2ϒ˙ = 0, Ai2 A = 0,

Aϒ 1˙ = Dϒ 2˙ , Ai1˙ A˙

=

D2i˙ A˙ ,

Aϒ 2˙ = 0, Ai2˙ A˙ = 0.

(4.8)

In addition, the fields {, , } are constrained by the differential equations

Dϒ 2˙  − Dϒ 1˙  − Dϒ 2˙ () = 0, i D2˙ A˙  − D1i˙ A˙  − D2i˙ A˙ () = 0,

D2i˙ A˙  = 0, (4.9a) Dϒ 2˙  = 0, D2ϒ˙  = 0, Di2 A  = 0, (4.9b) D2ϒ˙  − D1ϒ˙  − D2ϒ˙ () = 0, (4.9c) Di2 A  − Di1A  − Di2 A () = 0. (4.9d)

12 Note that such a transformation must be performed on all four patches simultaneously with the same (0,0) (0,0) −1 matrix ψ1 , i.e. ψ p  → (ψ1 ) ψp.

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Inserting (4.8) into the compatibility conditions of the linear system (4.4) and using (4.9), we obtain the equations D1ϒ˙ D2˙  − D1˙ D2ϒ˙  + [D2˙ , D2ϒ˙ ] D1ϒ˙ Di2 A  − Di1A D2ϒ˙  + [D2ϒ˙ , Di2 A ] Di1A D j2B  + D j1B Di2 A  + {Di2 A , D j2B } Dϒ 1˙ D2˙  − D1˙ Dϒ 2˙  + [Dϒ 2˙ , D2˙ ]

= 0, = 0, = 0, = 0,

(4.10a) (4.10b) (4.10c) (4.10d)

Dϒ 1˙ D2i˙ A˙  − D1i˙ A˙ Dϒ 2˙  + [Dϒ 2˙ , D2i˙ A˙ ] = 0,

(4.10e)

j D1i˙ A˙ D2˙ B˙ 

j D1˙ B˙ D2i˙ A˙ 

+ {D2i˙ A˙ ,

j D2˙ B˙ }

D1i˙ A˙ D2ϒ˙  + [D2ϒ˙ ,

D2i˙ A˙ ]

= 0,

(4.10f)

D1ϒ˙ D2˙  − D1˙ D2ϒ˙  + [D2ϒ˙ , D2˙ ] = 0,

(4.10g)

D1ϒ˙ D2i˙ A˙ 

(4.10h) (4.10i)

+

− = 0, Di1A Dϒ 2˙  − Dϒ 1˙ Di2 A  + [Di2 A , Dϒ 2˙ ] = 0,

which together with Eqs. (4.9) form the SYM(n) equations (the SYM hierarchy truncated up to level n) in light-cone gauge. As before, we obtain the full SYM hierarchy by taking the limit n → ∞.13 N = 3 SYM theory in light-cone gauge. Before continuing, we briefly explain why we called this gauge “light-cone gauge”. Recall that in terms of (4.1) the vector fields (2.35a) are given by the formulae (4.2). Therefore, the gauge potential Aα β˙ A B˙ which is associated with Dα β˙ A B˙ from (2.35a) is represented as a certain linear combination of the above introduced fields Aα ϒ˙ and Aϒ α˙ and, of course, similarly for Aα I˙ and A I α˙ . A short calculation then shows that A22˙ A B˙ = 0, A12˙ A B˙ = D22˙ A B˙ , A21A ˙ B˙ = D22˙ A B˙ , A11A = D  + (D ) + D  + (D ˙ B˙ ˙ B˙ 12˙ A B˙ 22˙ A B˙ 21A 22˙ A B˙ ) − D22˙ A B˙ , (4.11) A2 I˙ = 0, A1 I˙ = D2 I˙ , A I 2˙ = 0, A I 1˙ = D I 2˙ . Recall that for A = 1, B˙ = 1˙ the field A22˙ A B˙ is identified with the component A22˙ of the Yang-Mills potential Aα β˙ := Aα β1 ˙ 1˙ . Since A22˙ = 0, the terminology “light-cone gauge” becomes transparent from the viewpoint of N = 3 SYM theory (see also (3.22), ˙ where ix 22 = x 0 − x 3 ). Finally, we stress that (4.9) and (4.10) contain the equations Di1 D j2  + D j1 Di2  + {Di2 , D j2 } = 0, j

j

j

D1i˙ D2˙  + D1˙ D2i˙  + {D2i˙ , D2˙ } = 0, D2i˙  − D1i˙  − D2i˙ () = 0,

Di2  − Di1  − Di2 () = 0, D2i˙  = 0,

(4.12)

Di2  = 0

as a subset. These equations are equivalent to the field equations of N = 3 SYM theory in light-cone gauge. This is easily shown by substituting an expansion of the form (4.6) into the linear system (2.43) of N = 3 SYM theory as well as into the corresponding constraint equations (2.44). 13 One may also consider the case when all the indices run over all integers, i.e. over

.

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5. Hidden Symmetries of N = 3 SYM Theory In this section, we consider the SYM hierarchy written in light-cone gauge as an infinite set of equations on the prepotentials {, , } and discuss an algebra of hidden symmetries of the N = 3 SYM equations realized as point symmetries of the SYM hierarchy via derivatives with respect to extra bosonic and fermionic “times”. In particular, we reinterpret the results derived in the previous sections and in addition show how a “double” affinization of the algebra of supertranslations on Minkowski superspace can be realized as a hidden symmetry algebra of N = 3 SYM theory. Linearized N = 3 SYM equations. Let us consider the linearized form of (4.12), D(i1 D j)2 δ + {D(i2 , D j)2 δ} = 0, D2i˙ δ

−

j) (i j) (i D1˙ D2˙ δ + {D2˙ , D2˙ δ} D1i˙ δ − D2i˙ ((δ) + δ)

= 0,

(5.1a) (5.1b)

= 0,

(5.1c)

Di2 δ − Di1 δ − Di2 ((δ) + δ) = 0, D2i˙ δ = 0, Di2 δ = 0.

(5.1d) (5.1e)

Solutions {δ, δ, δ} to these equations for given {, , } satisfying (4.12) are called (infinitesimal) symmetries of the N = 3 SYM equations written in light-cone gauge. From the geometric point of view, solutions to (5.1) are vector fields on the solution space of the N = 3 SYM equations. Besides natural local symmetries (e.g. generated by superconformal and gauge transformations), the N = 3 SYM equations possess infinitely many nonlocal hidden symmetries, as we will show momentarily. Such symmetry transformations generate new solutions from old ones. Here, we describe an important subclass of such infinitesimal symmetry transformations associated with an affinization of supertranslations on 4|12 . These transformations generate the higher flows in the SYM hierarchy introduced earlier in this paper. Infinitesimal symmetries. Recall that the field equations of N = 3 SYM theory (4.12) form a subset of Eqs. (4.9), (4.10) of the SYM hierarchy.14 So, the prepotentials {, , } in (4.12), which we symbolically denote by , depend not only on coordinates of Minkowski superspace but also on an infinite number of additional moduli ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ 2˙ and θ ia+2 . Therefore, upon defining x α b+2 , x a+2 β , x a+2 b+2 , t a B , s Ab , ηia+ (x)

δ A B˙  := D A B˙ 1˙  =

∂ ∂ x A B˙

,

∂ ∂ (t) (s) δa B˙  := Da B˙ 1˙  = , δ Ab˙  := D1Ab˙  = , ˙ ∂t a B ∂s Ab˙ 

∂ ˙  =: Q i A , δi A  := Di A − 2ηiB ∂ x A B˙ 

∂  =: Q iA˙ , δ iA˙  := D iA˙ − 2θ i B ∂ x B A˙

(5.2a) (5.2b) (5.2c) (5.2d)

14 From now on, we assume that all indices in (4.9), (4.10) (besides R-symmetry indices i, j) run from 1 to ∞, i.e. they take values in
. In principle, one can also allow them to take values in .

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we obtain infinitesimal symmetries of Eqs. (4.12) since all vector fields appearing in (5.2) (anti) commute with Diα and Dαi˙ . The fermionic vector fields Q i A and Q iA˙ introduced above obey j j ∂ (5.3) Q i A , Q B˙ = −2δi ∂ x A B˙ and in addition they also anticommute with Di A and D iA˙ . Thus, the infinitesimal transformations  → δ defined in (5.2) as derivatives with respect to moduli parametrizing higher flows of the SYM hierarchy give solutions to the linearized N = 3 SYM equations in light-cone gauge (5.1), i.e. they define an infinite set of hidden (infinitesimal) symmetries. Symmetry equations as subset of the SYM hierarchy. Recall that the derivatives in (5.2) generate bosonic and fermionic flows on the solution space of N = 3 SYM theory. This hints that Eqs. (5.1) on infinitesimal symmetries defined by (5.2) may follow from a subset of the equations of the SYM hierarchy. We will show that this is indeed the case. Moreover, the remaining equations of the hierarchy describe the dependence of the symmetries δ on the additional moduli. This in fact is in the same spirit as for self-dual (S)YM hierarchies (see e.g. [3, 12, 14]). We exemplify all this by focussing on δ2ϒ˙  := D2ϒ˙ ,

(5.4)

which, due to (2.35) and (4.3), are linear combinations of the symmetries (5.2a), (5.2b). Let us substitute (5.4) into Eqs. (4.9a), (4.9c) and (4.10b) with A = 1. Remember also that Di1A ≡ Di A , Di2 A ≡ Di A+1 , D1i˙ A˙ ≡ D iA˙ and D2i˙ A˙ ≡ D iA+ ˙ 1˙ . Then we have the following equations on the symmetries (5.4): D j2 δ1ϒ˙  − D j1 δ2ϒ˙  + [δ2ϒ˙ , D j2 ] = 0, δ2ϒ˙  − δ1ϒ˙  − (δ2ϒ˙ ) = 0, δ2ϒ˙  = 0.

(5.5a) (5.5b) (5.5c)

Here,

δ1ϒ˙  := D1ϒ˙  (5.6) are also linear combinations of the symmetries (5.2a), (5.2b). Applying Di2 and D2i˙ to (5.5b), Di2 to (5.5a), and symmetrizing the R-symmetry indices i and j, we find D(i1 D j)2 δ2ϒ˙  + {D(i2 , D j)2 δ2ϒ˙ } = 0, (i

j)

(i

j)

D1˙ D2˙ δ2ϒ˙  + {D2˙ , D2˙ δ2ϒ˙ } = 0,

Di2 δ2ϒ˙  − Di1 δ2ϒ˙  − Di2 ((δ2ϒ˙ ) + δ2ϒ˙ ) = 0, D2i˙ δ2ϒ˙ 

−

D1i˙ δ2ϒ˙  −

D2i˙ ((δ2ϒ˙ ) + δ2ϒ˙ ) Di2 (δ2ϒ˙ ) = 0, D2i˙ (δ2ϒ˙ )

(5.7a) (5.7b) (5.7c)

= 0,

(5.7d)

= 0,

(5.7e)

where we used (4.9), (4.10) and (5.5c). Obviously, (5.7) is a special case of (5.1) for the symmetries (5.4). The remaining equations in (4.9), (4.10) on the symmetries (5.4), e.g. D1ϒ˙ δ2˙  − D1˙ δ2ϒ˙  + [D2˙ , δ2ϒ˙ ] = 0,

(5.8)

describe their dependence on the moduli along the higher flows. Similarly, one can show for all symmetries (5.2) (and their linear combinations) that the linearized N = 3 SYM equations (5.1) can be obtained from a subset of the SYM hierarchy (4.9), (4.10). Note that the infinitesimal transformations (5.2) were defined to be compatible with (4.7), which are the reality conditions for the flows.

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Symmetry algebra. As an immediate consequence of the transformations (5.2), we find j j (x) δi A , δ B˙ = −2δi δ A B˙ (5.9) upon acting on either ,  or . These relations are the only nonvanishing (anti) commutators of two successive infinitesimal symmetry transformations. We have thus obtained an infinite-dimensional (graded) algebra of hidden symmetries of the N = 3 SYM equations. To understand the nature of this algebra, let us consider the action of the vector fields entering in the definition of the transformations (5.2) on the transition functions f˜ = { f˜pq } of the holomorphic vector bundle E˜ → F M+2|N for finite n and eventually ˆ1 ∩W ˆ p we have put n → ∞. A short calculation reveals that on the intersection W ˙

µa λb ˙

µa λb ˙

µa λb a b˙

µ λ

∂ ∂ x 11˙ ∂

∂ x 12˙ ∂

∂ x 21˙ ∂

∂ x 22˙

= = = =

∂ ˙ 1˙ ∂ x a+1 b+

∂

˙ 2˙ ∂ x a+1 b+

∂

˙ 1˙ ∂ x a+2 b+

∂

˙ 2˙ ∂ x a+2 b+

+ + +

∂ ∂

∂ ∂s a+1 b˙

,

,

˙ 2˙ ∂t a b+ ∂

∂s a+2 b˙

, (5.10)

, ˙

µa Q i1 = Q ia+1 − 2ηiB λa˙ Q i1˙

+

˙ 1˙ ∂t a b+

∂

, ∂t a B˙ ∂ = Q ia+ − 2θ i B B a˙ , ˙ 1˙ ∂s

µa Q i2 = Q ia+2 , λa˙ Q i2˙ = Q ia+ , ˙ 2˙

where the above relations are understood upon action on f˜1 p . Similar expressions can be derived for the other patches. Therefore, we see that the vector fields on the righthand side of (5.10) are recursively generated from the generators of supertranslations on Minkowski superspace. Recall that for the real case, we have the relations (3.20)–(3.22) and µ = λ¯ . We see that the obtained algebra is related to the algebra T 4|12 [λ, µ] with generators ˙

µa λb Pα β˙ ,

µa Q iα



and

˙

˙

λb Q iβ˙

(5.11)

for a, b˙ ∈ . Here, Pα β˙ := ∂/∂ x α β and T 4|12 denotes the supertranslation algebra generated by Pα β˙ , Q iα and Q iβ˙ . As shown above, the algebra T 4|12 [λ, µ] is represented via (5.10) in terms of vector fields on the solution space of the N = 3 SYM equations. Altogether, we have thus obtained an affinization of the algebra of supertranslation T 4|12 on complexified Minkowski superspace. Moreover, in view of the self-dual bosonic case [3], it seems conceivable that by considering all four patches which cover generalized superambitwistor space and by extending the range of all indices a, a, ˙ . . . in the SYM hierarchy to all integers , the above algebra can be extended to the symmetry algebra T 4|12 [λ, λ−1 , µ, µ−1 ] which is generated by the same set of generators (5.11) but for a, b˙ ∈ . Finally, we note that in the real case, the graded double loop algebra T 4|12 [λ, λ−1 , µ, µ−1 ] becomes T 4|12 [λ, λ−1 , λ¯ , λ¯ −1 ] with the identity µa Q iα = (λa˙ Q iα˙ )† , etc. for a¯˙ = a.







Hidden Symmetries and Integrable Hierarchy of the N = 4 Supersymmetric Yang-Mills Equations

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6. Summary and Discussion The goal of this paper was to present a step towards an understanding of the appearance of nonlocal hidden symmetries in N = 4 SYM theory from first principles. Here, we were interested in symmetries related to particular space-time symmetries. For this, we 5|6 introduced a generalized superambitwistor space Lm,n fibred over P 1 × P∗1 together 5|6 with the space 4|12 × M−4|N −12 parametrizing subspaces ( P 1 × P∗1 )x˜ in Lm,n with 5|6 5|6 M|N x˜ ∈ . Then, by specializing to the case Ln := Ln,n , we discussed a Penrose-Ward 5|6 type transform relating holomorphic vector bundles over Ln and solutions to Yang2 Mills-Higgs type equations on M|N = 3n +2n−1|6(n+1) being termed the SYM(n) 5|6 equations which describe the SYM hierarchy truncated up to level n. Note that L1 coincides with the well-known superambitwistor space L5|6 and the SYM(1) equations are equivalent to the equations of motion of N = 3 SYM theory. The truncated SYM hierarchy turns into the full SYM hierarchy after taking the asymptotic limit n → ∞. The field equations of N = 3 SYM theory are embedded into this infinite system of partial differential equations as well as into the SYM(n) equations for finite n > 1. Hence, a given solution to the N = 3 SYM equations can be embedded into an infiniteparameter family of solutions to the whole hierarchy.15 The dependence of the fields on the extra parameters can be recovered by solving the equations of the SYM hierarchy. Furthermore, we reinterpreted the equations of the SYM hierarchy in the context of hidden symmetries of N = 3 SYM theory. In particular, we rewrote the constraint equations of the latter and those of the hierarchy in light-cone gauge. In this gauge, parametrized by three prepotentials ,  and , it became transparent that some equations of the SYM hierarchy are equations on hidden infinitesimal symmetries (i.e. solutions to the linearized field equations) of N = 3 SYM theory. We have shown that these nonlocal symmetries are related to a graded algebra T 4|12 [λ, µ] generated from the algebra T 4|12 of supertranslations on Minkowski superspace. In this respect, we emphasize again that the N = 3 and N = 4 SYM theories are physically equivalent theories. Thus, the twistor description gives a direct way of obtaining a particular set of infinitely many hidden symmetries in N = 4 SYM theory. Recall that the classical Green-Schwarz superstring on the curved background AdS5 × S 5 possesses an infinite set of hidden symmetries and nonlocal conserved charges [44] which are similar to those in two-dimensional models (see e.g. [45]). This is basically because of the possibility of interpreting the Green-Schwarz superstring on AdS5 × S 5 as a particular sigma model, where the fields take values in the coset space P SU (2, 2|4)/(S O(1, 4) × S O(5)). By virtue of the AdS/CFT correspondence [46], conserved nonlocal charges should also exist in N = 4 SYM theory (at least in the planar limit). In fact, within the spin chain approach, the authors of [47] were able to derive an analogous set of conserved charges in the superconformal Yang-Mills theory in the weak coupling limit. It is quite reasonable to expect that these Yangian symmetries and charges could be related to those derived in the present work. As was mentioned in Sect. 1, in recent years essential progress in our understanding of quantum properties of N = 4 SYM theory has been made with the help of twistor string theory [15]. Besides B-type topological string theory on the supertwistor space P 3|4 , Witten also mentioned the possibility of formulating a twistor string theory on the 15 This is a generic situation. However, for some concrete solutions, there could be obstructions of the same nature as for the SDYM hierarchy discussed in [3].

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superambitwistor space16 L5|6 . Within such a formulation the mechanism of reproducing perturbative N = 4 SYM theory would be completely different compared to the supertwistor space approach (i.e. no D-instantons are needed), as already at the classical level, holomorphic Chern-Simons theory on superambitwistor space yields all the interactions of N = 4 SYM theory. However, there are problems in formulating an action principle for holomorphic Chern-Simons theory on this space caused by the difficulty in making sense of an appropriate integration measure. Recently, some progress in this direction has been made in [26] for Euclidean signature in four dimensions. 5|6 Note that generalized superambitwistor space Lm,n is a Calabi-Yau supermanifold for 5|6 any values of m and n. One can therefore define a twistor string theory on Lm,n . It is also an interesting task to see how the ideas presented in [26] should be generalized in order to construct appropriate action functionals for the truncated SYM(n) hierarchies introduced in our paper. The next obvious step is the generalization of the twistor construction to all generators of the superconformal group. Besides questions associated with symmetry transformations, one in addition needs to write down the related conserved currents and charges (not only as superfields but also in components) in order to proceed further to quantum theory. It will then hopefully be possible to make contact with the quantum symmetry algebras considered in [47]. Moreover, it would also be interesting to see whether the symmetries described in this paper have an analog in the context of twistor string theory. Acknowledgements. We would like to thank O. Lechtenfeld and R. Wimmer for useful discussions. This work was partially supported by the Deutsche Forschungsgemeinschaft (DFG).

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29. Sämann, C.: Aspects of twistor geometry and supersymmetric field theories within superstring theory. Ph.D. thesis, Leibniz University of Hannover, 2006, http://arxiv.org/list/hep-th/0603098, 2006; Wolf, M.: On supertwistor geometry and integrability in super gauge theory. Ph.D. thesis, Leibniz University of Hannover, 2006 30. Witten, E.: An interpretation of classical Yang-Mills theory. Phys. Lett. B 77, 394 (1978) 31. Isenberg, J., Yasskin, P.B., Green, P.S.: Non-self-dual gauge fields. Phys. Lett. B 78, 462 (1978) 32. Manin, Yu.I.: Gauge field theory and complex geometry. New York: Springer Verlag, 1988 [Russian: Moscow: Nauka, 1984] 33. Eastwood, M.G.: Supersymmetry, twistors, and the Yang-Mills equations. Trans. Amer. Math. Soc. 301, 615 (1987) 34. Harnad, J.P., Hurtubise, J., Shnider, S.: Supersymmetric Yang-Mills equations and supertwistors. Annals Phys. 193, 40 (1989) 35. Howe, P.S., Hartwell, G.G.: A superspace survey. Class. Quant. Grav. 12, 1823 (1995) 36. Burns, D.: Some background and examples in deformation theory. In: Complex manifold techniques in theoretical physics. edited by D. E. Lerner, P. D. Sommer, London:Pitman, 1979 37. LeBrun, C.R.: Spaces of complex null geodesics in complex Riemannian geometry. Trans. Amer. Math. Soc. 278, 209 (1983) 38. Kodaira, K.: A theorem of completeness of characteristic systems for analytic families of compact submanifolds of complex manifolds. Ann. Math. 75, 146 (1962) 39. Ward, R.S., Wells, R.O.: Twistor geometry and field theory. Cambridge: Cambridge University Press, 1990 40. Griffith, P., Harris, J.: Principles of algebraic geometry. New York: John Wiley & Sons, 1978 41. Harnad, J.P., Hurtubise, J., Legare, M., Shnider, S.: Constraint equations and field equations in supersymmetric N = 3 Yang-Mills theory. Nucl. Phys. B 256, 609 (1985) 42. Rawnsley, J.H.: Flat partial connections and holomorphic structures in smooth vector bundles. Proc. Amer. Math. Soc. 73, 391 (1979) 43. Aref’eva, I.Ya., Volovich, I.V.: Reconstruction of superconnection from physical fields in the N = 4 supersymmetric Yang-Mills theory. Class. Quant. Grav. 3, 617 (1986) 44. Bena, I., Polchinski, J., Roiban, R.: Hidden symmetries of the AdS5 × S 5 superstring. Phys. Rev. D 69, 046002 (2004) 45. Lüscher, M., Pohlmeyer, K.: Scattering of massless lumps and nonlocal charges in the two-dimensional classical nonlinear sigma model. Nucl. Phys. B 137, 46 (1978); Lüscher, M.: Quantum nonlocal charges and absence of particle production in the two-dimensional nonlinear sigma model. Nucl. Phys. B 135, 1 (1978) 46. Maldacena, J.M.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)]; Gubser, S.S., Klebanov, I.R., Polyakov, A.M.: Gauge theory correlators from non-critical string theory. Phys. Lett. B 428, 105 (1998); Witten, E.: Anti-de Sitter space and holography. Adv. Theor. Math. Phys. 2, 253 (1998) 47. Dolan, L., Nappi, C.R., Witten, E.: A relation between approaches to integrability in superconformal Yang-Mills theory. JHEP 0310, 017 (2003); Yangian symmetry in D = 4 superconformal Yang-Mills theory. http://arxiv.org/list/hep-th/0401243, 2004; Dolan, L., Nappi, C.R.: Spin models and superconformal Yang-Mills theory. Nucl. Phys. B 717, 361 (2005) Communicated by G.W. Gibbons

Commun. Math. Phys. 275, 709–720 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0312-5

Communications in

Mathematical Physics

Extended Systems with Deterministic Local Dynamics and Random Jumps Elisha Kobre1,
, Lai-Sang Young2,

1 New York, NY, USA. E-mail: [email protected] 2 Courant Institute of Mathematical Sciences, 251 Mercer St., New York, NY 10012, USA.

E-mail: [email protected] Received: 18 September 2006 / Accepted: 9 January 2007 Published online: 15 August 2007 – © Springer-Verlag 2007

Abstract: We consider particle systems on lattices with internal dynamics at each site and random jumps between sites. Models with simple chaotic local dynamics, namely expanding circle maps, are considered. Results on mean drift rates, central limit theorems and dependences on jump parameters are proved. The aims of this paper are twofold. One is to study transport and diffusion properties of certain types of particle systems. The other is to bring to the attention of the dynamical systems community a large class of extended systems with rich dynamical properties and potential applications. The systems we have in mind have extended phase spaces; they are particle systems defined by microscopic rules that are part deterministic and part stochastic. Let Zd be the d-dimensional lattice. The dynamics occur on two different levels: internal to each site i ∈ Zd is a dynamical process describing what the particle does while at that site; another process governs the site-to-site movements of particles. These processes are coupled; they may be deterministic or stochastic; often both deterministic and stochastic components are present. For example, we may assume that at each site i, there is a dynamical system τi : Mi
. By default, a particle at x ∈ Mi remains at site i and goes to τi (x) as the system evolves, but under certain conditions there is some probability it will jump to another location at another site. The system as a whole is assumed to be Markov
(discrete or continuous time) with state space the disjoint union i∈Zd Mi . Indeed, the underlying structure need not be Zd ; it can be any network with any topology. The idea of extended systems defined by microscopic laws comes from statistical mechanics. While the local laws in most of the better understood models (e.g. the Ising model) are stochastic, deterministic laws (e.g. chain of oscillators) are viewed as equally viable. Among the known systems with deterministic or hybrid microscopic dynamics, a number are particle systems (e.g. Lorentz gases in R2 [Si], mechanical models of
A version of most of the results in this paper is contained in this author’s Ph.D. thesis [K].



This research is partially supported by a grant from the NSF.

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heat conduction [LLM, EY], random media [GBCB]). There is also a vast literature on purely probabilistic studies of particle systems on lattices. In these systems, each site tends to be “occupied” or “vacant”, with little or no internal dynamics; particles jump from site to site according to probabilistic laws that depend only on the local environment. Sometimes the environment is assumed to be random, i.e. it itself is driven by another (probabilistic) process. One of the aims of this paper is to promote (i) the study of general classes of extended dynamical systems defined by microscopic rules, and (ii) the development of techniques for dealing with hybrid deterministic-stochastic systems, in the hope that such systems will provide a larger class of viable models. To limit the scope, we consider in this paper only systems with deterministic local dynamics and random jumps. A multitude of questions can be asked. We focus on transport properties, and investigate questions of the following type: Start with
an initial density, and describe its time evolution. Now each point in the phase space i∈Zd Mi is described by 2 coordinates, i ∈ Zd and x ∈ Mi . We are particularly interested in the first, which we think of as a macroscopic observable. Our dynamics give rise, then, to an evolving distribution on Zd . What can be said about this distribution? Is there a well defined direction and rate of drift, and what can be said about fluctuations? Needless to say, there are no universal patterns of behavior in the generality above. For example, consider the situation where τi is a dissipative system and site-to-site jumping requires a certain energy threshold. Then an initial distribution may or may not remain localized (by which we mean confined to a finite-size region of Zd for all times) depending on the rate of energy loss and jumping threshold. At the other end of the spectrum, strong mixing properties of τi are likely to mimic stochastic local rules. Sufficiently chaotic local dynamics together with “ellipticity” in site-to-site transitions, therefore, are likely to lead to diffusive behavior on Zd . There is also the possibility of in-between scenarios leading to anomalous transport. In this paper, we focus on the chaotic case. We seek to understand how chaotic the local dynamics have to be to guarantee good statistical properties for macroscopic observables. Other relevant questions are the dependence of various quantities on parameters. We wish especially to demonstrate how such results can be proved using existing techniques for finite (meaning non-extended) dynamical systems. We identify a class of systems for which this approach is natural, and prove some results in a very simple situation free of technical estimates. Our results and proofs suggest many potential generalizations, but they will not be pursued here. 1. Setting and Statement of Results We first introduce a broader class of models, then specialize to a simple case for which some results are stated. 1.1. A class of extended systems. Let τ : M
be a self-map of a manifold M. At each site i ∈ Zd , we place a copy of τ : M
, referring to it as τi :
Mi
when we wish to emphasize its location in Zd . Each point in the phase space i∈Zd Mi is specified by two coordinates (i, x), where i ∈ Zd and x ∈ M (or Mi ). The following notation is needed to specify the transition rules: Let N be the set consisting of the 2d neighbors of 0 in Zd , i.e., each n ∈ N is a vector of the form (σ1 , . . . , σd ), where σk = 1 for some k ∈ {1, 2, . . . , d} and σ j = 0 for all j = k. Associated with each n ∈ N is a “jump map” φn : Un → M and a probability pn ∈ [0, 1] of jumping. We assume {Un , n ∈ N } is a collection of disjoint open subsets of M.

Extended Systems with Deterministic Local Dynamics and Random Jumps

Our system is described by a discrete-time Markov chain on follows. For k = 0, 1, 2, . . . , if (X k , Yk ) = (i, x), then (X k+1 , Yk+1 ) = (i, τ (x))

if

711




i∈Zd

Mi defined as

τ (x) ∈ ∪n∈N Un ,

and if τ (x) ∈ Un for some n ∈ N , then  (i + n, φn (τ (x))) with probability pn , (X k+1 , Yk+1 ) = (i, τ (x)) with probability 1 − pn . What we have described is a hybrid random-deterministic system. Formally, it is a Markov chain, but viewed purely as such, it is very degenerate: a good fraction of the time, the transition probability is a delta measure, i.e. the system is deterministic. 1.2. A model situation. We study in some detail the following special case of the systems introduced in Sect. 1.1. Let d = 1, and let τ : M
be a C 2 uniformly expanding map of S 1 . We specify two jump intervals U , Ur ⊂ S 1 , jump maps φ on U and φr on Ur , and jump probabilities p , pr ∈ [0, 1]. That is to say, if (i, x) is such that τ (x) ∈ U , then the particle jumps to the left, i.e. to site i − 1, with probability p , and so on. The only assumption we impose on φ and φr is that they be C 2 embeddings with min{|φ |, |φr |} · min |τ  | > λ0 for some λ0 > 1. For simplicity of exposition, all of our results are formulated and proved for the main case p , pr ∈ (0, 1). The cases corresponding to at least one of these probabilities being 0 or 1 are discussed at the end of Sect. 2.3. Let (X k , Yk ) with X k ∈ Z, Yk ∈ S 1 , be the process defined in Sect. 1.1. Lebesgue measure on S 1 is denoted by m. Theorem 1 (Drift rates). There exists α ∈ R such that (i) for m-a.e. x ∈ S 1 , if X 0 = 0 and Y0 = x, then n1 X n → α a.s., (ii) if E[|X 0 |] < ∞ and Y0 has a density on S 1 , then n1 E[X n ] → α. The quantity E[X n ] has the interpretation of being the center of mass of the distribution on the lattice Z at time n. Our next result describes the fluctuations of X n about this (moving) center of mass as n → ∞. Theorem 2 (Central Limit Theorem). Let X 0 and Y0 be as in Theorem 1(ii). Then for every interval J ⊂ R, we have    u2 X n − nα 1 lim P e− 2σ du ∈J = √ √ n→∞ n 2π σ J for some σ > 0. Next we turn to the dependence of drift rates on parameters. For definiteness, we fix p , vary p = pr , and let α( p) denote the drift rate of the system. Theorem 3 (Dependence on parameter). α( p) is an analytic function of p. It is natural to ask if α( p) is a monotonic function of p. One would expect the answer to be “yes”, but the problem has turned out to be a little more delicate. This is discussed in Sect. 4. Open question: Does α( p) increase monotonically with p?

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2. Random Maps and Drift Rates We recast the problem in a way that enables us to deduce Theorem 1 from known results.

2.1. Reformulation as random maps. Observe first that the process Yk is itself a Markov chain on S 1 ; its transition probabilities are independent of X k . Indeed, if Y0 = 0, then for each sample path of Yk , X k can be reconstructed – provided φ (y), φr (y) = y for all y. This simplifies the problem, but still, viewed purely as a Markov chain, the transition probabilities of Yk are very degenerate, and the connection to Lebesgue measure is not apparent. The connection to Lebesgue measure comes from the expanding properties of τ . To use that effectively, it is advantageous to view the process as a random map, by which we mean the following: Let ( , P) be a probability on the space of self-maps of S 1 . Corresponding to each i.i.d. sequence (T0 , T1 , T2 , . . . ) in ( , P), we consider compositions of the form T (n) := Tn−1 ◦ · · · ◦ T1 ◦ T0 ,

n = 1, 2, . . . .

This generates what we will refer to as the random maps system defined by ( , P). Sometimes it is convenient to modify the phase space so that the number of jumps can be equated with the number of visits to certain “jump regions”. In the setting of Sect. 1.2, we adjoin two intervals to S 1 , so that each time a jump occurs, the trajectory passes through one of these intervals. More precisely, let S = S 1 ∪ I ∪ Ir , where I and Ir are intervals and the union is disjoint. We assume there is a map ι which maps U isometrically onto I , and a map φ˜  : I → S 1 defined so that φ = φ˜  ◦ ι ; mappings associated with Ir are defined analogously. We introduce 4 maps, T∗ , T , Tr and Tr , from S to itself defined as follows: (i) For x ∈ τ −1 U , T (x) = Tr (x) = ι (τ (x)) ∈ I ; for x ∈ τ −1 Ur , Tr (x) = Tr (x) = ιr (τ (x)) ∈ Ir . (ii) For x ∈ S 1 and T ∈ {T∗ , T , Tr , Tr } not covered by (i), T (x) = τ (x). (iii) For x ∈ I , T (x) = T (φ˜  (x)) for all T ∈ {T∗ , T , Tr , Tr }; for x ∈ Ir , T (x) = T (φ˜r (x)) for all T ∈ {T∗ , T , Tr , Tr }. In (iii), T (φ˜  (x)) and T (φ˜r (x)) are to be interpreted as defined in (i) and (ii); notice that φ˜  (x), φ˜r (x) ∈ S 1 . Now let = {T∗ , T , Tr , Tr }, and let the probability P of each of these 4 constituent maps be (1 − p )(1 − pr ),

p (1 − pr ),

pr (1 − p ) and p pr

respectively. We denote the resulting random maps system by T. We claim that T faithfully represents the dynamics of the Markov
chain (X k , Yk ) defined in Sect. 1.2. Clearly, T is equivalent to a Markov chain on i∈Z Si with transition probabilities ⎧ ⎪ ⎨ (i, T (x)) (i, x) → (i − 1, T (x)) ⎪ ⎩ (i + 1, T (x))

if x ∈ S 1 , if x ∈ I , if x ∈ Ir .

T ∈ ,

Extended Systems with Deterministic Local Dynamics and Random Jumps

starting from Si0 for some (fixed) i 0 ∈ Z. Let π : ⎧ ⎪ ⎨ (i, x) π(i, x) = (i − 1, φ˜  (x)) ⎪ ⎩ (i + 1, φ˜ (x)) r




i∈Z Si

→

713




1 i∈Z Si

be given by

if x ∈ S 1 , if x ∈ I , if x ∈ Ir .

Via
π , we have a one-to-one correspondence between sample paths of the Markov chain on i∈Z Si and those of (X k , Yk ): we have merely introduced two “holding intervals”, namely I and Ir , to make it easier to count the number of jumps later on, and delayed the change in the X -coordinate by one time-step (which is harmless). 2.2. Invariant measures for piecewise expanding random maps. We recall some standard language: Let  be a metric space, and f a random maps system on  defined by { f i , i = 1, . . . , n} with P( f i ) = pi . A probability measure µ on  is called an invariant measure of f if for all Borel subsets E ⊂ , µ(E) =

n

pi · µ( f i−1 (E)).

i=1

Assuming pi > 0 for all i, we say µ is ergodic if there is no E ⊂ Y with 0 < µ(E) < 1 such that modulo sets of µ-measure 0, E is invariant under the random maps process. This is easily seen to be equivalent to f i−1 (E) = E mod 0 for all i. Invariant probability measures absolutely continuous with respect to Lebesgue measure are abbreviated as a.c.i.p.m.. Proposition 2.1. The following hold for the system T defined in Sect. 2.1: (1) T has a unique a.c.i.p.m., which we call µ; dµ (2) dm has bounded variation and is ≥ c0 everywhere for some c0 > 0. The proof uses the idea of transfer operators. For  and f as above, suppose there is a reference measure m on  with respect to which all the f i are nonsingular. Let L fi d f ∗ν

dϕ i denote the transfer operator associated with f i , i.e. if ν = dm , then L fi (ϕ) = dm , −1 ∗ ∗ where f i (ν) is the measure defined by f i (ν)(E) = ν( f i (E)). The transfer operator for the random map f is defined similarly, i.e.

Lf =

k

pi L fi .

i=i

Proof. (1) Our hypotheses ensure that each of the 4 constituent maps T of T is piecewise C 2 and uniformly expanding. It follows from the well known inequality of Lasota and  Yorke [LY] that if ϕ denotes the total variation of ϕ on S, then there exist a positive integer N and constants c ∈ (0, 1) and C > 0 such that for every ϕ of bounded variation, LTN (ϕ) ≤ c ϕ + Cϕ1 . (1) Since LT is a weighted average of these LT , (1) holds all the more for LT . From this point on, the proof of existence of an a.c.i.p.m. is identical to that of a single map: Iterating, one deduces that sup LkT (1) < ∞. k≥0

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n−1 k This gives the relative compactness of the sequence n1 i=0 LT (1). Any accumulation point is an invariant density of T. To prove uniqueness, let µ be an invariant measure of T with a density, such as the one constructed in the last paragraph, and let E ⊂ S be an invariant subset with m(E) > 0. Since T∗ (I ∪ Ir ) ⊂ S 1 , we have m(E ∩ S 1 ) > 0. Now τ : S 1
is well known to have an a.c.i.p.m. ν with a strictly positive density, and (τ, ν) is ergodic. By iterating T∗ , therefore, we deduce that E ⊃ S 1 mod 0. Since T (S 1 ) ⊃ I and Tr (S 1 ) ⊃ Ir , we conclude that E has full m-measure. (2) To prove that the density h of µ is bounded away from 0 everywhere on S, it suffices to show that (i) there is an interval J ⊂ S 1 on which h is bounded away from 0, and (ii) there is a sequence of maps T0 , T1 , . . . , Tk−1 such that T (k) (J ) = S. (i) is true because h has bounded variation and is > 0 m-a.e. (ii) is true because there exists n = n(J ) such that T∗n (J ) = S 1 , and as before, T (S 1 ) ⊃ I , Tr (S 1 ) ⊃ Ir .   2.3. Drift rates. Let χ A denote the characteristic function of a set A. Proof of Theorem 1. Let T be as above, and let ϕ = χ Ir − χ I . Proposition 2.1 together with the Ergodic Theorem tells us that for m-a.e. x ∈ S and a.e. sequence of maps, n−1 1

ϕ(T (i) (x)) → µ(Ir ) − µ(I ) n

as n → ∞.

(2)

i=0

As explained earlier, X k increases by 1 exactly when T (k) (x) ∈ Ir and decreases by 1 when T (k) (x) ∈ I . For the initial condition X 0 = 0 and Y0 = x, then, the left side of (2) is the net displacement of X k in the first n iterates. This implies that for m-a.e. x, the a.s. drift rate α is well defined and is equal to µ(Ir ) − µ(I ). The second assertion of Theorem 1 follows by integrating.   Remarks on the cases where p or pr ∈ {0, 1}. If p , pr = 1, the situation is as in the main case; we may set I = ∅ if p = 0, and so on. If p or pr = 1, then P(T∗ ) = 0, and the ergodicity argument can, in principle, fail. A well defined drift rate α(x) exists for m-a.e. x, but if the system is not ergodic, these rates may not be constant a.e. 3. Statistical Properties 3.1. Spectral properties of LT . To prove the central limit theorem and other statistical properties, it is useful to have the following strengthening of Proposition 2.1.  1 ϕ˜ < ∞}. As before,  Let X = {ϕ ∈ L (S) : ∃ϕ˜ be such that ϕ = ϕ˜ mod 0 and ϕ denotes the total variation of ϕ on S.We consider the following two norms on X: |ϕ|1 is the L 1 norm, and ϕ := |ϕ|1 + ϕ. The spectrum of an operator is denoted σ (·) and its spectral radius ρ(·). Proposition 3.1. (a) LT is a quasi-compact operator on the Banach space (X,  · ); (b) ρ(LT ) = 1; 1 is the unique eigenvalue of modulus 1, and its eigenspace is 1D. Proof. (a) follows from Eq. (1) and [H]. (b) follows once we prove the ergodicity of (Tn , µ) for all n ≥ 1. The case n = 1 is contained in Proposition 2.1. The proof for n > 1 is very similar: If E ⊂ S is invariant under Tn , then E ⊃ S 1 by iterating T∗n , and if T (n) is such that Tn−1 = T and Tk = T∗ for k < n − 1, then T (n) (S 1 ) ⊃ I , and so on.  

Extended Systems with Deterministic Local Dynamics and Random Jumps

715

The eigenfunction corresponding to eigenvalue 1 isthe density h of µ. From Proposition 3.1, it follows that for ϕ ∈ L ∞ and ψ ∈ X with ψdµ = 1,             (ϕ ◦ Tn )ψdµ − ϕdµ ψdµ =  ϕ · Ln (ψh)dm − ϕhdm  T      ≤ |ϕ|∞ |LnT (ψh) − h| dm ≤ |ϕ|∞ · C(ψ)θ n , where C(ψ) is a constant and sup{|z| : z ∈ σ (LT ) \ {1}} < θ < 1. 3.2. CLT for random maps. Let T be the random maps system defined in Sect. 2.1. As a prelude to Theorem 2, we prove  Proposition 3.2. Let f ∈ X be such that f dµ = 0. Then the CLT holds for { f ◦ Tn , n = 1, 2, . . . } with respect to the probability µ. Its variance is given by  ∞ 

2 2 f dµ + 2 f · ( f ◦ Tn )dµ. σ =− n=0

The route we intend to take involves comparing our random variables f ◦ T (n) to a sequence of reverse martingale differences, and to appeal to the central limit theorem (CLT) for such sequences. We recall the following known result: Theorem 3.1. [N]. Let Z n , n ≥ 1, be a stationary, ergodic sequence of martingale (resp. reverse martingale) differences with respect to an increasing (resp. decreasing) filtration Fn . If Z 1 ∈ L 2 , then the CLT holds with σ 2 = E[Z 12 ]. As we will see, the CLT for f ◦ Tn is a consequence of the chaotic properties of the constituent maps of T, not the Markov property of the system. For this reason, we again pursue a dynamical systems formulation. The following general result relates dynamical observations to martingale differences: Let (, F, ν) be a probability space and let T : 
be a measure-preserving transformation. Let Tˆ : L 2 (, ν)
be the operator defined by Tˆ f = T ◦ f , and let Tˆ ∗ denote the adjoint of Tˆ . Lemma 3.1. [L]. Let (, F, ν) and T be as above, and assume in addition that T is not invertible, so that Fi := T −i F, i = 0, 1, 2, . . . is a decreasing filtration. Let f ∈ L ∞ .

ˆ∗ n If the sum ∞ n=0 (T ) f converges a.e. to a function g, then Z i := Tˆ i f − (Tˆ i g − Tˆ i−1 g),

i = 1, 2, . . . ,

is a reverse martingale difference with respect to the filtration Fi . Returning to our model, we represent T as a skew-product over a shift space as is often done for random maps. More precisely, let s : ( N , P N )
be the Bernoulli shift on the symbol space = {T∗ , T , Tr , Tr }, where ( , P) is as in Sect. 2.1. That is to say, elements of N are infinite sequences ω = (ω0 , ω1 , . . . ), where each ωk ∈ , P N is the product measure of P, and s(ω) = (ω1 , ω2 , . . . ). Define F : N × S
by F(ω, x) = (s(ω), ω0 (x)). Observe that

PN

× µ is F-invariant.

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E. Kobre, L.-S. Young

In the discussion below, we will confuse functions defined on S with functions defined on N × S depending only on the second variable. Recall that X is the space of functions of bounded variations on S. 

ˆ∗ n Lemma 3.2. Let f ∈ X be such that f dµ = 0. Then ∞ n=0 (F ) f converges a.e. ∞ N and in L with respect to P × m. Proof. Notice first that F is nonsingular with respect to P N ×m with Jacobian J F(ω, x) = P(ω0 )−1 · J ω0 (x). It follows that for functions f on N × S depending only on the S-variable, LF f also depends only on the S-variable, and is equal to LT f . Now with d(P N × µ) dµ = = h ≥ c0 > 0 N dm d(P × m) (Proposition 2.1), we have 1 1 Fˆ ∗ f = LF ( f h) = LT ( f h), h h  and hence (Fˆ ∗ )n f = h1 LnT ( f h). Proposition 3.1 together with f hdm = 0 then gives n n LT ( f h) ≤ const·θ for some θ < 1. The assertion in the lemma follows.   Proof of Proposition 3.2. It suffices to prove the result for F, with f ∈ X. The con ˆ∗ n vergence of ∞ n=0 (F ) f needed in Lemma 3.1 is verified in Lemma 3.2. We know, therefore, that Z i := Fˆ i f − (Fˆ i g − Fˆ i−1 g) is a reverse martingale difference. To apply Theorem 3.1, we need to show that E[Z 12 ] < ∞. A straightforward computation gives ˆ − g))2 ] = − E[(Fˆ f − (Fg

 f 2 dµ + 2

∞ 

f · Fˆ n ( f ) d(P n × µ).

n=0

The convergence of the sum on the right follows from the exponential decay of covariances

n

n noted in Sect. 3.1. Finally, to go from √1n i=1 Z i to √1n i=1 Fˆ i f , observe that n 1 ˆi 1 (F g − Fˆ i−1 g) = √ (Fˆ n g − g) → 0 as n → ∞. √ n n i=1

  3.3. Proof of Theorem 2. The proof of Theorem 2 follows from Proposition 3.2 with a suitable choice of f together with a sequence of approximations. The relevant choice of f : S → R here is f (x) = (χ Ir − χ I ) − {µ(Ir ) − µ(I )}, where µ is the invariant measure of T. Lemma 3.3. Let f be as above. Then the variance σ 2 in Proposition 3.2 is > 0.

Extended Systems with Deterministic Local Dynamics and Random Jumps

717

Proof. This is equivalent to proving Z 1 ≡ 0, i.e. there is no g ∈ L ∞ (S) such that for a.e. x ∈ S and every T ∈ , f (T x) = g(T x) − g(x). Letting T = T∗ and summing the relation f (T i x) = g(T i x) − g(T i−1 x) over i = 1, 2, . . . , n, we see that g(T n x) = g(x) + na. Thus if a = 0, the boundedness of g is violated for large n. If a = 0, the same contradiction can be achieved by applying only T and T a large but finite number of times. Thus no such g exists; hence σ > 0.   Deducing Theorem 2 from Proposition 3.2 and Lemma 3.3. We may assume X 0 = 0, √ for the difference is negligible after dividing by n and letting n → ∞. Let µ0 be the distribution of Y0 . We think of µ0 as a probability on S 1 ⊂ S. Given an interval J ⊂ R and a small number ε > 0, we must show, via the approximations below, that for all large enough n, 

n−1 1

(P × µ0 ) √ f (T i x) ∈ J n

 ∈ (b − ε, b + ε),

n

i=0

where 1

b= √ 2π σ



u2

e− 2σ du. J

(1) By Proposition 3.2, there exist intervals J − ⊂ J ⊂ J + , both containments being strict, and N1 ∈ Z+ such that for all n > N1 , 

 n−1 1

1 i − (P × µ) √ f (T x) ∈ J > b − ε, 2 n i=0   n−1 1

1 n i + (P × µ) √ f (T x) ∈ J < b + ε. 2 n n

i=0

(2) Let h 0 be the density of µ0 , and choose N2 such that for all n > N2 , 

1 ε. 2

|LnT (h 0 ) − h|dm <

Let µn denote the measure whose density is LnT (h 0 ). (3) Choose N >> N1 , N2 such that for all n > N , 1 √ N √

N2

+N −1

1 N2 + N

f (T i x) ∈ J − =⇒ √

i=N2 N2

+N −1 i=0

N2

+N −1

1 N2 + N

1 f (T i x) ∈ J =⇒ √ N

N2

+N −1 i=N2

f (T i x) ∈ J,

i=0

f (T i x) ∈ J + .

718

E. Kobre, L.-S. Young

From (1)–(3), we deduce that for n > N ,   N2

+n−1 1 N2 +n i (P × µ0 ) √ f (T x) ∈ J N2 + n i=0   n−1

1 ≤ (P n × µn ) √ f (T i x) ∈ J + n i=0   n−1

1 1 ≤ (P n × µ) √ f (T i x) ∈ J + + ε < b + ε. 2 n i=0

A lower bound is proved similarly.

 

Remarks. 1. The spectral gap in Proposition 3.1 is not needed for the arguments in this section. Summability of covariances will suffice. 2. Many other statistical properties, such as correlation decay and large deviations, can be treated in a similar way. 3. An interesting question is recurrence, which can be formulated as whether or not a.s. X n = 0 infinitely often. This question is especially interesting in 2-dimensions; see recent results [C, Sch]. 4. Dependence of Drift Rate on Jump Rates In this section we fix p and let p = pr vary. All objects are viewed as functions of p. Proof of Theorem 3. For p ∈ C, we define LT ( p) = (1 − p )(1 − p)LT∗ + p (1 − p)LT + p(1 − p )LTr + p pLTr . Then p → LT ( p) is a analytic family of operator-valued functions. Let p0 ∈ (0, 1) ⊂ R be fixed. It follows from Proposition 3.1 and [DS] that there is a neighborhood N of p0 in C such that for all p ∈ N , LT ( p) has a unique eigenvalue λ( p) whose modulus is strictly larger than that of the rest of the spectrum. Moreover, the eigenspace of λ( p) is one-dimensional, and E( p), the projection onto this eigenspace, is analytic. Hence h( p) := E( p)(1), where 1 here denotes the constant function, is also analytic. For p ∈ N ∩ R, λ( p) = 1, and h( p) is the eigenfunction of LT ( p) with h( p)dm ¯ = 1. Now for p ∈ (0, 1), the drift rate is given by α( p) = µ( p)(Ir ) − µ( p)(I ). To prove the analyticity of this function, it suffices

to show the analyticity of p → h( p)χ I dm for I = I and I = Ir . Write h( p) = an ( p − p0 )n . Since an χ I  ≤  an , we are n near p . Thus h( p)χ dm = assured of the uniform convergence of (a χ )( p− p ) n I 0 0 I 

  an χ I dm ( p − p0 )n .  Monotonic dependence of α( p) on p? Intuitively, it seems obvious that with p fixed, the higher the probability of jumping to the right, the larger the drift rate should be. This naïve argument, however, contains the following difficulty: Consider two particles with identical trajectories through step k, and Yk = Yk ∈ τ −1 (Ur ). Suppose further that  X k+1 = X k , i.e. the first particle does not jump, and X k+1 = X k + 1, i.e. the second  one jumps to the right at the next step. Since Yk+1 = Yk+1 , it is difficult to compare the Y -coordinates of the two trajectories from this point on. In particular, it is possible that the second particle may reach U sooner, thus having the possibility of jumping to

Extended Systems with Deterministic Local Dynamics and Random Jumps

719

the left sooner; it may even make several jumps to the left in quick succession. If this happens, then increasing pr may in fact lead to a greater tendency to go to the left, and hence a smaller drift rate. We do not know, however, that such a scenario can actually happen. Looked at from a different angle, monotonicity of α( p) is equivalent to the monotonicity of the quantity µ( p)(Ir ) − µ( p)(I ), a proof of which is also not apparent to us except in certain special cases. 5. Summary and Conclusions We have demonstrated in the preceding pages that for systems with deterministic local dynamics and random jumps, (i) one can sometimes take better advantage of the local dynamics if the system is viewed as a random map (rather than a Markov chain); (ii) sometimes it is possible to reduce the extended system, i.e. the system on Zd , to a smaller system derived from the local dynamics τ : M
at each site; (iii) assuming (i) and (ii), if the probability of jumping is a function of the local coordinate x ∈ M, then the constituent maps of the random map are likely to be obtained by replacing τ on certain domains U ⊂ M by “sewing maps” φ : U → M; (iv) the statistical properties of the large system hinge on the properties of these surgically altered maps {T }. For example, if τ is Anosov, then T is piecewise hypberbolic (though not necessarily 1–1) if φ respects the invariant cones of τ . We remark that (iii) can sometimes be arranged even when the probability of jumping is given by a continuous function. If, for example, in the system studied in Sects. 1–4, the probability of jumping to the right is given by a probability density function β with supp(β) ⊂ (−b, b), β increasing from −b to 0 and decreasing from 0 to b, then one considers an infinite family of maps surgically altered on increasing domains Uγ with {0} ⊂ Uγ ⊂ (−b, b). These and similar modifications of the techniques in this paper provide a means to analyze a large class of extended dynamical systems. References [C] [DS] [EY] [GBCB] [H] [K] [LY] [LLM] [L] [N]

Conze, J.-P.: Sur un critere de recurrence en dimension 2 pour les marches stationnaires, applications. Ergodic Theory Dynam. Sys. 19(5), 12331245 (1999) Dunford, N., Schwartz, J.: Linear Operators, Part I. New York: Wiley, 1957 (Sect. VII.6 in particular) Eckmann, J.-P., Young, L.-S.: Nonequilibrium energy profiles for a class of 1-d models. Commun. Math. Phys. 262, 237–267 (2006) Grosfils, P., Boon, J.P., Cohen, E.G.D., Bunimovich, L.: Propagation and Organization in Random Media. J. Stat. Phys. 97, 575–608 (1999) Hennion, H.: Sur un theoreme spectral et son application aux noyaux lipchitziens. Proc. Am. Math. Soc. 118, 627–634 (1993) Kobre, E.: Rates of diffusion in dynamical systems with random jumps. Ph.D. thesis, New York University (2005) Lasota, A., Yorke, J.A.: On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186, 481–488 (1973) Larralde, H., Leyvraz, F., Mejia-Monasterio, C.: Transport properties of a modified lorentz gas. J. Stat. Phys. 113, 197–231 (2003) Liverani, C.: Central limit theorem for deterministic systems. In: International Conference on Dynamical Systems (Montevideo 1995), Pitman Research Notes in Math. Series 362, London: CRC Press, 1997 Neveu, J.: Mathematical foundations of the calculus of probability. San Francisco CA: Holden-Day, 1965

720

[Sch] [Si]

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Schmidt, K.: On joint recurrence. Comptes Rendu Acad. Sci. Paris Ser. I Math. 327(9), 837–842 (1998) Sinai, Ya.G.: Dynamical systems with elastic relections: ergodic properties of dispersing billiards. Russ. Math. Surv. 25(2), 137–189 (1970)

Communicated by A. Kupiainen

Commun. Math. Phys. 275, 721–748 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0313-4

Communications in

Mathematical Physics

Complexity for Extended Dynamical Systems Claudio Bonanno1 , Pierre Collet2 1 Dipartimento di Matematica Applicata, Università di Pisa, via F. Buonarroti 1/c, 56127 Pisa, Italy.

E-mail: [email protected]

2 Centre de Physique Théorique, École Polytechnique, CNRS UMR 7644, F-91128 Palaiseau Cedex, France.

E-mail: [email protected] Received: 22 September 2006 / Accepted: 2 March 2007 Published online: 15 August 2007 – © Springer-Verlag 2007

Abstract: We consider dynamical systems for which the spatial extension plays an important role. For these systems, the notions of attractor,
-entropy and topological entropy per unit time and volume have been introduced previously. In this paper we use the notion of Kolmogorov complexity to introduce, for extended dynamical systems, a notion of complexity per unit time and volume which plays the same role as the metric entropy for classical dynamical systems. We introduce this notion as an almost sure limit on orbits of the system. Moreover we prove a kind of variational principle for this complexity. 1. Introduction Dynamical systems are called “extended” when the spatial extension plays an important role. They occur for example, in nonlinear partial differential equations of parabolic or hyperbolic type when the size of the domain is much larger than the typical size of the structures developed by the solutions. As in Statistical Mechanics, one can try to use the infinite volume limit as an approximation. For several classes of such systems, it has been shown that one can define the semiflow of evolution in unbounded domains acting on bounded functions with some regularity (see for example [4, 15, 20, 7]). This is particularly convenient when studying traveling solutions or waves, since one would not like to fix some particular boundary conditions which restrict the nature of the solution (for example fixing a particular spatial period). Once the dynamics has been defined in an unbounded domain, one can ask for a notion of attractor. Such a notion was introduced by Feireisl (see [14] and [21]) by observing the system in bounded windows and inferring the result for the unbounded domain. When the evolution equation does not depend explicitly on space (homogeneous system), the attractor is translation invariant and often non-compact of infinite dimension. However, if restricted to a finite window it is often a compact set. A situation which occurs in several examples is that the functions in the attractor are analytic and

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bounded in a strip around the real domain (see for example [5, 27]). Compactness follows in bounded (real) regions when using C k norms for example. For such systems with non-compact translation invariant infinite dimensional attractors, one can try to define extensive quantities as in statistical mechanics. A notion of dimension per unit volume can be defined from the
-entropy per unit volume of Kolmogorov, see [18], where it was used in particular to quantify the fact that some function spaces are larger than others. Looking for example at an attractor composed of functions analytic in a strip and of infinite dimension, since an analytic function is completely determined by its data in a finite domain, the dimension observed in any finite window will always be infinite. To avoid this uninteresting result, one first fixes a precision
> 0. One then counts for
of balls of radius
needed to cover the attractor in the example the minimal number N
/|| exists, finite window . The next step is to prove that H (
) = lim||→∞ log2 N −1 and then to consider the quantity H (
)/ log2
for small
. As mentioned above, the order in which the limits in  and
are taken is important. If for fixed  one lets first
tend to zero, the result is in general infinite, while in the other order, one can get finite results. These ideas were applied to the attractors of various extended systems (see [8, 13, 6]). These ideas can also be adapted to give a definition of the topological entropy per unit volume (see [9, 10, 6, 30 and 31]). One first fixes a finite precision, considers
(T ) of different trajectories one can observe in a finite window the maximal number N  on the time interval [0, T ] at this given precision. One then considers the limits
(T ) log2 N 1 lim .
0 ||→∞ || T →∞ T

h top = lim lim

Here again the order of the limits is crucial, otherwise one may get an infinite quantity. Regarding similar approaches, angular limits have been proposed in [23] for cellular automata, and for entropies, topological and metric, along subspaces we refer to [22] and [28] and references therein. In [9] a similar definition was proposed for the metric entropy along the same limits as for the topological entropy. However this definition involves several limits which are up to now not known to exist. In order to circumvent this difficulty we deal in the present paper with the Kolmogorov complexity. For dynamical systems on a compact phase space with an ergodic invariant measure, it is known that the complexity per unit time of a typical trajectory is equal to the metric entropy (see [3, 29]). A first advantage of the complexity per unit time is that it can be defined for each trajectory with initial condition on a full measure set. We will also see below that the complexity satisfies some useful sub-additivity properties allowing to define a complexity per unit time and unit volume. The strategy is the same as for the topological entropy. We first fix a precision
. We then consider the complexity per unit time of a coding of these trajectories in the window  using a covering by balls of radius at most
. We then show that this quantity grows like the volume and define a complexity per unit time and unit volume at a fixed precision, finally letting the precision become infinite. We will deal in the present paper with systems satisfying some hypothesis inspired by the known results on extended systems. In particular we will not assume that the attractor is compact but that it is translation invariant. We will also assume a space time invariant ergodic measure is given. In other words, our results apply to R × R actions satisfying the hypothesis given below. In particular, we assume that the semi-flow on the function space is a flow when restricted to the attractor of the system. This follows for example from the analyticity in time of the solutions of the evolution equations. The

Complexity for Extended Dynamical Systems

723

procedure described above differs with the more standard approach to the space time entropy which uses boxes of roughly the same size in the space and time direction. It is however more natural from the point of view of the definition of the attractors of such systems. Moreover, the order of the limits and the property of being pointwise make our notion of complexity particularly suitable for numerical applications (see [1] for a discussion about dynamical systems). Regarding similar approaches, as we mentioned earlier and as far as we know, it is not known if one can define a metric entropy for extended systems using this more natural sequence of very anisotropic space time domains. One can also ask if there is an extension of Brudno’s theorem in the present context, namely if the complexity defined in the present paper equals the entropy per unit space time volume defined for R2 actions along space time boxes of bounded aspect ratio (in the case of the topological entropy, the various notions coincide, see [31]). In order to open the possibility of using other types of complexities, we have tried to isolate the properties we need without reference to a particular example, although the Kolmogorov complexity satisfies all the requirements. In order to simplify the proofs, we only discuss the case of one space dimension, although most results extend easily to higher dimension. The paper is organised as follows. In Sect. 2 we first state the required hypothesis on the dynamical system and on the complexity, and show that these hypotheses are satisfied by Kolmogorov complexity. We then state the main results. In Sect. 3 we prove that under these hypotheses one can define a complexity per unit time and unit volume. This is done following the scheme briefly mentioned above of fixing first a finite precision and removing it only at the end. In Sect. 4 we prove a variational principle for the complexity which shows that in the concrete examples of extended systems studied up to now, the complexity we have defined is finite. In fact, we show that computing the supremum of the complexity for functions in the supports of the invariant measures of the system, one obtains the topological entropy defined in [9]. In particular, by the classical variational principle (see [24 and 28]), this implies that the supremum of the complexity coincides with the supremum of the metric entropies along space time boxes of bounded aspect ratio. However it should be further investigated whether the two notions of complexity and metric entropy always coincide. 2. Settings and Results Let F be a set of real functions defined on R and consider the following actions on F: the space translation R  y → (ζ y u)(x) := u(x + y) and a flow of time evolution ϕt : F → F defined for t ∈ R. We assume that the two actions commute. We assume that the set F is endowed with a translation invariant metric d and that there exists a probability measure µ on F, such that µ is invariant and ergodic with respect to the (ζ, ϕ) action. We make the following assumptions on the set F and the flow ϕ. Let us assume that for any interval  ⊂ R the set F| := {g :  → R : ∃ f ∈ F with f | ≡ g}

724

C. Bonanno, P. Collet

is endowed with a metric d| such that (F, d) is the projective limit of (F| , d| ) as || → ∞. If for example F ⊂ Cb0 (R), the set of real bounded continuous functions on R, and d is the sup-norm, then for every  ⊂ R we have d| (g1 , g2 ) = supx∈ |g1 (x) − g2 (x)|. We assume that for any interval  ⊂ R we have (A1)

|| < ∞ =⇒ F| is pre-compact :

By Assumption (A1), for any
> 0 and || < ∞ we can define the set
= {finite open coverings of F| with balls of radius <
} , C

(2.1)


an element of C
. Fixed two finite intervals  and  with and we denote by U 1 2  disjoint interiors let  be the union  := 1 ∪ 2 , then we assume that
∈ (A2) there exists an integer q depending only on the metric d such that, for any U 1




C1 and U2 ∈ C2 and two balls B1 ∈ U1 and B2 ∈ U2 , either the intersection
∈ C
. B1 ∩ B2 is empty or can be covered by q balls of a covering U 

The last assumption on the system is about the separation speed of two nearby functions with time. We assume that there are constants γ > 0,  > 1 and C > 0 such that, for any || < ∞ and any
> 0 satisfying diam() > 2C
−1 and for any initial conditions f 1 and f 2 in F such that d| ( f 1 , f 2 ) <
, we have (A3)

d|\{d(x,∂)
for any t ∈ (0, C −1 diam()
) (cfr. [9]). Under these assumptions a notion of topological entropy for the flow ϕ has been defined in [9]. Let



N (T ) := max card(S (T )) : S (T ) is made of (, T,
)-distinguishable orbits , (2.2) where we say that f and g in F| have (, T,
)-indistinguishable orbits up to time T and with resolution
if d| (ϕt ( f ), ϕt (g)) <


∀ t ∈ (0, T ).

In [9, 10 and 30] it is proved that h top := lim

lim


0 ||→∞


(T ) log2 N 1 lim || T →∞ T

(2.3)

exists and is finite under some additional assumptions, in fact it is bounded by γ Dup , where Dup is called the upper local dimension per unit length of the set F in [9] or capacity per unit length in [18]. The aim of this paper is to introduce a measure of the complexity of the action of the flow ϕ on F which would be the analogue of the metric entropy for dynamical systems. To this aim we need to define a notion of complexity. Our definition is inspired by the notion of Kolmogorov complexity ([19]). Let A∗ be the set of finite words on a finite alphabet A, and for a word s let us denote by |s| its length. We say that K : A∗ → R+ , defined for any alphabet A, is a “good” complexity function if it satisfies the following hypotheses (H1)–(H4).

Complexity for Extended Dynamical Systems

725

The first hypothesis is about the behaviour of the complexity function on sub-words and a sub-additivity property. Let s = uv be the concatenation of two words u and v, then K (u) ≤ K (s) + log2 |u| + const

(H1.a)

for a constant independent on s and u. Moreover let us assume that there exists a function h : N → R+ satisfying limn→∞ h(n) n = 0 such that (H1.b)

K (s) ≤ K (u) + K (v) + h(|u|) + h(|v|).

Let now A1 and A2 be two different alphabets, with ri := card(Ai ). Moreover let A˜ ˜ = qr1 r2 for some integer number q ≥ 1, and we assume be an alphabet with card(A) that there exists a surjective map π : A˜ → A1 × A2 , with coordinate maps π1 and π2 on A1 and A2 , respectively. Let s ∈ A˜ ∗ and πi (s) ∈ Ai∗ be its projections. Then (H2.a) (H2.b)

K (πi (s)) ≤ K (s) + const

K (s) ≤ K (π1 (s)) + K (π2 (s)) + |s| log2 q + const,

where the constants are independent on s. The third hypothesis is an estimate on K that comes from observations by Shannon for his definition of information content ([26]). Let E ⊂ A∗ × N be a recursively enumerable set (for a definition see for example [19]), and for any n ∈ N let L n := {s ∈ A∗ : (s, n) ∈ E} be a set of finite cardinality. Then we assume that for all n ∈ N, (H3)

K (s) ≤ log2 (card(L n )) + log2 n + const

∀ s ∈ Ln

holds, where the constant only depends on the set E. Finally we ask for a relation between the bound of the complexity on a set of words and the cardinality of this set. We assume that 
∀c ∈ R. (H4) card s ∈ A∗ : K (s) < c ≤ 2c By using a “good” complexity function let us now define the complexity of the flow ϕt . Consider a fixed probability measure µ which is invariant and ergodic for the action of (ζ, ϕ). For a given
> 0 and an interval  ⊂ R with || < ∞, we consider
. We will use such coverings to code the orbit of a on F| the set of coverings C function f ∈ F under ϕ. To this aim, we introduce a time step τ > 0 and consider the orbits ( f, ϕτ ( f ), ϕ2τ ( f ), . . . ). By the method of symbolic dynamics we can asso
) on a finite alphabet U ciate to an orbit (ϕ jτ ( f ))n−1 j=0 a set of n-long words ψ( f, n,  

) = 1, . . . , card(U
) . If we denote U
:= U , . . . , U
) , we A = A(U 1 card(U   define1  

ψ( f, n, U ) := ω0n−1 ∈ A(U ) : ϕ jτ ( f ) ∈ Uω j ∀ j = 0, . . . , n − 1 . 1 Since the covering is made by open sets, the methods of symbolic dynamics give more than one word.

726

C. Bonanno, P. Collet


) as the set of In the same way we can define in the general case ψ(ϕmτ ( f ), n − m, U n−1 possible symbolic representations of the orbit (ϕ jτ ( f )) j=m . At this point we can use a complexity function K to define  
n−1 n−1
, m, n) := min K (ωm ) : ωm ∈ ψ(ϕmτ ( f ), n − m, U ) . (2.4) K ( f, τ, U

To simplify notations, for m = 0 we will write  

K ( f, τ, U , n) := min K (ω0n−1 ) : ω0n−1 ∈ ψ( f, n, U ) .

(2.5)

We can define the asymptotic linear rate of increase in n by
K ( f, τ, U ) := lim

n→∞


, n) K ( f, τ, U . n

To get rid of the dependence on the covering we define



K ( f, τ,
, ) := inf K ( f, τ, U . ) : U ∈ C

(2.6)

(2.7)

The next step will be to study the asymptotic rate of increase in ||. We restrict ourselves to a class of intervals defined as follows. Definition 2.1. A sequence of sets
= {k }k is called admissible if k = [ak , bk ] for two sequences {ak }k and {bk }k satisfying ak < bk for all k ≥ 1 and lim (bk − ak ) = +∞,

k→∞

la := lim inf

bk −ak max{ak ,0}

lb := lim inf

bk −ak − min{bk ,0}

k→∞

k→∞

> 0, > 0.

(2.8) (2.9) (2.10)

Intuitively, this definition says that these sequences do not move too fast to the left or to the right. If
is an admissible sequence of sets, let us define K µ ( f, τ,
) := lim

k→∞

K ( f, τ,
, k ) . |k |

(2.11)

Given these definitions, we will prove that Theorem 2.2. For a given ergodic probability measure µ, if the complexity function K satisfies (H1) and (H2), the limits in (2.6) and (2.11) exist almost surely and K ( f, τ,
) is almost surely equal to a constant K µ (τ,
) not depending on the admissible sequence
of sets. Moreover the function K µ (τ,
) is not decreasing in
, hence the limit K µ (τ ) := lim K µ (τ,
)
→0

exists and moreover there exists a constant K µ such that for all τ > 0, K µ (τ ) = Kµ. τ

Complexity for Extended Dynamical Systems

727

Theorem 2.3. If the complexity function satisfies also (H3) and (H4), then
 sup K µ : µ invariant probability measures = h top , where h top is defined in (2.3). Before giving the proofs of these theorems, we recall that for a finite word s ∈ {0, 1}∗ , the Kolmogorov complexity or Algorithmic Information Content of s is defined as
 C(s) := min |w| : w ∈ {0, 1}∗ , U (w) = s , (2.12) where | · | denotes the length of a word, and U is a universal Turing machine. For more details we refer to [19]. Theorem 2.4. The Kolmogorov complexity satisfies hypotheses (H1)–(H4). Proof. We recall that the translation of a finite word from the binary alphabet to any other finite alphabet A requires only a constant amount of information content not dependent on the word. Hence we assume that these constants are included in the hypotheses (H1)–(H3). Hypotheses (H1) and (H2) follow from [19], Eq. (2.2) and arguments used in [19], Sect. 2.1.2. Hypotheses (H3) is a corollary of Theorem 2.1.3 in [19]. Hypotheses (H4) is Theorem 2.2.1 in [19].   3. Proof of Theorem 2.2 Let us consider any fixed probability measure µ which is invariant and ergodic for the action of (ζ, ϕ). The first part of the proof relies on the application of arguments related to the subadditivity property to define the quantities in (2.6) and (2.11). Let X = (X m,n ) be a family of real random variables with indexes m, n ∈ N. We recall that X is almost subadditive if there exists a family of random variables U = (U j ), with j ∈ N, defined on the same probability space of X such that X m,n ≤

k−1 

(X ji , ji+1 + U ji+1 − ji )

(3.1)

i=1

for all 1 ≤ m < n and all partitions m = j1 < j2 < · · · < jk = n. In [25] the following result is proved Theorem 3.1 ([25]). Let X and U be jointly stationary and let X be almost subadditive + ∈ L 1 and that there exists an increasing sequence with respect to U . Assume that X 0,1 + of integers (m k )k with m 1 ≥ 1 such that lim inf k→∞

X 0,n+m k X 0,m k ≥ lim inf k→∞ n + mk mk

for all n ≥ 1 and lim

k→∞

almost surely

Um k = 0 almost surely. mk

(3.2)

(3.3)

728

C. Bonanno, P. Collet

Then lim

k→∞

X 0,m k = x¯ mk

exists almost surely

with −∞ ≤ x¯ < ∞ almost surely.
, n) as defined in (2.5), identifying X We first apply this theorem to K ( f, τ, U m,n
with K ( f, τ, U , m, n). Then, since for all ω ∈ AN it holds K (ω0 ) ≤ const, we have




K ( f, τ, U , 1) = min K (ω0 ) : ω0 ∈ ψ( f, 1, U ) ∈ L 1. Moreover by (H1.a) we have

, k) ≤ K ( f, τ, U , n + k) + log2 k + const K ( f, τ, U

for all n ≥ 1 and all f ∈ F, hence condition (3.2) of the previous theorem is satisfied with m k = k. We now show the sub-additivity property with respect to a family of random variables.
∈ C
, the family Lemma 3.2 For any fixed τ > 0,
> 0, || < ∞ and U 
, m, n)) (K ( f, τ, U is almost subadditive with respect to the family of functions m,n h = h( j) defined in (H1.b).

Proof. Without loss of generality we can assume m = 0 because of stationarity. Let us
) by (H1.b) we have fix a function f ∈ F. For all ω0n−1 ∈ ψ( f, n, U K (ω0n−1 ) ≤

k−1  j (K (ω jii+1 ) + h( ji+1 − ji )) i=1

for any partition 0 = j1 < j2 < · · · < jk = n − 1. With any such partition fixed, let j ω¯ jii+1 , i = 1, . . . , k − 1, be a collection of finite words such that
, ji , ji+1 ) = K (ω¯ jii+1 ). K ( f, τ, U j

Then, if we denote by ω¯ 0n−1 the concatenation j

j

j

k ω¯ 0n−1 := ω¯ j12 ω¯ j23 . . . ω¯ jk−1

it holds k−1 


(K (ω¯ jii+1 ) + h( ji+1 − ji )) ≥ K (ω¯ 0n−1 ) ≥ K ( f, τ, U , n), j

i=1

hence the sub-additivity property is proved.

 

Complexity for Extended Dynamical Systems

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Since condition (3.3) is verified by the function h(n) and the probability measure µ
) exists and is is invariant, we can apply Theorem 3.1 to obtain that the limit K ( f, τ, U τ τ c finite µ almost surely. Let us denote by YU
⊂ F a set with µ((YU
) ) = 0 on which   the limit exists. Then we define ⎧
,n) K ( f,τ,U ⎪ ⎨ lim if f ∈ YUτ
n n→∞ 
) := K ( f, τ, U . (3.4) ⎪ ⎩ +∞ otherwise We can then prove τ ⊂ F with µ((Y τ )c ) = 0 such that Lemma 3.3 There exists a set Y
,
, 



∈ C K ( f, τ,
, ) := inf K ( f, τ, U ) : U τ . Moreover there exists a sequence {V ˜ s }s of is well defined and finite for all f ∈ Y
,
coverings in C such that

lim K ( f, τ, V˜ s ) = K ( f, τ,
, )

s→∞

τ ∀ f ∈ Y
,

(3.5)

τ . and the sequence {K ( f, τ, V˜ s )}s is non-increasing for all f ∈ Y
,



⊂ C
. Let G = g Proof. We first restrict to a countable set of coverings D j ⊂ F 
as the set of finite open coverings be a countable set dense in F| , and define D



:= V ∈ C : the centers are in G and radii are rational . (3.6) D


∈ D
we can define Restricting to coverings V  



K˜ ( f, τ,
, ) := inf K ( f, τ, V ) : V ∈ D

(3.7)

τ ⊂ F with µ((Y τ )c ) = 0 defined by on the set Y
,
,

τ Y
, := YVτ
.
∈D
V 



τ . To this aim it is We now show that K˜ ( f, τ,
, ) is equal to K ( f, τ,
, ) on Y
,



enough to prove that for any U ∈ C there exists V ∈ D such that

K ( f, τ, V ) ≤ K ( f, τ, U ).

(3.8)

τ we have that Indeed from this and (3.4), on Y
,

K˜ ( f, τ,
, ) ≤ K ( f, τ,
, )
⊂ C
. and the other inequality is obtained by using D 

with c = Let us now prove (3.8). Let U = {U1 , . . . , Uc } be a covering in C

card(U ), and define r = max r (U j ) : j = 1, . . . , c <
, where r (U j ) is the radius of the ball B j . Then by of the set of functions G in F| , we can find a covering 
density
∈ D
with balls V such that U j ⊂ V j for all j = 1, . . . , c. Indeed it is V j  j=1,...,c

730

C. Bonanno, P. Collet

enough to choose balls V j with centres in functions of the set G at distances less than
−r 2 from the centres of the balls U j . For this choice of coverings for all n ≥ 1,

ψ( f, n, U ) ⊂ ψ( f, n, V ) τ , holds, hence from (2.5) and (3.4) it follows that for all f ∈ Y
,
) = lim K ( f, τ, V

n→∞


, n)
, n) K ( f, τ, V K ( f, τ, U
≤ lim = K ( f, τ, U ). n→∞ n n

We now prove the
second part of the assertion. Let us consider an enumeration of the 
= V coverings in D , then we define j j V˜ s :=



Vj,

1≤ j≤s

where, for two finite open coverings U and V, by U ∧ V we denote the finite open cov
ering which contains all the balls of U and V. By definition, it is clear that Vs ∈ D ˜ for all s ≥ 1. Moreover, since Vs contains all the balls of the coverings V1 , . . . , Vs , we have that, modulo a renumbering of the balls of V˜ s , ψ( f, n, V j ) ⊂ ψ( f, n, V˜ s ) for all j = 1, . . . , s and all f ∈ F. Hence for all j = 1, . . . , s, K ( f, τ, V˜ s , n) ≤ K ( f, τ, V j , n) + const

∀ f ∈ F,

where the constant is independent on the length n of the symbolic words. Dividing by n and taking the limit as n → ∞, we obtain for all f ∈ F, K˜ ( f, τ,
, ) ≤ lim inf K ( f, τ, V˜ s ) ≤ lim sup K ( f, τ, V˜ s ) ≤ K˜ ( f, τ,
, ), s→∞

s→∞

where the first two inequalities come from the definition of the upper and lower limit. Hence (3.5) is proved. By the same argument as above, it is immediate to verify that for all f ∈ F the sequence {K ( f, τ, V˜ s )}s is non-increasing. Hence the lemma is proved.   The next step is to show the existence of the limit in (2.11) for an admissible sequence of intervals to define K ( f, τ,
). We need the following general lemma Lemma 3.4 Let T : (X, ν) → (X, ν) be a measure preserving invertible transformation of a probability space (X, ν). Let ϑ and ξ be two real functions on X in the space L 1 (X, ν) and let ξ(x) ≥ 0 for all x ∈ X . Then there exists a set Y ⊂ X with ν(Y c ) = 0 such that for any sequences {ak }k and {bk }k of integers satisfying conditions (2.8)–(2.10) we have b k −1 1 ¯ ϑ(x) := lim ϑ(T j (x)) (3.9) k→∞ bk − ak j=ak

exists, is finite for all x ∈ Y and it is in L 1 (X, ν). Moreover it satisfies ¯ ϑ(x)dν. ϑ(x)dν = X

X

(3.10)

Complexity for Extended Dynamical Systems

For the function ξ we have lim

k→∞

731

ξ(T bk (x)) =0 bk − ak

(3.11)

for all x ∈ Y . This result is in the spirit of results in [17] and [16], where it is proved that we cannot ask for weaker conditions on the sequences {ak }k and {bk }k . However we could not relate directly our lemma to their results, hence in the appendix we give a proof. We will use this lemma for the space translation action to show that there exists a set Y
τ ⊂ F, with µ((Y
τ )c ) = 0, such that the limit along any admissible sequence of intervals
= {k }, lim

k→∞

K ( f, τ,
, k ) |k |

exists and is finite for all f ∈ Y
τ . The ergodicity of the measure µ will imply that this limit is independent on f and it is a constant K µ (τ,
). Moreover from the proof it will follow that the limit does not depend on the admissible sequence
of sets, indeed it will be given by (3.13). We will study separately the superior and the inferior limits. For the superior limit we use the functions K ( f, τ,
, [0, p]) for p ∈ N. We first prove that we can apply Theorem 3.1 to this sequence of functions. We start by verifying that K ( f, τ,
, [0, 1]) ∈ L 1 . For

any V[0,1] ∈ D[0,1] and the associated alphabet A, we can write
K ( f, τ, V[0,1] , n) ≤ n (max K (α) + const), α∈A

hence for all f ∈ Y[0,1] ,

K ( f, τ,
, [0, 1]) ≤ inf

 max


α∈A(V[0,1] )

K (α) :


V[0,1]

∈


D[0,1]

+ const.

(3.12)

Assumption (A1) implies that K ( f, τ,
, [0, 1]) ∈ L 1 . Note that the bound depends only on the length of the interval  = [0, 1]. Let 1 and 2 be two fixed intervals with disjoint interiors and denote their union
⊂ C
be the set of coverings of F| built as in (A2) by two  := 1 ∪ 2 , let C˜  
∈ C
and U
∈ C
. For any U˜
∈ C˜
we can write, by using (H2.a), coverings U      1 1 2 2
), n)
, n) K ( f, τ, U˜ K ( f, τ, π1 (U˜ || const ≤ + , n |1 | n || |1 | n |1 |
) denotes the “projection” of the covering U˜
onto C
. where π1 (U˜  1 Applying this argument with 1 = [0, p] and 2 = [ p, m + p] for all m ≥ 1, and by taking the limit as n → ∞ and the infimum limit on |1 | = p → ∞, we obtain condition (3.2) for any fixed |2 | = m.

Lemma 3.5 For any fixed
> 0 and τ > 0, the family (K ( f, τ,
, [0, p])) p∈N is almost subadditive with respect to the constant function u(||) ≡ log2 q, where q is the constant defined in (A2).

732

C. Bonanno, P. Collet

Proof. Let us consider two disjoint intervals 1 = [a, b] and 2 = [b, c] and the union
) for a covering  = [a, c]. Let us fix a function f ∈ F and let ω˜ 0n−1 ∈ ψ( f, n, U˜
∈ C˜
. By (H2.b) we have U˜  K (ω˜ 0n−1 ) ≤ K (π1 (ω˜ 0n−1 )) + K (π2 (ω˜ 0n−1 )) + n log2 q + const, since the map π = (π1 , π2 ) is surjective. Then by the same argument as in Lemma 3.2 we have for all f ∈ YU˜
∩ YU
∩ YU
, 

1

2




, n) ≤ K ( f, τ, U , n) + K ( f, τ, U , n) + n log2 q + const K ( f, τ, U˜ 1 2
= π (U
). Then we divide by n and take the limit as n → ∞. These limits exist for U i  i as proved above, and we get


K ( f, τ, U˜ ) ≤ K ( f, τ, U ) + K ( f, τ, U ) + log2 q 1 2
∈ C
and the special covering U˜
∈ C˜
built from the two. For for all coverings U i   i
satisfying any fixed δ > 0 let us choose two coverings U i
K ( f, τ, U ) ≤ K ( f, τ,
, i ) + δ, i

then
K ( f, τ,
, 1 ) + K ( f, τ,
, 2 ) + 2δ + log2 q ≥ K ( f, τ, U˜ ) ≥ K ( f, τ,
, )

and sub-additivity is proved since it holds for all δ > 0.

 

Since u(||) = log2 q obviously satisfies condition (3.3), we have that there exists a set Y˜
τ ⊂ F with µ((Y˜
τ )c ) = 0 on which K ( f, τ,
, [0, p]) is defined for all p ∈ N, and the limit K ( f, τ,
, [0, p]) =: K µ ( f, τ,
) (3.13) lim p→∞ p exists, is finite almost surely and it is in L 1 (F, µ). Moreover the limit holds also in L 1 and we denote K µ (τ,
) :=

F

K µ ( f, τ,
) dµ.

(3.14)

We remark that if the measure µ is ergodic then K µ ( f, τ,
) is almost surely constant and equal to K µ (τ,
). Following the notation of Lemma 3.4, we denote K¯ ( f, τ,
, [0, p]) := lim

N →∞

N −1 1  K (ζ j p f, τ,
, [0, p]), N j=0

where it exists. Then we prove the following lemma

(3.15)

Complexity for Extended Dynamical Systems

733

Lemma 3.6 For any fixed τ and
, there exists a set Y¯
τ with µ((Y¯
τ )c ) = 0 such that for all f ∈ Y¯
τ and any admissible sequence of intervals
= {k }, K ( f, τ,
, k ) K¯ ( f, τ,
, [0, p]) ≤ lim inf p→∞ |k | p

lim sup k→∞

holds. If moreover the measure µ is ergodic then lim sup k→∞

K ( f, τ,
, k ) ≤ K µ (τ,
). |k |

Proof. Let us consider an admissible sequence of intervals with integer boundary points. For a fixed integer p ∈ N, we can use the sub-additivity property (H2.b) as in Lemma 3.5 to show that for all f ∈ Y˜
τ we have, by setting  bpk  =: b˜k and  apk  + 1 =: a˜ k , K ( f, [ak , bk ]) ≤

b˜ k −1



K ( f, [ j p, ( j + 1) p]) + log2 q



j=a˜ k

+K ( f, [ak , a˜ k p]) + K ( f, [b˜k p, bk ]) + 2 log2 q, where the dependence on τ and
has been ignored to simplify notations. First of all, by repeating the same argument we used to prove (3.12), we can prove that there exists a constant depending only on p, see the remark after (3.12), that is a bound from above for K ( f, [ak , a˜ k p]) and K ( f, [b˜k p, bk ]) for all f ∈ Y˜
τ . Moreover we can write K ( f, [ j p, ( j + 1) p]) = K (ζ j p f, [0, p]), hence ˜

b k −1   K ( f, [ak , bk ]) 1 const K (ζ j p f, [0, p]) + log2 q + ≤ . bk − ak bk − ak bk − ak j=a˜ k

We now apply Lemma 3.4 to the action of the space translation ζ and with K ( f, [0, p]) having the role of the L 1 function ϑ. Let Y p ⊂ F be the full measure set given for K ( f, [0, p]) by Lemma 3.4, then we conclude by (3.9) that for all f ∈ Y¯
τ := Y˜
τ ∩ (∩ p Y p ), we have lim sup k→∞

K ( f, [ak , bk ]) K¯ ( f, [0, p]) log2 q + ≤ bk − ak p p

for all p ∈ N. Hence we obtain the first part of the assertion. The second part follows by first applying Lemma 3.4 to an ergodic measure, from which we get that for all p ∈ N, ¯ K ( f, [0, p]) dµ K ( f, [0, p]) = X

almost surely. Then we use (3.13) and (3.14) to conclude. The result for the sequences {ak }k and {bk }k follows by writing K ( f, [ak , bk ]) ≤ K ( f, [ak , ak  + 1]) + K ( f, [ak  + 1, bk ]) +K ( f, [bk , bk ]) + 3 log2 q and reducing to the above argument.

 

734

C. Bonanno, P. Collet

To prove a similar result for the inferior limit we use the following result proved in [12]: Theorem 3.7 ([12]). Let T : (X, ν) → (X, ν) be a measure preserving invertible transformation of a probability space (X, ν). Let {βn }n be a sequence of integrable real functions on X such that 1 inf βn (x) dν(x) > −∞, (3.16) n n X and for all n, k, (3.17) βn+k (x) − βn (x) − βk (T n (x)) ≤ h k (T n (x))  for a sequence of functions {h k }k satisfying h k ≥ 0 and X h k dν ≤ const. Then there exists a function ϑ ∈ L 1 (X, ν) such that 1 ϑ(x) dν = lim βn (x) dν, (3.18) n→∞ n X X and a function ξ ∈ L 1 (X, ν) such that ξ ≥ 0 and n−1 

ϑ(T j (x)) ≤ βn (x) + ξ(T n (x))

(3.19)

j=0

for almost all x ∈ X and all n ∈ N. Lemma 3.8 Under the hypothesis of ergodicity for the probability measure µ on F, for any fixed τ and
, there exists a set Y τ
with µ((Y τ
)c ) = 0 such that for all f ∈ Y τ
and any admissible sequence of intervals
= {k }, lim inf k→∞

K ( f, τ,
, k ) ≥ K µ (τ,
) |k |

holds. Proof. Let us consider first the case of sequences {ak } and {bk } of integers. We apply Theorem 3.7 to the sequence β p ( f ) := K ( f, τ,
, [0, p]) for f ∈ Y˜
τ and p ∈ N. Indeed condition (3.16) is easily verified since K ( f, [0, p]) is non-negative,2 and condition (3.17) is the subadditive property we proved in Lemma 3.5. Hence there exists a full measure set Y (ξ, ϑ) ⊂ F on which (3.19) is verified for two given functions ϑ and ξ . Moreover from (3.13) and (3.14), for an ergodic measure µ we have that ϑ( f ) dµ = K µ (τ,
). F

Now let us define the set Yζ (ξ, ϑ) :=



{ζn f : f ∈ Y (ξ, ϑ)} ,

n∈Z

since µ is ζ -invariant it holds µ((Yζ (ξ, ϑ))c ) = 0. On Yζ (ξ, ϑ) we can write K ( f, [ak , bk ]) = K (ζak f, [0, bk − ak ]) 2 To simplify notations we neglect the dependence on τ and
.

Complexity for Extended Dynamical Systems

735

and, using (3.19), bk −a k −1 

ϑ(ζ j+ak ( f )) ≤ K ( f, [ak , bk ]) + ξ(ζbk ( f )).

j=0

At this point we apply Lemma 3.4 to ϑ and ξ , and we obtain using the ergodicity of the measure µ, K µ (τ,
) = lim

k→∞

b k −1 1 K ( f, [ak , bk ]) ϑ(ζ j ( f )) ≤ lim inf k→∞ bk − ak bk − ak j=ak

for almost all f ∈ F. Let us call this full measure set Y τ
. As in Lemma 3.6, the general case for real sequences {ak } and {bk } follows by K ( f, [ak , bk ]) ≥ K ( f, [ak , bk  + 1]) − K ( f, [ak , ak ]) − +K ( f, [bk , bk  + 1]) − 3 log2 q. The lemma is proved.

 

Putting together Lemmas 3.6 and 3.8 we have that, for any fixed τ and
, there exists a set Y
τ := Y¯
τ ∩ Y τ
⊂ F with µ((Y
τ )c ) = 0, such that for all f ∈ Y
τ the limit lim

k→∞

K ( f, τ,
, k ) = K µ ( f, τ,
) |k |

exists and is finite for all admissible sequence of intervals
= {k }. Moreover, for an ergodic measure µ, for all f ∈ Y
τ and any admissible sequence of intervals
, it is equal to the constant K µ (τ,
) defined in (3.14). Hence the limit in (2.11) exists. To finish the proof of Theorem 2.2, we first have to prove that K µ (τ,
) is nondecreasing in
. Let
1 <
2 , then according to the above arguments, we can define K ( f, τ,
1 , ) and K ( f, τ,
2 , ) for all || < ∞ as in (2.7) and all f ∈ Y
τ1 ∩ Y
τ2 .
1
2 ⊂ C we have Y
τ1 , ⊂ Y
τ2 , and Moreover, since C K ( f, τ,
2 , ) ≤ K ( f, τ,
1 , ). Restricting to  = [ak , bk ] with ak < bk for all k ≥ 1 and satisfying (2.8)–(2.10), dividing by || = (bk − ak ) and taking the limit as in (2.11) gives K µ (τ,
2 ) ≤ K µ (τ,
1 ) < ∞ Y
τ1

on the set ⊂ Q+ and define

Y
τ2 .

Let us now consider a monotonically vanishing sequence {
k }k ⊂ Y τ :=



Y
τk .

k≥1

µ((Y τ )c )

Then = 0 and K µ (τ,
) is finite on Y τ for all
> 0 and non-decreasing on
. Hence we can define K µ (τ ) = lim K µ (τ,
)
→0

Yτ.

In Theorem 2.3 we prove that it is finite for on the full measure set of functions complexity functions satisfying also (H3) and (H4). K (τ ) Now it remains to prove that µτ does not depend on τ . We will use Assumption (A3).

736

C. Bonanno, P. Collet

Lemma 3.9 The full measure set Y := Y τ does not depend on τ , and there exists a K (τ ) constant K µ such that µτ = K µ on Y for all τ > 0. Proof. Let us fix constants γ , , C as in (A3), an interval || < ∞,
> 0 and a time step τ ∈ (0, C −1 diam()
). Then for all τ  > 0 we denote
 := eγ τ
,    :=  \ d(x, ∂) < C
−1 (τ + 1) .
∈ C
, we consider the covering U¯
 ∈ C
 which has Moreover, given a covering U   
and radius increased by a factor eγ τ . By balls with the same centres as those in U Assumption (A3), for any function f ∈ F, we can build a symbolic orbit 
 ) by using the information contained in a symbolic orbit ω¯ 0n −1 ∈ ψ( f, n  , U¯ 
), with n = n  τ  by defining ω0n−1 ∈ ψ( f, n, U τ   ⎧ ⎨ j  +  j  τ  − 1  if τ  > τ τ ω¯ j  = ω j with j = , ⎩  τ j τ  if τ  < τ

hence 

K (ω¯ 0n −1 ) ≤ K (ω0n−1 ) + n 

τ + const τ

(3.20) 

with a constant dependent only on the complexity function, and the term n  ττ that contains the information we need each time that the difference between j and j  increases by one unit. Let now ω¯ 0n−1 be the symbolic orbit on which the minimum for the function K ( f, τ,
U , n) is attained. Then by (3.20) we have 

 τ
 
¯ 0n −1 ) ≤ K ( f, τ, U , n) + n  + const K ( f, τ  , U¯  , n ) ≤ K (ω τ

and dividing by n and taking the limit for n → ∞ we have
 )
) K ( f, τ  , U¯ K ( f, τ, U 1  + ≤  τ τ τ

(3.21)




be a covering such that for all f ∈ Y τ ∩ Y τ . At this point, let us fix a δ > 0 and let U
K ( f, τ, U ) < K ( f, τ,
, ) + δ,
 and (3.21) we have then using the induced covering U¯  


) K ( f, τ  , U¯ K ( f, τ,
, ) + δ + 1 K ( f, τ  ,
 ,  )  ≤ <   τ || τ || τ ||

for all  = [ak , bk ] as in (2.8)–(2.10). Then the limit for || → ∞ gives K µ (τ  ,
 ) K µ (τ,
) < <∞ τ τ

(3.22)

Complexity for Extended Dynamical Systems

737 



| on Y τ ∩ Y τ , where we have used | || → 1 as || → ∞, and we have suppressed the dependence on the function f because of the ergodicity of the measure µ. The final step is the limit for
, and since
 → 0 as
→ 0 we have

K µ (τ ) K µ (τ  ) <∞ <  τ τ 

on Y τ ∩ Y τ = Y τ . Repeating the argument interchanging the roles of τ and τ  , the lemma is proved.   Hence Theorem 2.2 is proved. 4. Proof of Theorem 2.3 K (τ )

Since we proved that µτ = K µ for all τ > 0, in this proof we can fix τ = 1 for simplicity of notation and drop it from formulas. The first inequality
 sup K µ : µ invariant probability measures ≤ h top (4.1) follows by showing that for any (ζ, ϕ)-invariant probability measure µ, and for any fixed
> 0 and any finite interval ,


/4

F

log2 (N (T )) T →∞ T

K ( f,
, ) dµ ≤ lim

(4.2)

holds, where the right-hand side is defined as in (2.3). Indeed (4.2) implies (4.1) just dividing by ||, and using the L 1 convergence proved in Theorem 2.2 for the limits as || → ∞ and
→ 0.
) holds for any covering To prove (4.2), since for any f ∈ F, K ( f,
, ) ≤ K ( f, U

U ∈ C , it is enough to prove the following lemma
∈ C
such that Lemma 4.1 There exists a covering U¯ 

F


/4

log2 (N (n)) . n→∞ n


K ( f, U¯ ) dµ ≤ lim

(4.3)


∈ C
with balls of radius
, and a new covering Proof. Let us consider a covering U  3
U¯ with balls with the same centres as before and radius 2
3 . We recall that the Lebesgue number lemma states that for a finite open covering of a compact metric space, there is a finite number γ > 0 such that the γ -neighbourhood of any point is contained in at least one open set of the covering. The number γ is called the Lebesgue number of the covering. Since F| is a metric compact set, the covering
has a finite Lebesgue number, and by its construction we conclude that its Lebesgue U¯ number is not less than 4
. The idea of the proof is to use (H3), hence we need to count all possible symbolic orbits. For all n let us consider a minimal (n, 4
)-spanning set (n, 4
) for F| , that is for any f ∈ F there exists g ∈ (n, 4
) such that

d| (ϕk ( f ), ϕk (g)) ≤


4

∀ k = 0, . . . , n − 1.

738

C. Bonanno, P. Collet

For any g ∈ (n, 4
) let us denote by Rg ⊂ F the set of functions f ∈ F which are

4 -spanned by g. We can make R g g a partition of F just by choosing for each f ∈ F only one function spanning it.
 For any function g ∈ (n, 4
) we can consider the sequence of balls B(ϕk (g), 4
) k
, we with k = 0, . . . , n−1. At the same time, by our result on the Lebesgue number of U¯ n−1
). can associate to each such sequence of balls a symbolic orbit ω0 (g) ∈ ψ(g, n, U¯ Then we have
, n) K ( f, U¯ 1  dµ ≤ K (ω0n−1 (g)) µ(Rg ). (4.4) n n F
g∈(n, 4 )



) , the set Let us now consider for the alphabet A = 1, . . . , card(U¯   E := (ω0n−1 (g), n) ⊂ A∗ × N. n∈N g∈(n, 4
)

It is a recursively enumerable set, hence we can apply Hypothesis (H3) getting  
 + log2 n + const, K (ω0n−1 (g)) ≤ log2 σ n, (4.5) 4     where σ n, 4
:= card( n, 4
). Since this estimate is uniform on g, applying it to (4.4) we get     
, n) log2 σ n, 4
+ log2 n + const K ( f, U¯ dµ ≤ (4.6) n n F  since g µ(Rg ) = 1. To finish the proof of the lemma we use the inequality 

/4 ≤ N (n) σ n, 4 which is well known in ergodic theory, see for example [11].   Let us now prove the other inequality. For any fixed
> 0, from the definition of topological entropy (2.3) we define3
(n) log2 N , (4.7) n→∞ n h  (
) h(
) := lim . (4.8) ||→∞ || The quantity h(
) is non-decreasing in
and its limit as
→ 0 is h top . With respect to the quantities defined above, given any fixed δ > 0 there exist
0 > 0, λ0 (
) > 0 and n 0 (
, ) > 0 such that

h  (
) := lim

h top − δ < h(
) < h top

∀
<
0 ,

(h(
) − δ)|| < h  (
) < (h(
) + δ)||
(n) < 2n(h  (
)+δ||) 2n(h  (
)−δ||) < N We first state a lemma we need in the following 3 We recall that we consider fixed τ = 1.

∀ || > λ0 (
), ∀ n > n 0 (
, ).

(4.9) (4.10) (4.11)

Complexity for Extended Dynamical Systems

739


 Lemma 4.2 Let us consider a fixed
> 0 and a finite interval
. If ρ j j is a sequence of probability measures on F, invariant with
respect to the time evolution ϕ1 , then there exists a sub-sequence { jh }h such that ρ jh h is weakly convergent to a probability measure ρ, invariant with respect to ϕ1 , and lim sup K ( f,
, ) dρ jh ≤ K ( f,
, ) dρ. h→∞

F

F

Proof. The existence of the ϕ1 -invariant probability measure ρ follows
by the  compactness of the space F| . For simplicity of notations, let us assume that ρ j j is weakly convergent to ρ. By the monotonicity of the sequence {K ( f, V˜ s )}s proved in Lemma 3.3, we have K ( f,
, ) dρ j = lim K ( f, V˜ s ) dρ j (4.12) s→∞ F

F

for all j ∈ N and also for the ϕ1 -invariant measure ρ. By the sub-additive ergodic theorem in L 1 it holds K ( f, V˜ s , n) K ( f, V˜ s , n) ˜ K ( f, Vs ) dρ j = lim dρ j = inf dρ j (4.13) n→∞ F n→∞ F n n F for all s ∈ N. Moreover, for each fixed n ∈ N, the function f → K ( f, V˜ s , n) is upper semi-continuous, hence using weak convergence of ρ j we get for all s ∈ N (see for example [2]) K ( f, V˜ s , n) K ( f, V˜ s , n) dρ j ≤ inf dρ = K ( f, V˜ s ) dρ, (4.14) n→∞ F n n F F

inf

n→∞

where for the last equality we have used (4.13) for the ϕ1 -invariant measure ρ. The assertion follows by putting together (4.13) and (4.14), and by applying (4.12) to both sides.   Lemma 4.3 Given any δ > 0, there exists
0 > 0 such that for any
<
0 there exists λ0 (
) > 0 such that for any finite interval || > λ0 (
): there exists a probability measure µ
 on F, invariant with respect to the time evolution ϕ1 , such that 1 K ( f,
/2, ) dµ
 ≥ h top − 4δ || F holds. Proof. For any fixed δ > 0 let us consider
0 > 0 as defined for (4.9), and let us fix


(n) := f N (n)
<
0 . Referring to (2.2), for any finite interval  let us denote by S j j=1 the functions of a maximal set of (, n,
)-different orbits. On this set we consider the sequence of probability measures on F given by



ν,n

N (n)  1 :=
δfj, N (n) j=1

(4.15)

740

C. Bonanno, P. Collet


(n), for any where δ· denotes the usual Dirac mass. Hence by definition of the set S
/2
/2 open covering U ∈ C we have

1 ||






N (n)
/2  K ( f j , U
/2 , n) K ( f, U , n)
1 1  dν,n = .
(n) n || N n F  j=1

We now use Hypothesis (H4). For any given δ > 0 we have  
/2
card f ∈ S (n) : K ( f, U , n) < n||(h(
) − 3δ) ≤ 2n||(h(
)−3δ) . Putting together (4.16) and (4.17) we obtain  

/2 N (n) − 2n||(h(
)−3δ) (h(
) − 3δ) K ( f, U , n)
1 dν,n ≥ .
(n) || F n N

(4.16)

(4.17)

(4.18)

For any fixed m ∈ N let us write n = tm +r with 0 ≤ r < m. Using the sub-additivity
/2 property for the family of functions (K ( f, U , m, n))m,n proved in Lemma 3.2, we write for any f ∈ F, 
/2
/2 K ( f, U , n) ≤ t−2 j=0 K (ϕl ( f ), U , jm, ( j + 1)m) (4.19)
/2
/2 +K ( f, U , l) + K ( f, U , (t − 1)m + l, n) + th(m) for all l = 0, . . . , m − 1, where h(·) is the function defined in (H1.b). Moreover for all j = 0, . . . , t − 2 and all l = 0, . . . , m − 1,
/2
/2
∗
K (ϕl ( f ), U , jm, ( j + 1)m) dν,n = K ( f, U , m) d(ϕl+ jm ν,n ) F

F

holds, and for all f ∈ F the uniform estimate
/2


/2


/2

K ( f, U , l) + K ( f, U , (t − 1)m + l, n) ≤ 2m log2 (card(U )) + const holds. Hence letting µ
,n

⎛ ⎞ m−1 t−2 1  ⎝ 1  ∗ 1
⎠ := ϕl+ jm ν,n = m t −1 (t − 1)m l=0

j=0

(t−1)m−1 


ϕi∗ ν,n

i=0

we obtain
/2
/2 K ( f, U , n)
K ( f, U , m) (t − 1)m th(m) + o(n) dν,n ≤ dµ
,n + . n tm + r m tm + r F F (4.20) Let µ
 be an accumulation point for the sequence of probability measures {µ
,n }n given by Lemma 4.2. It follows that µ
 is a probability measure on F which is invariant for the time action ϕ1 . Using (4.20) and the upper semi-continuity of the function
/2 f → K ( f, U , m) for all m ∈ N, we have
/2
/2 K ( f, U , n)
K ( f, U , m) h(m) lim sup dν,n ≤ dµ
 + (4.21) n m m n→∞ F F for all m ∈ N.

Complexity for Extended Dynamical Systems

741

Let now λ0 (
) > 0 and n 0 (
, ) > 0 be defined as in (4.10) and (4.11). Then from (4.18) we have that if || > λ0 and n > n 0 , 1 ||



  K ( f, U , n)
dν,n ≥ 1 − 2−n||δ (h(
) − 3δ) n F
/2

holds. Hence for all m ∈ N, if
<
0 and || > λ0 (
) we have


/2

K ( f, U , m) h(m) dµ
 + ≥ (h top − 4δ)|| m m F
/2

(4.22)


/2

for all coverings U ∈ C , where we have used (4.9). By the sub-additive ergodic theorem in L 1 we have that
/2 K ( f, U , m)
/2 lim dµ
 = K ( f, U ) dµ
 , m→∞ F m F
/2

where K ( f, U ) is defined as in (3.4). Hence, since h(m) = o(m), from (4.22) we have that given any δ > 0 there exists
0 > 0 such that for any
<
0 there exists λ0 (
) > 0 such that for any || > λ0 ,
/2 K ( f, U ) dµ
 ≥ (h top − 4δ)|| (4.23) F


/2


/2

holds for any finite open covering U ∈ C . To finish the proof of the lemma, we use the sequence of coverings {V˜ s }s defined in Lemma 3.3 to obtain K ( f,
/2, ) dµ
 = lim K ( f, V˜ s ) dµ
 ≥ (h top − 4δ)||, (4.24) F

s→∞

F

where the last inequality is given by (4.23).

 

Lemma 4.4 For any given δ > 0 there exists
0 > 0 such that for any
<
0 the following holds: there exists a probability measure ρ
on F, invariant with respect to the (ζ1 , ϕ1 ) action such that K ρ
( f,
/2) dρ
≥ h top − 4δ. K ρ
(
/2) := F

Proof. In Theorem 2.2 we proved that K ρ
(
), as defined in (3.14), is not dependent on the admissible sequence  of intervals. Hence we will restrict to the family of intervals  = {[0, p]} p for p ∈ N. For a fixed q ∈ N, writing p = tq + r with 0 ≤ r < q and using the sub-additivity property proved in Lemma 3.5, we have for all f ∈ F,  K ( f,
/2, [0, p]) ≤ t−2 j=0 K (ζl ( f ),
/2, [ jq, ( j + 1)q]) +K ( f,
/2, [0, l]) + K ( f,
/2, [(t − 1)q + l, p]) + t log2 q (4.25) for all l = 0, . . . , q −1, where K ( f,
/2, [0, l]) and K ( f,
/2, [(t −1)q +l, p]) are O(q) as shown in (3.12). Fixing
<
0 , for all p > λ0 (
), where λ0 (
) is given as in (4.10)

742

C. Bonanno, P. Collet

and in Lemma 4.3, we denote by µ
p the ϕ1 -invariant probability measure associated to [0, p]. For all p > λ0 we write using (4.25),

 t−2 t log2 q + o( p) K ( f, 2
, [0, p]) K (ζl ( f ), 2
, [ jq, ( j + 1)q])
dµ p ≤ dµ
p + p p p F F j=0 (4.26) for all l = 0, . . . , q − 1. Moreover for all j = 0, . . . , t − 2 and all l = 0, . . . , q − 1
∗
K (ζl ( f ),
/2, [ jq, ( j + 1)q]) dµ p = K ( f,
/2, [0, q]) d(ζl+ jq µ p ) (4.27) F

F

holds, hence we define ⎛ ⎞ q−1 t−2   1 1 1 ∗
⎠ ⎝ ρ
p := ζl+ = jq µ p q t −1 (t − 1)q l=0

j=0

(t−1)q−1 

ζi∗ µ
p

(4.28)

i=0

and obtain, using Lemma 4.3 and (4.26) K ( f, 2
, [0, q])
t log2 q + o( p) (t − 1)q h top − 4δ ≤ dρ p + . (4.29) tq + r F q tq + r   We now apply Lemma 4.2 to the sequence of measures ρ
p , and we obtain a probap

bility measure ρ
, invariant with respect to the (ζ1 , ϕ1 ) action, which satisfies K ( f, 2
, [0, q])
log2 q dρ + ≥ h top − 4δ q q F

(4.30)

for all q ∈ N. Letting q → ∞ we have K ρ
(
/2) ≥ h top − 4δ

(4.31)

by using the definition of K ρ
(
/2) as given in (3.13) and (3.14).   In Theorem 2.2 we have proved that K µ (
) is non-decreasing in
for any probability invariant measure µ. Hence, from Lemma 4.4, we obtain that for all
<
0 (δ), K ρ
≥ K ρ
(
 ) ≥ h top − 4δ

∀
 <


2

(4.32)

holds, where we recall that ρ
are probability measures invariant with respect to the (ζ1 , ϕ1 ) action. To finish the proof of the theorem, we only need to construct a probability measure satisfying (4.32), which is invariant with respect to space translation and time evolution for all (x, t) ∈ R × R. Let us choose a fixed
<
0 and denote ρ := ρ
. The probability measure 1 1 ∗ ζx∗ (ϕ−t ρ) dtd x (4.33) ν := 0

0

is invariant with respect to space translation and time evolution for all (x, t) ∈ R × R by definition. We now prove

Complexity for Extended Dynamical Systems

743

Lemma 4.5 The probability measure ν defined in (4.33) satisfies K ν ≥ h top − 4δ.

(4.34)

Proof. It is enough to prove that for
 small enough K ν (
 ) ≥ h top − 4δ

(4.35)

holds. For all p ∈ N let us write  1 1  K ( f,
 , [0, p]) K ( f,
 , [0, p]) ∗ dν = d(ζx∗ (ϕ−t ρ)) dtd x (4.36) p p F F 0 0 and for the moment consider (x, t) fixed. The first step is to write    ∗ ∗  ∗ F K ( f,
, [0, p]) d(ζx (ϕ−t ρ)) = F K (ζx ( f ),
, [0, p]) d(ϕ−t ρ)  ∗ ρ). = F K ( f,
 , [x, p + x]) d(ϕ−t

(4.37)

By using the sub-additivity property proved in Lemma 3.5, we write ∗ ∗ K ( f,
 , [0, p + 1]) d(ϕ−t ρ) ≤ K ( f,
 , [x, p + x]) d(ϕ−t ρ) + o( p), F

F

where the term o( p) contains the constant 2 log2 q, and the integral of the terms K ( f,
 , [0, x]) and K ( f,
 , [ p + x, p + 1]) which are bounded as in (3.12). Hence K ( f,
 , [0, p + 1]) K ( f,
 , [x, p + x]) ∗ ∗ d(ϕ−t d(ϕ−t lim ρ) ≤ lim inf ρ), p→∞ F p→∞ F p p (4.38) ∗ ρ is (ζ , ϕ )where on the left-hand side we know that the limit exists because ϕ−t 1 1 invariant. We now want to estimate the left hand side. Let us start by writing ∗ K ( f,
 , [0, p]) d(ϕ−t ρ) = K (ϕ−t ( f ),
 , [0, p]) dρ F

F

= lim

lim

s→∞ n→∞

K (ϕ−t ( f ), V˜ s , n) dρ, n F


 where we used the sequence of open coverings {V˜ s } in C[0, p] defined in Lemma 3.3. ˜ By definition of K (ϕ−t ( f ), Vs , n) we look at the complexity of the symbolic words in ψ(ϕ−t ( f ), n, V˜ s ) (see (2.5)). Hence we have

K (ϕ−t ( f ), V˜ s , n) = K ( f, ϕt (V˜ s ), n),

(4.39)

where ϕt (V˜ s ) is a covering of F|[0, p] . Indeed, for g ∈ F there exists V j ∈ V˜ s such that ϕ−t (g) ∈ V j , because ϕ is invertible. Hence g ∈ ϕt (V j ) ∈ ϕt (V˜ s ). Moreover, by Assumption (A3), we have that there are constants γ > 0,  > 1 and C > 0 such that for p > 2C(
 )−1 if d|[0, p] ( f 1 , f 2 ) <
 then d|[2C(
 )−1 , p−2C(
 )−1 ] ( f 1 , f 2 ) < eγ t
 .

744

C. Bonanno, P. Collet

This implies that for any t ∈ [0, 1], the covering ϕt (V˜ s ) is a covering of F|[2C(
 )−1 , p−2C(
 )−1 ] and each of its sets is contained in a ball of radius η = eγ t
 . η Hence there exists a covering U˜s ∈ C[2C(
 )−1 , p−2C(
 )−1 ] such that K ( f, ϕt (V˜ s ), n) ≥ K ( f, U˜s , n)

(4.40)

for all f ∈ F and all n ∈ N. By using (4.39) and (4.40) we get for all s ∈ N, K (ϕt ( f ), V˜ s , n) K ( f, U˜s , n) dρ ≥ lim dρ ≥ lim n→∞ F n→∞ F n n ≥

F

K ( f, η, [2C(
 )−1 , p − 2C(
 )−1 ]) dρ

hence lim



p→∞ F

≥

K ( f,
 ,[0, p+1]) p

d(ϕt∗ ρ) ≥



p+1−4C(
 )−1 K ( f,η,[2C(
 )−1 , p+1−2C(
 )−1 ]) lim p p+1−4C(
 )−1 p→∞ F

(4.41) dρ = K ρ (η).

At this point, we would put together (4.37), (4.38) and (4.41), and use Lemma 3.6 to get K ν (
 ) ≥ K ρ (η). The assertion would then follow by choosing
 small enough to have η < 2
and use (4.32). The only problem to this argument is that we have tacitly assumed that it is possible to exchange the order of limit in p and integrations in (x, t) in (4.36). However, since by Lemma 3.5 we have K ( f,
 , [0, p]) ≤ K ( f,
 , [0, 1]) + log2 q ≤ const p for all f ∈ F, we can integrate with respect to ζx∗ (ϕ−t ρ) and apply the Lebesgue dominated convergence theorem.   Hence Theorem 2.3 is proved. Appendix A. Proof of Lemma 3.4 Let us denote as usual (Sn ϑ)(x) :=

n−1 

ϑ(T j (x)).

(A.1)

j=0

From the Birkhoff ergodic theorem we have that there exists a set Y1 ⊂ X with ν(Y1c ) = 0 such that for any diverging sequence (n k )k ⊂ Z, ¯ ϑ(x) := lim

k→∞

n k −1 (Sn k ϑ)(x) 1  = lim ϑ(T j (x)) k→∞ |n k | |n k | j=0

(A.2)

Complexity for Extended Dynamical Systems

745

exists, is finite for all x ∈ Y1 and is in L 1 (X, ν). Moreover ϑ¯ satisfies (3.10). Let Y1 be such that the same holds for |ϑ|. To prove (3.9), given any sequence of integers {ak }k and {bk }k as in the hypothesis, we write for all x ∈ Y1 , b k −1 1 (Sbk ϑ)(x) (Sak ϑ)(x) bk ak ϑ(T j (x)) = − , bk − ak bk − ak bk bk − ak ak

(A.3)

j=ak

and to study the convergence of (A.3) we divide the indices k ∈ N into four sets:     I1 := k ∈ N : |ak | ≥ bk − ak ; |bk | ≥ bk − ak ,     I2 := k ∈ N : |ak | ≥ bk − ak ; |bk | < bk − ak ,     I3 := k ∈ N : |ak | < bk − ak ; |bk | ≥ bk − ak ,     I4 := k ∈ N : |ak | < bk − ak ; |bk | < bk − ak . First of all we can neglect I4 since it contains only a finite number of indices by (2.8). Moreover we introduce for i = 1, 2, 3 the notation (++)

Ii

:= {k ∈ Ii : ak ≥ 0 ; bk ≥ 0}

and analogously for the other two possible combinations, (−+) and (−−). We remark (−+) (−−) (++) (−+) ∪ I2 and I3 = I3 ∪ I3 . that I2 = I2 (++) Let us consider (A.3) for the indices k ∈ I1 . By (2.8) and (A.2), for any given x ∈ Y1 and any fixed η > 0 there exists k0 ∈ N such that for all k ≥ k0 we have    (Sbk ϑ)(x)   ¯ − ϑ(x) < η,  bk    (Sak ϑ)(x)   ¯ − ϑ(x) < η.  a k

Also by (2.9) we can assume that for k ≥ k0 we have ak bk −ak bk bk −ak

k < lim sup bka−a +η = k

k→∞

1 la

+ η,

k < 1 + lim sup bka−a +η =1+ k

k→∞

1 la

+ η.

Applying these inequalities to (A.3) we have that for all k ≥ k0 ,        b k −1   1 1 1 j   ¯ ϑ(T (x)) − ϑ(x) < η 1 + + η +η . b − a la la k   k j=ak

(A.4) (A.5)

746

C. Bonanno, P. Collet (++)

This proves (3.9) for all sequences ak and bk with k ∈ I1 . The same argument applies (−+) (−−) and I1 by writing the right-hand side of (A.3) respectively as to k in I1 (Sbk ϑ)(x) bk bk +|ak | bk

+

|ak | (Sak ϑ)(x) bk +|ak | |ak | ,

(A.6)

(Sak ϑ)(x) |ak | |ak |−|bk | |ak |

−

(Sbk ϑ)(x) |bk | |ak |−|bk | |bk | ,

(A.7)

and using conditions (2.8)-(2.10). (−+) Let us consider now k ∈ I2 . First of all 0 ≤ lim

k→∞

bk 1 < lim √ =0 bk + |ak | k→∞ bk + |ak |

holds, hence lim

k→∞

Moreover we can apply (A.2) to

|ak | = 1. bk + |ak |

(Sak ϑ)(x) |ak |

and, since bk <

(A.8)

(A.9) √

bk + |ak |,

  (S√b +|a | |ϑ|)(x)  (Sbk ϑ)(x)   ≤ lim sup √ 1  =0 lim sup  √k k bk + |ak |  bk + |ak | bk + |ak | k→∞ k→∞

(A.10)

holds. Hence, using (A.6) and applying (A.10) to the first term and (A.9) and (A.2) to (−+) the second term, we prove (3.9) for k ∈ I2 . (−−) The same arguments apply also to I2 , and to I3 by interchanging the role of ak and bk . The proof of (3.11) follows by a similar argument. First we show that there exists a set Y2 ⊂ X with ν(Y2c ) = 0 such that for any diverging sequence of integers {n k }k lim

k→∞

ξ(T n k (x)) =0 |n k |

(A.11)

holds for all x ∈ Y2 . Since ξ ∈ L 1 and ξ ≥ 0, and since the transformation T is measure preserving, for all η > 0, ∞ 

∞    ν ξ(T k (x)) > kη = ν {ξ(x) > kη} =

k=1

k=1

∞ ∞ 1 1 kη ν {(k + 1)η ≥ ξ(x) > kη} < ξ(x) dν < = η η {(k+1)η≥ξ(x)>kη} k=1

k=1

1 < η

ξ(x)dν < ∞ X

holds, hence from the Borel-Cantelli Lemma it follows that the measure of the set on k which ξ(T k(x)) > η infinitely often is zero. Let us moreover assume that the function ξ(x) satisfies the Birkhoff theorem (condition (A.2)) for all x ∈ Y2 .

Complexity for Extended Dynamical Systems

747

√ We now use (A.11) as we used (A.2) before. If |bk | ≥ bk − ak then by (2.8) bk → ∞, hence we can apply (A.11) to bk . Hence by using (A.5), we have that for all x ∈ Y2 and for any given η > 0 there exists k0 (x) such that for all k ≥ k0 (x),   ξ(T bk (x)) 1 ≤η 1+ +η bk − ak la holds, hence (3.11) holds √ in Y2 for these indices. If instead |bk | < bk − ak , we can apply (A.2) by writing lim sup k→∞

(S√b −a ξ )(x) ξ(T bk (x)) 1 ≤ lim sup √ =0 √k k bk − ak b − a bk − ak k→∞ k k

using ξ ≥ 0. Hence (3.11) holds in Y2 also in this case. This finishes the proof of the lemma by choosing Y := Y1 ∩ Y2 .   Acknowledgements. The first named author would like to acknowledge support and kind hospitality by the Centre de Physique Théorique during his stay at the École Polytechnique, Palaiseau Cedex, France.

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Commun. Math. Phys. 275, 749–789 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0311-6

Communications in

Mathematical Physics

Full-Wave Invisibility of Active Devices at All Frequencies Allan Greenleaf1 , Yaroslav Kurylev2 , Matti Lassas3 , Gunther Uhlmann4 1 2 3 4

Department of Mathematics, University of Rochester, Rochester, NY 14627, USA Department of Mathematics, University College London, Gower Street, WC1E 6BT, London, UK Helsinki University of Technology, Institute of Mathematics, P.O. Box 1100, 02015, Helsinki, Finland Department of Mathematics, University of Washington, Seattle, WA 98195, USA. E-mail: [email protected]

Received: 7 November 2006 / Accepted: 14 March 2007 Published online: 15 August 2007 – © Springer-Verlag 2007

Abstract: There has recently been considerable interest in the possibility, both theoretical and practical, of invisibility (or “cloaking”) from observation by electromagnetic (EM) waves. Here, we prove invisibility with respect to solutions of the Helmholtz and Maxwell’s equations, for several constructions of cloaking devices. The basic idea, as in the papers [GLU2, GLU3, Le, PSS1], is to use a singular transformation that pushes isotropic electromagnetic parameters forward into singular, anisotropic ones. We define the notion of finite energy solutions of the Helmholtz and Maxwell’s equations for such singular electromagnetic parameters, and study the behavior of the solutions on the entire domain, including the cloaked region and its boundary. We show that, neglecting dispersion, the construction of [GLU3, PSS1] cloaks passive objects, i.e., those without internal currents, at all frequencies k. Due to the singularity of the metric, one needs to work with weak solutions. Analyzing the behavior of such solutions inside the cloaked region, we show that, depending on the chosen construction, there appear new “hidden” boundary conditions at the surface separating the cloaked and uncloaked regions. We also consider the effect on invisibility of active devices inside the cloaked region, interpreted as collections of sources and sinks or internal currents. When these conditions are overdetermined, as happens for Maxwell’s equations, generic internal currents prevent the existence of finite energy solutions and invisibility is compromised. We give two basic constructions for cloaking a region D contained in a domain Ω ⊂ Rn , n ≥ 3, from detection by measurements made at ∂Ω of Cauchy data of waves on Ω. These constructions, the single and double coatings, correspond to surrounding either just the outer boundary ∂ D + of the cloaked region, or both ∂ D + and ∂ D − , with metamaterials whose EM material parameters (index of refraction or electric permittivity and magnetic permeability) are conformal to a singular Riemannian metric on Ω. For the single coating construction, invisibility holds for the Helmholtz equation, but fails for Maxwell’s equations with generic internal currents. However, invisibility can be restored by modifying the single coating construction, by either inserting a physical surface at ∂ D − or using the double coating. When cloaking an infinite cylinder, invisi-

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bility results for Maxwell’s equations are valid if the coating material is lined on ∂ D − with a surface satisfying the soft and hard surface (SHS) boundary condition, but in general not without such a lining, even for passive objects. 1. Introduction There has recently been considerable interest [AE, MN, Le, PSS1, MBW] in the possibility, both theoretical and practical, of a region or object being shielded (or “cloaked”) from detection via electromagnetic (EM) waves. [GLU1, §4] established a non-tunnelling result for time-independent Schrödinger operators with highly singular potentials. This can be interpreted as cloaking, at any frequency, with respect to solutions of the Helmholtz equation using a layer of isotropic, negative index of refraction material. [GLU2, GLU3] raised the possibility of passive objects being undetectable, in the context of electrical impedance tomography (EIT). Motivated by consideration of certain degenerating families of Riemannian metrics, families of singular conductivities, i.e., not bounded below or above, were given and rigorous results obtained for the conductivity equation of electrostatics, i.e., for waves of frequency zero. A related example of a complete but noncompact two-dimensional Riemannian manifold with boundary having the same Dirichlet-to-Neumann map as a compact one was given in [LTU]. More recently, there has been exciting work based on the availability of metamaterials which allow fairly arbitrary behavior of EM material parameters. The constructions in [Le] are based on conformal mapping in two dimensions and are justified via change of variables on the exterior of the cloaked region. [PSS1] also proposes a cloaking construction for Maxwell’s equations based on a singular transformation of the original space, again observing that, outside the cloaked region, the solutions of the homogeneous Maxwell equations in the original space become solutions of the transformed equations. The transformation used is the same as used previously in [GLU2, GLU3] in the context of Calderón’s inverse conductivity problem. The paper [PSS2] contains analysis of cloaking on the level of ray-tracing, while full wave numerical simulations are discussed in [CPSSP]. Striking positive experimental evidence for cloaking from microwaves has recently been reported in [SMJCPSS]. Since the metamaterials proposed to physically implement these constructions need to be fabricated with a given wavelength, or narrow range of wavelengths, in mind, it is natural to consider this problem in the frequency domain. (As in the earlier works, dispersion, i.e., dependence of the EM material parameters on k, which is certainly present for metamaterials, is neglected.) The question we wish to consider is then whether, at some (or all) frequencies k, these constructions allow cloaking with respect to solutions of the Helmholtz equation or time-harmonic solutions of Maxwell’s equations. We prove that this indeed is the case for the constructions of [GLU3, PSS1], as long as the object being cloaked is passive; in fact, for the Helmholtz equation, the object can be an active device in the sense of having sources and sinks. On the other hand, for Maxwell’s equations with generic internal currents, invisibility in a physically realistic sense seems highly problematic. We give several ways of augmenting or modifying the original construction so as to obtain invisibility for all internal currents and at all frequencies. Due to the degeneracy of the equations at the surface of the cloaked region, it is important to rigorously consider weak solutions to the Helmholtz and Maxwell’s equations on all of the domain, not just the exterior of the cloaked region. We analyze various constructions for cloaking from observation on the level of physically meaningful EM

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waves, i.e., finite energy distributional solutions of the equations, showing that careful formulation of the problem is necessary both mathematically and for correct understanding of the physical phenomena. It turns out that the cloaking structure imposes hidden boundary conditions on such waves at the surface of the cloak. When these conditions are overdetermined, finite energy solutions typically do not exist. The time-domain physical interpretation of this is not entirely clear, but it seems to indicate an accumulation of energy or blow-up of the fields which would compromise the desired cloaking effect. As mentioned earlier, the example in [PSS1] turns out to be a special case of one of the constructions from [GLU2, GLU3], which gave, in dimensions n ≥ 3, counterexamples to uniqueness for Calderón’s inverse problem [C] for the conductivity equation. (Such counterexamples have now also been given for n = 2 [V, KSVW].) Thus, since the equations of electromagnetism (EM) reduce at frequency k = 0 to the conductivity equation with conductivity parameter σ (x), namely ∇ · (σ ∇u) = 0, for the electrical potential u, the invisibility question has already been answered affirmatively in the case of electrostatics. The present work addresses the invisibility problem at all frequencies k = 0. We wish to cloak not just a passive object, but rather an active “device”, interpreted as a collection of sources and sinks, or an internal current, within D. Since the boundary value problems in question may fail to have unique solutions (e.g., when −k 2 is a Dirichlet eigenvalue on D), it is natural, as in [GLU1], to use the set of Cauchy data at ∂Ω of all of the solutions, rather than the Dirichlet-to-Neumann operator on ∂Ω, which may not be well-defined. The basic idea of [GLU3, Le, PSS1] is to form new EM material parameters by pushing forward old ones via a singular mapping F. Solutions of the relevant equations, Helmholtz or Maxwell, with respect to the old parameters then push forward to solutions of the modified equations with respect to the new parameters outside the cloaked region. However, when dealing with a singular mapping F, the converse is not in general true. This means that outside D, depending upon the class of solutions considered, there are solutions to the equations with respect to the new parameters which are not the push forwards of solutions to the equations with the old parameters. Furthermore, it is crucial that the solutions be dealt with on all of Ω, and not only outside D. Especially when dealing with the cloaking of active devices, this gives rise to the question of what are the proper transmission conditions on ∂ D, which allow arbitrary internal sources to be made invisible to an external observer. To address these issues rigorously, one needs to make a suitable choice of the class of weak solutions (on all of Ω) to the singular equations being considered. For both mathematical and, even more so, physical reasons, the weak solutions that are appropriate to consider seem to be the locally finite energy solutions; these belong to the Sobolev space H 1 with respect to the singular volume form |
g |1/2 d x 2 1/2 on Ω for Helmholtz, and L (Ω, |
g | d x) for Maxwell. These considerations are absent from [Le, PSS1, PSS2], where the cloaking is justified by appealing to both the transformation of solutions on the exterior of D under smooth mappings F (essentially the chain rule) and the fact, in the high frequency limit, that rays originating in Ω\D avoid ∂ D and do not enter D. As we will show, analysis of the transmission conditions at ∂ D shows that the constructions of [PSS1, PSS2], although adequate for cloaking active devices for scalar fields, i.e., for Helmholtz, and cloaking passive devices for vector-valued waves, i.e., for Maxwell, fail to admit finite energy solutions to Maxwell when generic active devices are present. Furthermore, analysis of cloaking of an infinite cylinder, which was numerically explored in [CPSSP] and provides a model of the experimental verification of cloaking in [SMJCPSS], shows that

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even cloaking a passive object may be problematic. Fortunately, it is possible to remedy the situation by augmenting or modifying this construction. We now describe the results of this paper. For what we call the single coating, which is the construction of [GLU3], and apparently that intended in [PSS1], we establish invisibility with respect to the Helmholtz equation at all frequencies k = 0. In fact, one can not only cloak a passive object in a region D ⊂⊂ Ω, containing material with nonsingular index of refraction n(x), from all measurements made at the boundary ∂Ω, but also an active device, interpreted as a collection of sources and sinks within D. Among the phenomena described here is that finite energy solutions to the single coating construction must satisfy certain “hidden” boundary conditions at ∂ D. For the Helmholtz equation, this is the Neumann boundary condition at ∂ D − , and it follows that waves which propagate inside D and are incident to ∂ D − behave as if the boundary were perfectly reflecting. Thus, the cloaking structure on the exterior of D produces a “virtual surface” at ∂ D − . However, for Maxwell’s equations with electric permittivity ε(x) and magnetic permeability µ(x), the situation is more complicated. The hidden boundary condition forces the tangential components of both the electric field E and magnetic field H to vanish on ∂ D − . For cloaking passive objects, for which the internal current J = 0, this condition can be satisfied, but for generic J , finite energy time-harmonic solutions fail to exist, and thus the single coating construction is insufficient for invisibility. In practice, even for cloaking passive objects, this may degrade the effective invisibility. We find two ways of dealing with this difficulty. One is to simply augment the single coating construction around a ball by adding a perfect electrical conductor (PEC) lining at ∂ D, in order to make the object inside the coating material to appear like a passive object. (Such a lining was apparently incorporated, although claimed to be unnecessary, into the code used in [CPSSP] in an effort to stabilize the numerics.) However, for the sake of brevity, the necessary weak formulation of the boundary value problem for this setup will not be considered in this paper. Alternatively, one can introduce a more elaborate construction, which we refer to as the double coating. Mathematically, this corresponds to a singular Riemannian metric which degenerates in the same way as one approaches ∂ D from both sides; physically it would correspond to surrounding both the inner and outer surfaces of D with appropriately matched metamaterials. We show that, for the double coating, no lining is necessary and full invisibility holds for arbitrary active devices, at all nonzero frequencies, for both Helmholtz and Maxwell. It is even possible for the field to be identically zero outside of D while nonzero within D, and vice versa. Finally, we also analyze cloaking within an infinitely long cylinder, D ⊂ R3 . In the main result of §7 and §8, we show that the cylinder D becomes invisible at all frequencies if we use a double coating together with the so-called soft and hard (SHS) boundary condition on ∂ D. For the origin and properties of the SHS condition and a description of how the SHS condition can be physically implemented, see [HLS, Ki1, Ki2, Li]. We point out that there is some confusion in the physical literature concerning the theoretical possibility of invisibility. By this we mean uniqueness theorems for the inverse problem of recovering the electromagnetic parameters from boundary information (near field) or scattering (far field) at a single frequency, or for all frequencies. There is a vast literature on this subject. We only mention here mathematical results directly related to the one mentioned in [Le, SMJCPSS]. The Helmholtz operator at non-zero energy for isotropic media is given by ∆ + k 2 n(x), where n(x) is the index of refraction and k = 0. Unique determination of n(x) from boundary data for a single frequency k, under suitable regularity assumptions on n(x) and in dimension n ≥ 3, was proved

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in [SyU], with a similar result for the acoustic wave equation in [N]. (See [U] for a survey of related results). The article [N] was referred to in [Le, SMJCPSS] as showing that perfect invisibility was not possible. However, the results of [SyU, N] for the Helmholtz equation are valid only under the assumption that the medium is isotropic and that the index of refraction is bounded. This does not contradict the possibility of invisibility for an anisotropic index of refraction, nor for an unbounded isotropic index of refraction. The constructions of [GLU3, Le, PSS1] and the present paper violate both of these conditions. We also point out that the counterexamples given in [GLU1, Sect. 4] yield invisibility for the Helmholtz equation, in dimension n ≥ 3, for certain isotropic negative indices of refraction which are highly singular (and negative) on Ω \ D. We also note here that, at fixed energy, the Cauchy data is equivalent to the inverse scattering data. The connection between the fixed energy inverse scattering data, the Dirichlet-to-Neumann map and the Cauchy data is discussed, for instance, in [N] for the Schrödinger equation and in [U] for the Helmholtz equation in anisotropic media. The scattering operator is well defined for the degenerate metrics defined here; see, e.g., [M]. There is a large literature (see [U]) on uniqueness in the Calderón problem for isotropic conductivities under the assumption of positive upper and lower bounds for σ . It was noted by Luc Tartar (see [KV] for an account) that uniqueness fails badly if anisotropic tensors are allowed, since if F : Ω −→ Ω is a smooth diffeomorphism with F|∂Ω = id, then F∗ σ and σ have the same Dirichlet-to-Neumann map (and Cauchy data.) Note that since ε and µ transform in the same way, this already constitutes a form of invisibility, i.e., from the Cauchy data one cannot distinguish between the EM material parameter pairs ε, µ and
ε = F∗ ε,
µ = F∗ µ. Thus, uniqueness for anisotropic media, in the mathematical literature, has come to mean uniqueness up to a pushforward by a (sufficiently regular) map F. Such uniqueness in the Calderón problem is known under various regularity assumptions on the anisotropic conductivity in two dimensions [S, N1, SuU, ALP] and in three dimensions or higher [LaU, LeU, LTU], but for all of these results it is assumed that the eigenvalues of σ (x) are bounded below and above by positive constants. Related to the Calderón problem is the Gel’fand problem, which uses Cauchy data at all frequencies, rather than at a fixed one; for this problem, uniqueness results are also available, e.g., [BeK, KK], with a detailed exposition in [KKL]. For example, in the anisotropic inverse conductivity problem as above, Cauchy data at all frequencies determines the tensor up to a diffeomorphism F : Ω −→ Ω. Thus, a key point in the current works on invisibility that allows one to avoid the known uniqueness theorems for the Calderón problem is the lack of positive lower and upper bounds on the eigenvalues of these symmetric tensor fields. In this paper, as in [GLU3, Le, PSS1], the lower bound condition is violated near ∂ D, and there fails to be a global diffeomorphism F relating the pairs of material parameters having the same Cauchy data. For Maxwell’s equations, all of our constructions are made within the context of the permittivity and permeability tensors ε and µ being conformal to each other, i.e., multiples of each other by a positive scalar function; this condition has been studied in detail in [KLS]. For Maxwell’s equations in the time domain, this condition corresponds to polarization-independent wave velocity. In particular, all isotropic media are included in this category. This seemingly special condition arises naturally from our construction, since the pushforward (
ε,
µ) of an isotropic pair (ε, µ) by a diffeomorphism need not be

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isotropic but does satisfy this conformality. For both mathematical and practical reasons, it would be very interesting to understand cloaking for general anisotropic materials in the absence of this assumption. We believe that our results suggest improvements which can be made in physical implementations of cloaking. In the very recent experiment [SMJCPSS], the configuration corresponds to a thin slice of an infinite cylinder, inside of which a homogeneous, highly conducting disk was placed in order to be cloaked. This corresponds to the single coating with the metric g2 (see §2) on D being a constant multiple of the Euclidian metric. The analysis here suggests that lining the inside surface ∂ D − of the coating with a material implementing the SHS boundary condition [HLS, Ki1, Ki2, Li] should result in less observable scattering than occurs without the SHS lining, improving the partial invisibility already observed. The paper is organized as follows. In §2 we describe the single and double coating constructions. We then establish cloaking for the Helmholtz equation at all frequencies in §3. The notion of a finite energy solution for the single coating is defined in §§3.2 and then the key step for showing invisibility is Proposition 3.5. We discuss the Helmholtz equation for the double coating in §§3.3; there we define the notion of a weak solution and the Neumann boundary condition at the inner surface of the cloaked region. The key step for invisibility for Helmholtz at all frequencies in the presence of the double coating is Proposition 3.11. In §4 we study invisibility at all frequencies for Maxwell’s equations. We define the notion of finite energy solutions for the single and double coatings. In §5 we demonstrate invisibility for Maxwell’s at all frequencies for the double coating; see Proposition 5.1. In §6 we show that, for the single coating construction, the Cauchy data for Maxwell’s equations must vanish on the surface of the cloaked region, showing that generically finite energy solutions for Maxwell’s equations in the cloaked region do not exist. In §7 we consider an infinite cylindrical domain and show invisibility at all frequencies for Maxwell’s equations for the double coating; the key result is Proposition 7.1. In §8, we consider how to cloak the cylinder, treating its surface as an obstacle with the SHS boundary condition. Finally, in §9, we briefly indicate how general the constructions can be made. In particular, we note that a modification the double coating allows one to change the topology of the domain and yet maintain invisibility. 2. Geometry and Basic Constructions The material parameters of electromagnetism, namely the conductivity, σ (x), electrical permittivity, ε(x), and magnetic permeability, µ(x), all transform as a product of a contravariant symmetric 2-tensor and a (+1)−density. That is, if F : Ω1 −→ Ω2 , y = F(x), is a diffeomorphism between domains in Rn , then σ (x) = (σ jk (x)) on Ω1 pushes forward to (F∗ σ )(y) on Ω2 , given by   n j k   ∂ F 1 ∂ F jk pq (x) q (x)σ (x) , (1) (F∗ σ ) (y) = p ∂F j ∂x det [ ∂ x k (x)] p,q=1 ∂ x  −1 x=F

(y)

with the same transformation rule for the other material parameters. It was observed by Luc Tartar (see [KV]) that it follows that if F is a diffeomorphism of a domain Ω fixing ∂Ω, then σ and
σ := F∗ σ have the same Dirichlet-to-Neumann map, producing infinite-dimensional families of indistinguishable conductivities. On the other hand, a

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Riemannian metric g = (g jk (x)) is a covariant symmetric two-tensor. Remarkably, in dimension three or higher, a material parameter tensor and a Riemannian metric can be associated with each other by σ jk = |g|1/2 g jk , or g jk = |σ |2/(n−2) σ jk , (g jk )

(2)

)−1

= (g jk and |g| = det (g). Using this correspondence, examples of sinwhere gular anisotropic conductivities in Rn , n ≥ 3, that are indistinguishable from a constant isotropic conductivity, in that they have the same Dirichlet-to-Neumann map, were given in [GLU3]. The two constructions there are based on two different types of degenerations of Riemannian metrics, whose singular limits can be considered as coming from singular changes of variables. The singular conductivities arising from these metrics via the above correspondence are then indistinguishable from a constant isotropic σ . In the current paper, we will continue to examine one of these constructions, corresponding to pinching off a neck of a Riemannian manifold; we refer to it as the single coating. We also introduce another construction, referred to as the double coating. We start by giving basic examples of each of these. For both examples, let Ω = B(0, 2) ⊂ R3 , the ball of radius 2 and center 0, be the domain at the boundary of which we make our observations; D = B(0, 1) ⊂ Ω the region to be cloaked; and Σ = ∂ D = S2 the boundary of the cloaked region. Single coating construction. We begin by recalling an example from [GLU3, PSS1]; the two dimensional examples in [Le, V] are either essentially the same or closely related in structure. For the single coating, we blow up 0 using the map x r F1 : B(0, 2)\{0} → Ω \ D, F1 (x) = ( + 1) , r = |x|, 0 < r ≤ 2. (3) 2 r On B(0, 2), let (ge )i j = δi j be the Euclidean metric, corresponding to constant isotropic material parameters; via the map F1 , ge pushes forward, i.e., pulls back by F −1 , to a metric on Ω \ D,
g1 = (F1 )∗ ge := (F1−1 )∗ (ge ). Introducing the boundary normal coordinates (ω, τ ) in the annulus Ω \ D, where ω = (ω1 , ω2 ) are local coordinates on Σ = S2 and τ > 0 is the distance in metric
g1 to Σ, we have, from (3),
g1 = τ 2 dω2 + dτ 2 , τ = 2(r − 1).

(4)

Here dω2 = h αβ (ω)dωα dωβ is the standard metric on S2 , induced by the Euclidian metric on R3 . Note that
g1 has the following properties: Consider a local ge -orthonormal frame (∂r , v, w) on Ω \ D consisting of the radial vector ∂r =

xj ∂ ∂ = ∂r r ∂x j

and two vector fields v, w. Then, g1 (∂r , v) =
g1 (∂r , w) = 0,
g1 (w, v) = 0,
g1 (∂r , ∂r ) = 4,

g1 (v, v)
g1 (w, w) ∈ [c1 , c2 ], ∈ [c1 , c2 ], (r − 1)2 (r − 1)2

(5)

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where c1 , c2 > 0. Thus,
g1 has one eigenvalue bounded from below (with eigenvector corresponding to the radial direction) and two eigenvalues that are of order (r −1)2 (with eigenspace span{v, w}). In Euclidean coordinates, we have that, for |
g1 | = det (
g1 ), |
g1 (x)|1/2 ∼ C1 (r − 1)2 , (6) 2x ij |
g1 νi | ≤ C2 , νi = = 2(∂r )i . r Here and below we use Einstein’s summation convention, summing over indices appearing both as sub- and super-indices in formulae, and ν = (ν1 , ν2 , ν3 ) denotes the unit co-normal vectors of the surfaces {x ∈ Ω \D : |x| = s}, 1 < s < 2, with respect to the metric
g. On D, we simply let
g2 be the Euclidian metric. Together, the pair (
g1 ,
g2 ) define a singular Riemannian metric on Ω, 
g1 , x ∈ Ω \ D,
g=
g2 , x ∈ D, which is singular on Σ + , i.e., as one approaches Σ from Ω\D; in the sequel, we will identify the metric
g and the corresponding pair (
g1 ,
g2 ). To unify notation for the two basic constructions, we will denote in the single coating case M1 = Ω, M2 = D and let M be the disjoint union M = M1 ∪ M2 . Also, for notational unity with the double coating, we let γ1 = {0} ⊂ M1 , γ2 = ∅ ⊂ M2 , and γ = γ1 ∪ γ2 . Moreover, we denote N1 = Ω \ D, N2 = D, Σ = ∂ D, and N = N1 ∪ Σ ∪ N2 := Ω ⊂ R3 . Double coating construction. The double coating refers to a metric on Ω that is degenerate on both sides of Σ and has the same limit as one approaches Σ from both sides. We now introduce notation, shared with the single coating, that will be used throughout for the double coating. Let M1 = Ω = B(0, 2), which is compact with boundary, and M2 := S31/π , the 3-sphere of radius 1/π , which is compact without boundary, and again let M = M1 ∪ M2 be their disjoint union. For the double coating, let γ1 = {0} ⊂ M1 , γ2 = {N P} ⊂ M2 , where N P is a chosen point, e.g., the North Pole of S31/π , and γ = γ1 ∪ γ2 . As in the single coating example, we let N1 = Ω \ D = B(0, 2)\ B(0, 1), N2 = D = B(0, 1), Σ = ∂ D, and N = N1 ∪ Σ ∪ N2 ⊂ R3 . We take the diffeomorphism F1 : M1\γ1 −→ N1 to be the blow-up of γ1 as in the single coating, while we blow-up γ2 by defining F2 : M2 \γ2 −→ N2 as follows. Denote by S P the point on Ω2 antipodal to N P. Then the Riemannian normal coordinates centered at S P are defined on B(0, 1) ⊂ TS P S3 R3 , exp S P : B(0, 1) → M2 \{N P}. Denote by F2 the diffeomorphism −1  F2 = exp S P : M2 \{N P} → B(0, 1). Introduce (local) spherical coordinates (ω, r ) on N2 = B(0, 1), considered as a subset of TS P (S3 ), with ω = (ω1 , ω2 ), ω ∈ Σ = ∂ B(0, 1) and 0 ≤ r ≤ 1. The standard metric g on S31/π in these coordinates takes the form sin2 (πr ) dω2 + dr 2 , π2 where dω2 is again the standard metric on S2 . g2 =

(7)

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Observe that
g2 = (F2 )∗ (g2 ), as one approaches Σ − on B(0, 1), has very similar properties to
g1 on B(0, 2)\ B(0, 1) as one approaches Σ + . Indeed, again consider the j ∂ radial vector ∂r = ∂r = xr ∂ ∂x j at x ∈ N2 and two vectors v, w such that in Euclidean metric (∂r , v, w) is a local orthonormal frame. Then, as follows from (7), at x ∈ N2 with, say, 1/2 < r < 1,
g2 (∂r , ∂r ) = 1,
g2 (∂r , v) =
g2 (∂r , w) = 0,
g2 (v, v)
g2 (w, w)
g2 (w, v) = 0, , ∈ [c1 , c2 ], (1 − r )2 (1 − r )2 g2 has one eigenvalue bounded from below (with eigenvector where c1 , c2 > 0. Thus,
corresponding to the radial direction) and two eigenvalues that are of order (1 − r )2 , and with respect to the Euclidean coordinates on N2 , ij

|
g2 (x)|1/2 ≤ C1 (1 − r )2 , |
g2 νi | ≤ C2 , νi = −

xi 1 = −(∂r )i , < r < 1. r 2

(8)

Set
g1 = (F1 )∗ ge on N1 , where F1 is defined as for the single coating example. Together, these define a singular metric
g = (
g1 ,
g2 ) on the entire ball N = N1 ∪ N2 ∪ Σ = B(0, 2). Comparing (4) and (7), we see that, in the Fermi coordinates 1 associated to Σ, |
g |1/2
g i j is Lipschitz continuous on N ; note also that |
g |1/2
gi j is not invertible at ∂ B(0, 1). Although they are distinct, each of these constructions may be summarized as follows. The domain Ω, which we will refer to as N , decomposes as N = N1 ∪ Σ ∪ N2 , where N1 = Ω \ D, N2 = D and Σ = ∂ D. N1 and N2 are manifolds with boundary, with ∂ N1 = ∂Ω ∪ ∂ D + = ∂ N ∪ Σ + and ∂ N2 = Σ − , where the superscripts ± are used when considering limits from the exterior or interior of the cloaked region. The singular electromagnetic material parameters on N will correspond to a singular Riemannian metric
g = (
g1 ,
g2 ), arising as the pushforward of a (nonsingular) Riemannian metric g = (g1 , g2 ) on a manifold with two components, M = M1 ∪ M2 , via a map F : M \γ −→ N ,  F1 (x), x ∈ M1 \γ , F(x) = F2 (x), x ∈ M2 \γ . Here, M1 and M2 are disjoint, with M1 diffeomorphic to N ; γ1 = γ ∩ M1 is either a point (the point being blown up) for the single and double coatings, or a line (for the cloaking of an infinite cylinder in §7,8); and γ2 = γ ∩ M2 is either empty (for the single coating) or a point (for the double coating) or a line (for the cylinder.) In §9, we will show that such constructions exist in great generality, and for this reason the proofs will be expressed in terms of analysis on M and N . In this generality, we say that (M, N , F, γ , Σ, g) is a coating construction if (M, g) is a (nonsingular) Riemannian manifold, γ ⊂ M and Σ ⊂ N are as above, and F : M \γ → N \Σ is a diffeomorphism of either type. This then defines a singular Riemannian metric
g everywhere on N \Σ = N1 ∪ N2 , by 
g1 := F1∗ g1 , x ∈ N1 ,
g=
g2 := F2∗ g2 , x ∈ N2 . 1 Recall that- the Fermi coordinates associated to Σ are (ω, τ ), where ω = (ω1 , ω2 ) are local coordinates on Σ and τ = τ (x) is the distance from x to Σ with respect to the metric
g , multiplied by +1 in N1 and −1 in N2 .

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If we introduce Fermi coordinates (ω, τ ) near Σ as above, the
g satisfies (5),(6) or (8), with r − 1 replaced by τ , for the single and double coatings, resp. From these, one sees that |
g |1/2 g jk has a jump discontinuity across Σ for the single coating and is Lipschitz for the double coating. Note that in both examples, N = Ω = B(0, 2), so that N and M1 have the same topology. However, in a direct extension of the double coating construction, described in §9, the domain N containing the cloaked region N2 need not even be diffeomorphic to M1 . 3. The Helmholtz Equation We are interested in invisibility of a cloaked region with respect to the Cauchy data of solutions of the Helmholtz equation, (∆g + k 2 )u = f in Ω,

(9)

k consists of where f represents a collection of sources and sinks. The Cauchy data Cg, f the set of pairs of boundary measurements (u|∂Ω , ∂ν u|∂Ω ), where u ranges over solutions to (9) in some function or distribution space (discussed below). Let (M, N , F, γ , Σ, g) be a single coating construction as in §2. For the moment, as in the Introduction, we continue to refer to N as Ω, N2 as D and Σ + as ∂ D + ; we may assume that M1 = N , M2 = D and F2 = id, so that
g2 = g2 is a (nonsingular) Riemannian metric on D. Thus,
g is a Riemannian metric on Ω, singular on Ω \ D, resulting from blowing up the metric g1 on Ω with respect to a point O and inserting the (D, g2 ) into the resulting “hole”. k k for all frequencies 0 < k < ∞, if supp(
We wish to show that C
= Cg,0 f) ⊂ D g,
f g , it is necesand k is not a Neumann eigenvalue of (D, g2 ). Due to the singularity of
sary to consider nonclassical solutions to (9), and we will see that the exact notion of weak solution is crucial. Furthermore, a hidden Neumann boundary condition on ∂ D − is required for the existence of finite energy solutions. Physically, this means that the coating on Ω \ D makes the inner boundary ∂ D − appear to be a perfectly reflecting “sound-hard surface” for waves propagating in D, while, from the exterior, the cloaked device is invisible; that is, measurements of solutions of the Helmholtz equation at ∂Ω cannot distinguish between (Ω,
g ) and (Ω, g).

3.1. k = 0 and weak solutions. First consider the case when k = 0 and f = 0. As described in the Introduction, this situation was treated in [GLU3] in the context of electrical impedance tomography. There, it sufficed to consider as weak solutions those L ∞ functions satisfying (9) (for the metric
g ) in the sense of distributions. It was shown that, for given Dirichlet data h on ∂Ω, (9) has a unique such solution,
u , which must, by removable singularity considerations, be constant on D. These same conclusions would have held if we had considered the larger class of spatial H 1 weak solutions (defined below). However, for k > 0 or f = 0, we will see that this notion of weak solution is inappropriate. 3.1.1. k > 0 and spatial H 1 solutions. Definition 1.
u is a spatial H 1 solution to the Dirichlet problem for the Helmholtz equation, 2 (∆
u=
f on Ω,
u |∂Ω = h g + k )


(10)

Full-Wave Invisibility of Active Devices at All Frequencies

759

if
u ∈ H 1 (Ω, d x)

(11)

∂i (|
g |1/2
gi j ∂ j
u ) + k 2 |
g |1/2
u = |
g |1/2
f in H −1 (Ω, d x).

(12)

and

Here, for s ∈ R, H s (Ω, d x) refers to the standard Sobolev space of distributions with s derivatives in L 2 (Ω, d x). Note that (11), together with the properties of the metric tensor given in §2, implies that |
g |1/2
g i j ∂i
u ∈ L 2 (Ω, d x). Later in our analysis (see (36)), we will see that (12) implies that the normal derivative of
u on ∂ D − vanishes, ∂r
u |∂ D − = 0. On the other hand, the fact that
u ∈ H 1 (Ω, d x) implies that
u |∂ D − =
u |∂ D + = constant := u(O), with u the solution to (∆g + k 2 )u = 0 in ∂Ω, u|∂Ω = h, where the first equality follows from the trace theorem for H 1 functions and the second from considerations similar to those in [GLU3, Prop. 1]. Note that, for generic k and h, u(O) = 0. Thus,
u 2 :=
u|D needs to be a solution of the overdetermined elliptic boundary value problem on (D,
g2 ), (∆ + k 2 )
u 2 = 0, ∂ν
u 2 |∂ D = 0,
u 2 |∂ D = constant = 0.

(13)

Clearly, for generic k > 0 there exists no solution to (13) and therefore there is no weak solution to (10) in the sense of Definition 1. Rather, one needs to use an H 1 norm adapted to the singular Riemannian metric
g ; this is in fact physically natural, being essentially the energy of the wave. We formulate the correct notion in the next section.

3.2. Finite energy solutions for the single coating. We now give a more satisfactory definition of weak solution, restricting the notion to those solutions that are physically meaningful in that they have finite energy. We now revert to the notation of M, N , . . . when discussing the single coating construction, i.e., let (M, N , F, γ , Σ, g) denote a single coating as in §2. Our first task is to understand in what sense the expression |
g |1/2
g i j ∂i
u is rigorously defined. ∞
To this end, define for φ ∈ C (N ), 
∂ j φ
+ |

|2 ) d x.
2X := g |1/2 |φ (|
g |1/2
g i j ∂i φ φ N

Let H 1 (N , |
g |1/2 d x) = X := cl X (C ∞ (N )) be the completion of C ∞ (N ) with respect to the norm  ·  X . We note that H 1 (N , |
g |1/2 d x) ⊂ L 2 (N , |
g |1/2 d x), so we can consider its elements as measurable functions on N .

760

A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann

Lemma 1. The map
= (D j φ
3
)3j=1 , φ ∈ C ∞ (N ), φ −→ D
g |1/2
g i j ∂i φ gφ
g ) j=1 = (|
has a bounded extension 1 g |1/2 d x) → M(N ; R3 ), D
g : H (N , |


where M(N ; R3 ) denotes the space of R3 -valued signed Borel measures on N . Moreover, for
u ∈ X , we have, in the sense of Borel measures (D
u )(Σ) = 0. g


(14)

j
∞
∈ C ∞ (N ) and

Proof. Let φ η ∈ C(N ). Then D
g φ ∈ L (N ). Let φ = φ ◦ F, η = ∞
η ◦ F ∈ L (Ω). Then,   j
j
φ )
η d x = η dx (D
(D
g g φ)
N

 =

N\Σ

M1\γ1

|g|

1/2 kl ∂ y

g

∂x

j

∂ φ η dx + l k

 |g|1/2 g kl M2

∂y j ∂k φ η d x. ∂ xl

As the metric g is bounded from above and below, and ∂∂ yx l = O(r −1 ) on M1 and = δl on M2 , we have      ≤ C0 (φ H 1 (M ,d x) η/r  L 2 (M ,d x) + φ H 1 (M ,d x) η L 2 (M ,d x) )  (D j φ
)
η d x
g   1 1 2 2 j

j

N


|| X ||
η||C(N ) d
η), Σ), ≤ C1 ||φ g (supp(
1/2

g . This shows the existence where d
g is the distance on N with respect to the metric
1 (N , |
1/2 d x) → M(N ; R3 ). Also, if we consider of the bounded extension D
: H g | g functions
η supported in small neighborhoods of Σ, we see that (14) follows.   We also need the following auxiliary result Lemma 2. Assume that
u is a measurable function on N such that
u ∈ L 2 (N , |
g |1/2 d x), 1
u | N\Σ ∈ Hloc (N \Σ, d x),  |
g |1/2
g i j ∂i
u∂ j
u d x < ∞.

(15) (16) (17)

N\Σ

g |1/2 d x). Then
u ∈ H 1 (N , |
Note that, due to the fact that
g is bounded and positive definite on any compact subset of N\Σ, condition (16) in fact follows from conditions (15), (17) and is included for the convenience of future references.

Full-Wave Invisibility of Active Devices at All Frequencies

761

Proof. Consider first the case when
u = 0 in N1 . First, the condition (17) implies that
v =
u | N2 ∈ H 1 (N2 , d x). Let f = v|Σ ∈ 1/2 f 1 H (Σ) and E ∈ H (N1 , d x) be an extension of f . Let χ ∈ C0∞ (R) be a cut-off function with χ (t) = 1 for |t| < 21 and χ (t) = 0 for |t| > 1. We introduce Fermi coordinates near Σ as in §2, (τ, ω), τ ∈ (0, 2), ω = (ω1 , ω2 ) ∈ Σ. Define, for ε > 0,  v(x), x ∈ N2 , wε (x) = χ ( τε )E f (x), x ∈ N1 . Then wε ∈ H 1 (N , d x) and, using (3), (5), we see that  lim |
g |1/2 [
g i j ∂i (wε −
u )∂ j (wε −
u ) + (wε −
u )2 ] d x ε→0 N\Σ  = lim |
g |1/2 [
g i j ∂i wε ∂ j wε + |wε |2 ] d x = 0.

(18)

ε→0 N1

To see this, observe that the integrand vanishes outside a neighborhood of Σ + of volume less than Cε. Next, divide the integral involving derivatives in the right-hand side of (18) into the terms involving components tangential and normal to the boundary, using the fact that τ = 2(r − 1):  τ αβ g ∂ωα E f ∂ωβ E f dτ dω1 dω2 , |
g |1/2 χ 2 ( )
ε N1\Σ and where α, β run over {1, 2},   2 τ   |
g |1/2 ∂τ [χ ( )E f ] dτ dω1 dω2 . ε N1\Σ g αβ is bounded, the integral involving tangential derivatives tends to 0 As, by (5), |
g |1/2
due to the volume of the domain of integration. Again, by (5) we have |
g |1/2 ≤ Cτ 2 ; τ −1 this, together with the volume estimate and the fact that |∂τ χ ( ε )| ≤ Cτ , implies that the integral involving normal derivatives tends to 0 when ε → 0. Similarly, we see that  τ |
g |1/2 |χ ( )E f |2 d x → 0 for ε → 0. ε N1\Σ The function wε ∈ H 1 (N , d x) can be approximated with an arbitrarily small error in H 1 (N , d x) by a C ∞ (N ) function, and we see that the same holds in the X -norm. Thus wε ∈ H 1 (N , |
g |1/2 d x), and the above limit shows that
u ∈ H 1 (N , |
g |1/2 d x). Now let
u be a measurable function in N satisfying (15), (16), and (17). Let χ N2 be the characteristic function of N2 . As χ N2
u ∈ H 1 (N , |
g |1/2 d x), it is enough to show that 1 1/2
u − χ N2
u ∈ H (N , |
g | d x). This means that it is enough to consider the case when
u = 0 in N2 . Clearly, we can restrict our attention to the case when
u vanishes also near ∂ N . Now let u 1 =
u ◦ F in M1 \γ1 . Then we see that  |g|1/2 g i j ∂i (u 1 )∂ j (u 1 ) d x < ∞. M1\γ1

762

A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann

Let w = ∇u 1 | M1\γ1 . Using a change of coordinates in integration and (15), we see that w ∈ L 2 (M1 \ γ1 , d x). Extending u 1 and w to functions u e1 and w e on γ1 , we obtain functions u e1 ∈ L 2 (M1 , d x) and the R3 –valued function w e ∈ L 2 (M1 , d x). Now ∇u e1 −w e ∈ H −1 (M1 , d x) is supported on γ1 . Since there are no non-zero H −1 (M1 , d x) distributions supported on γ1 , we see that ∇u e1 = w e ∈ L 2 (M1 , d x). Thus we see that u e1 ∈ H 1 (M1 , d x). In the following we identify u 1 and u e1 . As u 1 vanishes near ∂ M1 , and γ1 consists of a single point and thus is a (2, 1)-polar set [Ma, pp.393–7], there are φ j ∈ C0∞ (M1 \γ1 ) such that φ j → u 1 in H 1 (M1 , d x) as j → ∞, that is,  lim |g|1/2 [g ik ∂i (φ j − u)∂k (φ j − u) + (φ j − u)2 ] d x = 0. j→∞ M1


j ) ⊂ N1 and
j ∈ C ∞ (N ), with supp(φ Now let φ 0
j = φ



φ j ◦ F1−1 0

in N1 , in N2 .

Then the previous equation implies that 
j −

j −

j −
|
g |1/2 [
g ik ∂i (φ u )∂k (φ u ) + (φ u )2 ] d x = lim j→∞ N\Σ 
j −

j −

j −
lim |
g |1/2 [
g ik ∂i (φ u )∂k (φ u ) + (φ u )2 ] d x = 0, j→∞ N1

where we use that
u = 0 in N2 .
j is a sequence converging in the X -norm and that the limit is
This shows that φ u. 1 1/2 Thus
u ∈ H (N , |
g | d x), proving the claim.   Although in this section (M, N , F, γ , Σ, g) continues to denote a single coating, we will see later that the following definition is also appropriate for the double coating construction. Let
f ∈ L 2 (N , d x) be a function such that supp (
f ) ∩ Σ = ∅. Definition 2. Let (M, N , F, γ , Σ, g) be a coating construction. A measurable function
u on N is a finite energy solution of the Dirichlet problem for the Helmholtz equation on N , 2 (∆
u=
f on N , g + k )


u |∂ N = h,

(19)

if
u ∈ L 2 (N , |
g |1/2 d x); 1
u | N\Σ ∈ Hloc (N \Σ, d x);  |
g |1/2
g i j ∂i
u∂ j
u d x < ∞, N\Σ

h, ;
u |∂ N =


(20) (21) (22)

Full-Wave Invisibility of Active Devices at All Frequencies

763


∈ C ∞ (N ) with ψ
|∂ N = 0, and, for all ψ 


+ k 2

|
[−(D
u∂ j ψ uψ g |1/2 ]d x = g )
j

N





(x)|
f (x)ψ g |1/2 d x,

(23)

N

where the integral on the left hand side of (23) is defined by distribution-test function duality. Note as before that condition (21) follows from (20), (22). Cloaking by the single coating of arbitrary active devices, with respect to solutions of the Helmholtz equation at all frequencies, then follows from the following. Theorem 1. Let u = (u 1 , u 2 ) : M\γ → R and
u : N\Σ → R be measurable functions such that u =
u ◦ F. Let f = ( f 1 , f 2 ) : M\γ → R and
f : N\Σ → R be L 2 functions
supported away from γ and Σ such that f = f ◦ F, and
h : ∂ N → R, h : ∂ M1 → R be such that h =
h ◦ F1 . Then the following are equivalent: 1. The function
u , considered as a measurable function on N , is a finite energy solution to the Helmholtz equation (19) with inhomogeneity
f and Dirichlet data
h in the sense of Definition 2. 2. The function u satisfies (∆g + k 2 )u 1 = f 1

on M1 , u 1 |∂ M1 = h,

(24)

and (∆g + k 2 )u 2 = f 2

on M2 , g jk ν j ∂k u 2 |∂ M2 = b,

(25)

with b = 0. Here u 1 denotes the continuous extension of u 1 from M1 \γ to M1 Moreover, if u solves (24) and (25) with b = 0, then the function
u = u ◦ F −1 : N \Σ → R, considered as a measurable function on N , is not a finite energy solution to the Helmholtz equation. Remark. (i) It follows that the construction of [GLU1, PSS1] cloaks active devices from detection by unpolarized EM waves at all frequencies. (ii) Observe that in (24) the right hand side f 1 is zero near γ1 . Thus u 1 , considered as a distribution in a neighborhood of γ1 , has an extension on γ1 that is C ∞ smooth function in a neighborhood of γ1 . (iii) As noted previously, for the single coating case one may assume that N2 = M2 and F| M2 is the identity. Thus
u | N2 = u| M2 ; hence, if
u is a finite energy solution of the Helmholtz equation on N , we see that u| M2 satisfies the Neumann boundary condition on ∂ M2 and thus also
u | N2 automatically has to satisfy the Neumann condition on Σ − . The Neumann boundary condition that appears on ∂ N2 means that, observed from the inside of the cloaked region N2 , the single coating construction has the effect of creating a virtual sound hard, i.e., perfectly reflecting, surface at Σ. Similarly, we will see later that there are hidden boundary conditions for Maxwell’s equations in the presence of the single coating, but they are overdetermined and generally preclude such solutions existing.

764

A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann

Proof. First we proof that Helmholtz on M implies Helmholtz on N . Let f ∈ L 2 (M, d x) be a function such that supp ( f ) ∩ (γ ∪ ∂ M1 ∪ ∂ M2 ) = ∅. Assume that a function u on M is a classical solution of (24) and (25). Notice that we have required here that u 2 on ∂ M2 satisfies the Neumann boundary condition at ∂ M2 . Again, define
u = F∗ u and
f = f ◦ F −1 on N \Σ and extend it, e.g., by setting it equal to zero on Σ. Note that then
f ∈ L 2 (N , d x) is supported away from Σ, and 2 1/2
u ∈ L (N , |
g | d x) satisfies 2 (∆
u1 =
f1 =
f | N1 in N1 ,
u |∂ N =
h, g + k )


(26)

and 2 (∆
u2 =
f2 =
f | N2 in N2 . g + k )


(27)

Let Σ(ε) be the ε-neighborhood of Σ with respect to the metric
g . Let γ (ε) be the ε-neighborhood of γ ⊂ M1 with respect to the metric g. Let gbnd and
gbnd be the induced metrics on ∂γ (ε) and ∂Σ(ε), correspondingly. Clearly, the function
u satisfies conditions (20), (21), and (22). By Lemma 2, we have that
u ∈ H 1 (N , |
g |1/2 d x), and D
u is thus well defined. g
Using relations (5) for the normal component and (26), (27), and property (14) of
∈ C ∞ (N ), D
g u, we see that, for ψ 0  j
+ k 2

|

|
[−(D
u∂ j ψ uψ g |1/2 −
fψ g |1/2 ]d x (28) g )
N 
+ (k 2

)|
(−
g i j ∂i
u ∂jψ u−
f )ψ g |1/2 d x = lim ε→0 N\Σ(ε)  
)|
= lim ( + )(−
g i j ν j ∂i
uψ gbnd |1/2 d S ε→0 ∂Σ(ε)∩N2 ∂Σ(ε)∩N1 
)|
= lim gbnd |1/2 d S + (−
g i j ν j ∂i
u2ψ (29) ε→0 Σ(ε)∩N2 
◦ F))|gbnd |1/2 d S + lim (−g i j ∂i u 1 ν j (ψ (30) ε→0 ∂γ (ε)

= 0. Indeed, the integral (29) in the right-hand side of this equation tends to 0 due to the
◦ F. To analyze the integral boundary condition on Σ − (25), and boundedness of ψ (30) observe that, as supp f 1 ∩ γ1 = ∅, u 1 is infinitely smooth near γ1 . Thus all ∂i u 1 and
◦ F are bounded near γ1 , while the area of ∂γ (ε) is bounded by Cε2 . Hence we see ψ that (23) is valid and thus 2 (∆
u=
f in N g + k )


in the sense of Definition 2. Summarizing, so far we have proven that a (classical) solution to the Helmholtz equation on M yields, via the pushforward, a finite energy solution to the equation on N . Next we consider the other direction and prove that the Helmholtz equation on N implies the Helmholtz equation on M. Assume that
u satisfies the Helmholtz equation (19) on (N ,
g ) in the sense of Definition 2, with
f ∈ L 2 (N ) supported away from Σ. In particular,
u is a measurable function in N satisfying (15), (16), and (17).

Full-Wave Invisibility of Active Devices at All Frequencies

765

Let u =
u ◦ F and f =
f ◦ F on M \γ . Then we have (∆g + k 2 )u 1 = f 1 = f | M1\γ1 in M1 \γ1 , u 1 |∂ M1 = h

(31)

(∆g + k 2 )u 2 = f 2 = f | M2 in M2 .

(32)

and

By conditions (15), (16), and (17), we have that |u|2 ∈ L 1 (M1 \γ1 , |g|1/2 d x), g jk (∂ j u)(∂k u) ∈ L 1 (M1 \γ1 , |g|1/2 d x), and thus u 1 ∈ H 1 (M1 \γ1 , d x). As before, we see that (∆g + k 2 )u 1 = f 1 in M1 , u 1 |∂ M1 = h,

(33)

where f 1 extends to have the value 0 at γ1 and u 1 is smooth near γ1 . Let us now consider the boundary condition on M2 . Since
u satisfies (23), we see
∈ C ∞ (N ), that for ψ 0



j
+ k 2

|

|
[−(D
u∂ j ψ uψ g |1/2 −
fψ g )|1/2 ]d x g )

+ (k 2

) |
(−
g i j ∂i
u ∂jψ u−
f )ψ g |1/2 d x = lim ε→0 N\Σ(ε)  
)|
= lim ( + )(−
g i j ν j ∂i
uψ gbnd |1/2 d S ε→0 ∂Σ(ε)∩N2 ∂Σ(ε)∩N1  = (−g i j ν j ∂i u 2 |∂ M2 ψ)|gbnd |1/2 d S ∂ M2  (−g i j ∂i u 1 ν j ψ)|gbnd |1/2 d S + lim ε→0 ∂γ (ε)  = (−g i j ν j ∂i u|∂ M2 ψ)|gbnd |1/2 d S,

0=

(34)

N

(35)

∂ M2


◦ F. Here we use the fact that u 1 is a smooth function, implying that ∂i u 1 where ψ = ψ
◦ F is bounded. As ψ
|∂ M2 ∈ C ∞ (∂ M2 ) is arbitrary, this is bounded and that ψ = ψ shows that
g i j ν j ∂i
u 2 |∂ M2 = 0.

(36)

Thus, we have shown that the function u is a classical solution on M of (∆g + k 2 )u 1 = f 1 in M1 , u 1 |∂ M1 = h

(37)

(∆g + k 2 )u 2 = f 2 in M2 , g jk ν j ∂k u 2 |∂ M2 = 0.

(38)

and

This proves the claim, and finishes the proof of Theorem 1.

 

766

A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann

3.2.1. Operator theoretic definition of the Helmholtz equation. It is standard in quantum physics that a self-adjoint operator can be defined via the quadratic form corresponding to energy. In the case considered here, the energy associated with the wave operator is defined by the quadratic (Dirichlet) form A,  A[
u,
u ] := |
g |1/2
g i j ∂i
u∂ j
u d x,
u ∈ D(A). (39) N\Σ

As we deal with the sound-soft boundary ∂ N or, more generally, with the source on ∂ N of the form
u |∂ N =
h, the domain D(A) of the form A should be taken as D(A) = H01 (N , |
g |1/2 d x) ⊂ X. Thus, by standard techniques of operator theory, e.g., [Ka], the form A defines a positive D 2 selfadjoint operator, denoted A0 = −∆
g |1/2 d x). Next we recall this cong , on L (N , |
g |1/2 d x) is in the domain of A0 ,
u ∈ D(A0 ) if there struction. We say that
u ∈ H01 (N , |
1 2 1/2 1/2
is an f ∈ L (N , |
g | d x) such that for all
v ∈ H0 (N , |
g | d x), 
(40) A[
u ,
v] = f
v |
g |1/2 d x. N

In this case, we define A0
u=
f. From spectral theory, one knows that the operator A0 defines spaces D(As0 ) and s operators A0 : D(As+1 0 ) → D(A0 ), s ∈ R. D Proposition 1. Assume that −k 2 is not in the spectrum of ∆
u is a finite energy g . Then
solution to 2 u=
f,
u |∂ N =
h ∈ H 1/2 (∂ N ) (∆
g + k )


if and only if D 2 −1
2 h),
u = E
h + (∆
g + k )E
g + k ) ( f − (∆


where E
h is an

H 1 (N , d x)-extension

(41)

of
h to N satisfying supp (E
h) ⊂ ∂ N ∪ N1 .

Proof. Note that the above conditions on E
h imply that the right-hand side in (41) is a well-defined function in H 1 (N , |
g |1/2 d x). First we show that if
u satisfies the con
∈ C ∞ (N ), ψ
|∂ N = 0, imply that ditions of Definition 2 then it satisfies (41). As ψ
∈ H 1 (N , |g|1/2 d x), we see by (23) that
ψ u − E
h satisfies 0   j 2 (−D
u − E
h) ∂ j
v + k 2 (
u − E
h)
v) d x = |
g |1/2 (
f − (∆
h)
v d x, g + k )E
g (
N

for any
v∈

N\Σ

C0∞ (N ). 

By (14) and (22), this implies

 g i j ∂i (
|
g |1/2 −
u − E
h)∂ j
v + k 2 (
u − E
h)
v dx N\Σ  2 = |
g |1/2 (
f − (∆
h)
v d x, g + k )E


(42)

N\Σ

for any
v ∈ C0∞ (N ). We need to show that (42) is valid for all
v ∈ H01 (N , |g|1/2 d x).

Full-Wave Invisibility of Active Devices at All Frequencies

767

Observe that    g i j ∂i (E
|
g |1/2 −
h)∂ j
v + k 2 (E
h)
v dx = N\Σ

N\Σ

2 |
g |1/2 ((∆
h)
v d x, g + k )E


where we use that supp(E
h) ⊂ ∂ N ∪ N1 and
v |∂ N = 0. Thus, it remains to show that    1/2 ij 2 −
g ∂i
|
g| u∂ j
v+k
u
v dx = |
g |1/2
f
v dx (43) N\Σ

N\Σ

for
v ∈ H01 (N , |g|1/2 d x). Clearly, to show this it is enough to show that   lim g i j ∂i
|
g |1/2 −
u∂ j
v + k 2
u
v−
f
v d x = 0, ε→0 N\Σ1 (ε)

(44)

where Σ1 (ε) = N1 ∩ Σ(ε). Next we argue analogously to the reasoning that led to Eq. (28). Let v =
v ◦ F, f =
f ◦ F, and u =
u ◦ F in M\γ . To clarify notations, denote u 1 = u| M1 , u 2 = u| M2 , v1 = v| M1 , v2 = v| M2 , and f 1 = f | M1 , f 2 = f | M2 . Then, by Proposition 1, (∆g + k 2 )u 1 = f 1 , in M1 ,

(45)

(∆g + k )u 2 = f 2 , in M2 , ∂ν u 2 |∂ M2 = 0,

(46) (47)

2

and we see that   lim g i j ∂i
|
g |1/2 −
u∂ j
v + k 2
u
v−
f
v dx ε→0 N\Σ1 (ε)   = lim |g|1/2 −g i j ∂i u∂ j v + k 2 uv − f v d x ε→0 (M1\γ (ε))∪M2   = (−g i j ν j ∂i (u) v)|gbnd |1/2 d S + (−g i j ν j ∂i (u) v)|gbnd |1/2 d S. ∂γ (ε)

∂ M2

By (47), we have that  ∂ M2

Next we consider

(−g i j ν j ∂i (u) v)|
gbnd |1/2 d S = 0.

(48)

 I1 (ε) =

∂γ (ε)∩M1

(−g i j ν j ∂i (u) v)|
gbnd |1/2 d S.

Note that limε→0 I1 (ε) exists as the limits (44) and (48) exists. As supp ( f ) ∩ γ = ∅, we see that u 1 is smooth function near γ . Moreover, as
v ∈ X, we observe that v1 ∈ H 1 (M1 \γ , d x), and so v1 can be extended to v1 ∈ H 1 (M1 , d x). Hence, by the Sobolev embedding theorem, v1 ∈ L 6 (M1 , d x). This allows us to deduce that  −3/2 |v1 | d S = 0. (49) lim inf ε ε→0

∂γ (ε)

768

A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann

Indeed, 

ε

 ∂γ (r )

0

  |v1 |d S(x) dr = 1/6 



≤

|v1 | d x 6

γ (ε)

γ (ε)

γ (ε)

|v1 |d x

5/6

dx

= o(ε5/2 ).

Clearly, this inequality implies (49). Thus using boundedness of g i j ν j ∂i u we see that  lim inf ε→0

∂γ (ε)

(−g i j ν j ∂i (u) v|gbnd |1/2 )d S = 0.

As limε→0 I1 (ε) exists, this implies limε→0 I1 (ε) = 0. As
u |∂ N =
h by Definition 2 we have shown that Definition 2 implies (41). Next, consider the case when
u satisfies (41). Since
u ∈ X , we see by (14) that   j D
(
u )∂
v d x = |
g |1/2
g i j ∂i
u∂ j
v dx (50) j g N

N\Σ

for all
v ∈ C0∞ (N ). Thus, by (41) we see that (43) is valid for
v ∈ C0∞ (N ), which implies condition (22). The other conditions in Definition 2 follow easily from (41).   3.3. Helmholtz for the double coating. We now examine solutions to the Helmholtz equation in the presence of the double coating; we will establish full-wave invisibility at all nonzero frequencies. Unlike for the single coating, for the double coating no extra boundary conditions appear at Σ. Otherwise, the reasoning here parallels that in §3.2. Throughout this section, (M, N , F, γ , Σ, g) is a double coating construction. 3.3.1. Weak solutions for the double coating. Suppose that k ≥ 0 and
f ∈ L 2 (N , 1/2 |
g | d x). We use the same notion of weak solution as for the single coating, saying that
u is a finite energy solution of 2 (∆
u=
f in N ,
u |∂ N =
h g + k )


(51)

if
u is a solution of the Dirichlet problem in the sense of Definition 2. We start with analogues of the space H 1 (N , |
g |1/2 d x), and Lemmas 1 and 2. To this
∈ C ∞ (N ), end define, for φ 
∂ j φ
+ |

2Y :=
|2 ) d x. g |1/2 |φ φ (|
g |1/2
g i j ∂i φ N

Let H 1 (N , |
g |1/2 d x) = Y := cl X (C ∞ (N )) be the completion of C ∞ (N ) with respect to the norm  · Y . Note that H 1 (N , |
g |1/2 d x) ⊂ L 2 (N , |
g |1/2 d x), so we can consider its elements as measurable functions in N .

Full-Wave Invisibility of Active Devices at All Frequencies

769

Lemma 3. The map
= (D j φ
3
)3j=1 , φ ∈ C ∞ (N ), φ −→ D
g |1/2
g i j ∂i φ gφ
g ) j=1 = (|
has a bounded extension 1 D
g |1/2 d x) → M(N ; R3 ), g : H (N , |


where M(N ; R3 ) denotes the space of R3 -valued signed Borel measures on N . Moreover, for
u ∈ Y , we have u )(Σ) = 0. (D
g


(52)

If
u is a measurable function on N such that g |1/2 d x),
u ∈ L 2 (N , |
1
u | N\Σ ∈ Hloc (N \Σ, d x), and  |
g |1/2
g i j ∂i
u∂ j
u d x < ∞,

(53) (54) (55)

N\Σ

then
u ∈ H 1 (N , |
g |1/2 d x). Proof. The proof here is essentially the same as of Lemmas 1 and 2. The only difference is that, as described in §2, the map F : M \γ → N \Σ now consists of two maps, Fi : Mi \γ → Ni , i = 1, 2, having similar structure to each other, namely that of the map F1 in the single coating construction. (Recall that for the double coating construction, γ1 := γ ∩ M1 is a point O ∈ M1 and γ2 := γ ∩ M2 a point N P ∈ M2 .) Therefore, when proving that
u satisfying (53)–(55) is in H 1 (N , |
g |1/2 d x), we can use the fact that, in this case, both (1 − χ N1 )
u and (1 − χ N2 )
u satisfy (53)–(55) and carry out the proof for each of them as for the (1 − χ N2 )
u term in the proof of Lemma 2. Invisibility of active devices in the presence of the double coating with respect to the Helmholtz equation at all frequencies then follows from Theorem 2. Let u = (u 1 , u 2 ) : M\γ → R and
u : N\Σ → R be measurable functions such that u =
u ◦ F. Let f = ( f 1 , f 2 ) : M\γ → R and
f : N\Σ → R be L 2 functions supported away from γ and Σ such that f =
f ◦ F. Then the following are equivalent: 1. The function
u , considered as a measurable function on N , is a finite energy solution to the Helmholtz equation (51) with inhomogeneity
f and Dirichlet data
h in the sense of Definition 2. 2. We have (∆g + k 2 )u 1 = f 1

on M1 , u|∂ M = h :=
h◦F

(56)

and (∆g + k 2 )u 2 = f 2

on M2 .

(57)

770

A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann

Proof. As in the proof of Theorem 1, we first prove that Helmholtz on M implies Helmholtz on N . Let f ∈ L 2 (M, d x) be a function such that supp ( f )∩(γ ∪∂ M1 ∪∂ M2 ) = ∅. Assume that a function u = (u 1 , u 2 ) on M is a classical solution of (56) and (57). Define
u = F∗ u and
f = f ◦ F −1 on N \Σ and extend it, e.g., by setting it equal to zero on Σ. Note that then
f ∈ L 2 (N , d x) is supported away from Σ. Then
u ∈ L 2 (N , |
g |1/2 d x) satisfies 2 u1 =
f1 =
f | N1 in N1 ,
u |∂ N =
h, (∆
g + k )


(58)

and 2 u2 =
f2 =
f | N2 in N2 . (∆
g + k )


(59)

Let Σ(ε) be the ε-neighborhood of Σ with respect to the metric
g . Let γ1 (ε) be the ε-neighborhood of γ1 = {0} ⊂ M1 with respect to the metric g. Let γ2 (ε) be the εneighborhood of γ2 = {N P} ⊂ M2 with respect to the metric g. Let gbnd and
gbnd be the induced metrics on ∂γ (ε) and ∂Σ(ε), correspondingly. Clearly, the function
u satisfies conditions (20), (21), and (22). By Lemma 3, we have that
u ∈ H 1 (N , |
g |1/2 d x), and D
u is thus well defined. g
Using relations (5), (6) in M1 and (8) in M2 , it follows from (58), (59) that for
∈ C ∞ (N ), ψ 0  j
+ k 2

|

|
[−(D
u∂ j ψ uψ g |1/2 −
fψ g |1/2 ]d x g )
N 
+ (k 2

)|
= lim (−
g i j ∂i
u ∂jψ u+
f )ψ g |1/2 d x ε→0 N\Σ(ε)  
)|
= lim ( + )(−
g i j ν j ∂i
uψ gbnd |1/2 d S ε→0 ∂Σ(ε)∩N2 ∂Σ(ε)∩N1 
◦ F))|gbnd |1/2 d S = lim (−g i j ∂i u 1 ν j (ψ ε→0 ∂γ1 (ε) 
◦ F))|gbnd |1/2 d S + lim (−g i j ∂i u 2 ν j (ψ ε→0 ∂γ2 (ε)

= 0.

(60)

Indeed, of (60) tend to 0 by the same arguments as the both terms in the right-hand side1/2
◦ F))|gbnd | d S in (28). Hence we see that (23) is valid term ∂γ (ε) (−g i j ν j ∂i u 1 (ψ and thus 2 u=
f in N (∆
g + k )


in the sense of Definition 2. So far, we have proven that a (classical) solution to the Helmholtz equation on M yields a finite energy solution to the equation on N . Next, we prove the converse, i.e., that the Helmholtz equation on N implies the Helmholtz equation on M. Assume that
u satisfies the Helmholtz equation (19) on (N ,
g ) in the sense of Definition 2, with
f ∈ L 2 (N ) supported away from Σ. In particular,
u is a measurable function in N satisfying (15), (16), and (17). Let u =
u ◦ F and f =
f ◦ F on M \γ . Then we have (∆g + k 2 )u 1 = f 1 = f | M1\γ1 in M1 \γ1 , u 1 |∂ M1 = h

(61)

Full-Wave Invisibility of Active Devices at All Frequencies

771

and (∆g + k 2 )u 2 = f 2 = f | M2\γ2 in M2 \γ2 .

(62)

By conditions (15), (16), and (17), we have that |u i |2 ∈ L 1 (Mi \γi , |g|1/2 d x), jk

gi (∂ j u i )(∂k u i ) ∈ L 1 (Mi \γi , |g|1/2 d x), i = 1, 2. Thus u i ∈ H 1 (Mi \γi , d x). As before, we see that then (∆g + k 2 )u 1 = f 1 in M1 , u 1 |∂ M1 = h, (∆g + k 2 )u 2 = f 2 in M2 ,

(63)

where f i is extended to have the value 0 at γi and u i are smooth near γi .
∈ C ∞ (N ), Since
u satisfies (23), we see that for ψ 0  j
+ k 2

|

|
0= [−(D
u∂ j ψ uψ g |1/2 −
fψ g )|1/2 ]d x g )
N 
+ (k 2

) |
(−
g i j ∂i
u ∂jψ u+
f )ψ g |1/2 d x = lim ε→0 N\Σ(ε)

  
)|
= lim (−
g i j ∂i
gbnd |1/2 d S(x) + u |∂Σ(ε) ν j ψ ε→0 ∂Σ(ε)∩N2 ∂Σ(ε)∩N1  = lim (−g i j ∂i u 1 ν j ψ)|gbnd |1/2 d S(x) ε→0 ∂γ1 (ε)  ij + lim (−gs ∂i u 2 ν j ψ)|gbnd |1/2 d S(x) ε→0 ∂γ2 (ε)

= 0,
◦ F. Here as in the proof of Proposition 1, we use the fact that u 1 is a where ψ = ψ smooth function implying that ∂i u 1 is bounded. Thus, we have shown that the function u is a classical solution on M of (∆g + k 2 )u 1 = f 1 in M1 , u 1 |∂ M1 = h

(64)

(∆g + k 2 )u 2 = f 2 in M2 .

(65)

and

This proves the claim.

 

Next we prove a result that is not necessary for the proof but gives, in the case of the u , simpler than before. double coating, an alternative treatment of the distribution D
g
Lemma 4. In the double coating construction, the term |
g |1/2
g i j ∂i
u ∈ D (N , d x),

(66)

u is well-defined as a sum of products of Sobolev appearing in Definition 2 as D
g
distributions and Lipschitz functions.

772

A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann

Proof. The problem we need to consider here is that L 2 (N , |
g |1/2 d x) contains functions that are not locally integrable with respect to measure d x and thus we do not immediately see that distribution derivatives ∂ j
u in N are well defined. We deal with this by applying condition (22). To do this, let u =
u ◦ F : M \γ → R. Using (20), (21), (22) and changing variables in the integration, one sees that  |g|1/2 g i j (∂i u)∂ j u) d x < ∞. M\γ

As g is bounded from above and below, this implies that u ∈ H 1 (M \γ , d x) ⊂ L 6 (M \ γ ), d x). Furthermore, changing variables again implies that  |
g |1/2 |
u |6 d x < ∞, N\Σ

L 6 (N ,

det (
g )1/2 d x).

Now in the boundary normal coordinates (ω, τ ) near so that
u∈ Σ, τ (x) = distR3 (x, Σ), we have τ −2 |
g |1/2 ∈ [c1 , c2 ], c1 , c2 > 0, and thus





|
u |τ (x)1/3 τ (x)−1/3 d x

|
u| d x = N

N

≤ 
u τ 1/3  L 6 (N ,d x) τ (x)−1/3  L 6/5 (N ,d x) ≤ 
u  L 6 (N ,τ 2 d x) τ (x)−2/5  L 1 (N ,d x) ≤ C
u  L 6 (N ,|
g |1/2 d x) < ∞, cf. the discussion at the end of §§3.2.3. A similar computation shows that
u ∈ L p (N , d x) −1, p for some p > 1, and thus ∂ j
u∈W (N , d x). As is shown at the end of §2 that g jk ∈ C 0,1 (N ), |
g |1/2
multiplication by |
g |1/2
g jk maps W |
g|

1, p 

−→ W


g ∂ j
u∈W

1/2 jk

1, p 

−1, p

(67)

and thus, by duality,

(N , d x),

i.e., the distribution (66) is defined as a sum of products of Lipschitz functions and W −1, p -distributions.   3.4. Coating with a lining: a physical surface. In the previous sections we have considered the Helmholtz equation in a domain N ⊂ R3 , equipped with a metric
g that is singular at a surface Σ. Later, for Maxwell’s equations, we will need to consider Σ as a “physical” surface, i.e., an obstacle on which we have to impose a boundary condition. To motivate these constructions, we consider next, for the Helmholtz equation, what happens when we have such a physical surface at Σ. More precisely, we consider the Helmholtz equation in the domain N \Σ = N1 ∪ N2 , where, on both sides of the boundary of Σ, that is, on Σ+ = ∂ N1 \∂ N and on Σ− = ∂ N2 , we impose a degenerate boundary condition of Neumann type. In physical terms, this corresponds to having a material, sound hard surface located at Σ, separating space into two open components, N1 and N2 . Although we will not need this, it can in fact be shown that
u is a solution in the sense of Def. 3 iff it is in the sense of Def. 2.

Full-Wave Invisibility of Active Devices at All Frequencies

773

3.4.1. Weak solutions for the double coating with Neumann boundary conditions. In the following, we consider a double coating (M, N , F, Σ, g). Suppose that k ≥ 0 and
f ∈ L 2 (N , |
g |1/2 d x). Definition 3. We say that
u is a finite energy solution of the boundary value problem with degenerate Neumann boundary conditions at Σ, 2 (∆
u=
f in N \Σ, g + k )
h
u |∂ N =


|
g|

1/2

∂ν
u |Σ+ = 0, |
g|

1/2

(68)

∂ν
u |Σ− = 0,

(69) (70)

if
u is a measurable function in N \Σ such that g |1/2 d x);
u ∈ L 2 (N \Σ, |
1 ∂ j
u ∈ Hloc (N \Σ, d x);  |
g |1/2
g i j ∂i
u∂ j
u d x < ∞;

(71) (72) (73)

N\Σ 2 u=
f in some neighborhood of ∂ N , (∆
g + k )


u |∂ N = h;

and finally,




+ (k 2 −

|
−
g i j ∂i
u ∂jψ f )
uψ g |1/2 d x = 0

(74)

(75)

N\Σ

for all
= ψ




1 (x), ψ
ψ2 (x),

x ∈ N1 , x ∈ N2 ,


2 ∈
1 ∈ C ∞ (N 1 ) vanishing near the exterior boundary ∂ N = ∂ N1 \ Σ and ψ with ψ ∞ C (N 2 ). Invisibility for the double coating with a physical surface at Σ, with respect to the Helmholtz equation at all frequencies, is a consequence of the following analogue of Theorem 2: Theorem 3. Let u = (u 1 , u 2 ) : M\γ → R and
u : N\Σ → R be measurable functions such that u =
u ◦ F. Let f = ( f 1 , f 2 ) : M\γ → R and
f : N\Σ → R be L 2 functions
supported away from Σ and γ such that f = f ◦ F, and
h : ∂ N → R, h : ∂ M1 → R be such that h =
h ◦ F1 . Then the following are equivalent: 1. The function
u , considered as a measurable function on N \ Σ, is a finite energy solution of (68) with Neumann boundary conditions at Σ and inhomogeneity
f in the sense of Definition 3. 2. The function u satisfies (∆g + k 2 )u 1 = f 1

on M1 , u|∂ M1 = h :=
h◦F

(76)

and (∆g + k 2 )u 2 = f 2

on M2 .

(77)

774

A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann

Proof. The proof is identical to that of Theorem 2.   Remark. Let
g be a singular metric on N corresponding to a double coating. The implication of Theorems 2 and 3 is that the solutions
u in N\Σ coincide in the following cases: 1. 2.

We have the metric
g on N , singular at the virtual surface Σ. We have the metric
g on N \Σ and a sound hard physical surface at Σ, in the sense of Definition 3. Similar results can be proven when the metric
g in N corresponds to a single coating.

4. Maxwell’s Equations 4.1. Geometry and definitions. Let us start with a general Riemannian manifold (M, g), possibly with a non-empty boundary, and consider how to define Maxwell’s equations on M. We follow the treatment in [KLS], using, however, slightly different notation. Using the metric g, we define a permittivity and permeability by setting ε jk = µ jk = |g|1/2 g jk , on M. Although defined with respect to local coordinates, ε and µ are in fact invariantly defined, and transform as a product of a (+1)−density and a contravariant symmetric two-tensor. Remark. In R3 with the Euclidean metric g jk = δ jk , we have ε jk = µ jk = δ jk . If we would like to define a generalization of isotropic media on a general Riemannian manifold, it would be as ε jk = α(x)−1 |g|1/2 g jk , µ jk = α(x)|g|1/2 g jk , on M, where α(x) is a positive scalar function. However, in the following we assume for simplicity that α = 1. In the following we consider the electric and magnetic fields, E and H , as differential 1-forms, given in some local coordinates by E = E jdx j,

H = Hjdx j,

and J , the internal current, as a 2-form. Now consider the time harmonic Maxwell’s equations on (M, g) at frequency k. They can be written invariantly as d E = ik ∗g H, d H = −ik ∗g E + J,

(78)

where ∗g : C ∞ (Ω j M) −→ C ∞ (Ω 3− j M) denotes the Hodge-operator on j-forms, 0 ≤ j ≤ 3, given on 1-forms by 1 1/2 jl |g| g E j slpq d x p ∧ d x q 2 1 = ε jl E j slpq d x p ∧ d x q , 2

∗g (E j d x j ) =

(79)

Full-Wave Invisibility of Active Devices at All Frequencies

775

where slpq is the Levi-Civita permutation symbol, slpq = 1 if (l, p, q) even permutation of (1, 2, 3), slpq = −1 if (l, p, q) odd permutation of (1, 2, 3), and zero otherwise. Thus ∗g (E j d x j ) = (ε j3 E j ) d x 1 ∧ d x 2 − (ε j2 E j ) d x 1 ∧ d x 3 + (ε j1 E j ) d x 2 ∧ d x 3 . Next, we want to write these equations in arbitrary coordinates so that they resemble the traditional Maxwell equations. The idea is that we want to have expressions which specialize, in the case of the Euclidean metric on R3 , to expressions involving curl and the matrices ε jk and µ jk . To write equations in such a form, let us introduce, for H = H j d x j , the notation (curl H )l = s lpq

∂ Hq . ∂x p

The exterior derivative d(H j d x j ) =

∂ Hj dxk ∧ dx j ∂xk

may then be written as 1 (80) (curl H )l slpq d x p ∧ d x q . 2 Combining (79) and (80) we see that Maxwell equations (78) can be written as dH =

(curl E)l = ik µ jl H j , (curl H )l = −ik ε jl E j + J l . Below, we denote also (∇ × E) j = (curl E) j , and usually denote the standard volume element of R3 by d V0 (x). There are many boundary conditions that makes the boundary value problem for Maxwell’s equations on a domain well posed. For example: –

Electric boundary condition: ν × E|∂ M = 0,

–

where ν is the Euclidean normal vector of ∂ M. Physically this corresponds to lining the boundary with a perfectly conducting material. Magnetic boundary condition: ν × H |∂ M = 0,

–

where ν is the Euclidean normal vector of ∂ M. In other words, the tangential components of the magnetic field vanish. Soft and hard surface (SHS) boundary condition [HLS, Ki1, Ki2, Li]: ζ · E|∂ M = 0 and ζ · H |∂ M = 0, where ζ = ζ (x) is a tangential vector field on ∂ M, that is, ζ × ν = 0. In other words, the part of the tangential component of the electric field E that is parallel to ζ vanishes, and the same is true for the magnetic field H . This can be physically realized by having a surface with thin parallel gratings [HLS, Ki1, Ki2, Li].

776

A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann

4.2. Definition of solutions of Maxwell equations. Assume that k ∈ R \ {0}. We will define finite energy solutions for Maxwell’s equations in the same way for both the single and double coatings. Let (M, N , F, γ , Σ, g) be either a single or double coating construction, as in §2, denoting as usual
g = F∗ g on N \Σ. On M and N \Σ, we then define permittivity and permeability tensors by setting ε jk = µ jk = |g|1/2 g jk , on M,
ε jk =
µ jk = |
g |1/2
g jk , on N \Σ. Let J be a smooth internal current 2-form on M that is supported away from ∂ M. 4.3. Finite energy solutions for single and double coatings. The definition of finite energy solution is the same for both coatings. On M, the parameters ε and µ are bounded from below and above, so Maxwell’s equations, ∇ × E = ikµ(x)H, ∇ × H = −ikε(x)E + J

in M,

R(ν, E, H )|∂ M = b

(81)

are defined in the sense of distributions in the usual way. Here, ν denotes the Euclidean unit normal vector of ∂ M and R(·, ·, ·) is a boundary value operator corresponding to the boundary conditions of interest, e.g., R(ν, E, H ) = ν × E for the electric boundary condition. If J is smooth, Maxwell’s equations imply that E, H ∈ C ∞ (M). Next, we consider Maxwell’s equations on N . Let J
be a smooth 2-form on N that is supported away from ∂ N ∪ Σ. Definition 4. Let (M, N , F, γ , Σ, g) be either a single or double coating. We say that
H
) is a finite energy solution to Maxwell’s equations on N , ( E,
= ik

, ∇ × H
= −ik

+ J
on N , ∇×E µ(x) H ε(x) E
H
, D
:=

and

are forms in N with if E, εE B :=
µH  2

j E
k d V0 (x) < ∞, =
ε jk E  E L 2 (N ,|
g |1/2 d V0 (x)) N  2

j H
k d V0 (x) < ∞; =
µ jk H  H  L 2 (N ,|
g |1/2 d V (x))

(82)

L 1 (N , d x)-coefficients satisfying

0

(83) (84)

N


H
) is a classical solution of Maxwell’s equations on a neighborhood U ⊂ N of ( E, ∂N:
= ik

, ∇ × H
= −ikε(x) E
+ J
in U, ∇×E µ(x) H
H
)|∂ N =
R(ν, E, b; and finally,

 


− ik

) d V0 (x) = 0, ((∇ ×
h) · E h ·
µ(x) H N


+

− J
)) d V0 (x) = 0 ((∇ ×
e) · H e · (ik
ε(x) E N

for all
e,
h ∈ C0∞ (Ω 1 N ).

Full-Wave Invisibility of Active Devices at All Frequencies

777

Here, C0∞ (Ω 1 N ) denotes smooth 1-forms on N whose supports do not intersect ∂ N , and the inner product “·” denotes the Euclidean inner product.
H
are solutions of (82) in the sense of Def. 4 implies that Remark. The fact that E, they are distributional solutions in the usual sense. Thus they also satisfy the divergence equations,
= 1 ∇· J
, ∇·

= 0, ∇·
εE µH ik

(85)

in the sense of distributions. 5. Full Wave Invisibility for the Double Coating In this section, (M, N , F, γ , Σ, g) denotes a double coating construction. Invisibility for active devices enclosed in the double coating, with respect to Maxwell’s equations at all frequencies, is a consequence of:
and Theorem 4. Let E and H be 1-forms with measurable coefficients on M \γ and E ∗


. H be 1-forms with measurable coefficients on N \Σ such that E = F E, H = F ∗ H
Let J and J be 2-forms with smooth coefficients on M \γ and N \Σ that are supported away from γ and Σ. Then the following are equivalent:
and H
on N form a finite energy solution of Maxwell’s equations 1. The 1-forms E
= ik

, ∇ × H
= −ik

+ J
on N , ∇×E µ(x) H ε(x) E
H
)|∂ N = b. R(ν, E,

(86)

2. The 1-forms E and H on M satisfy Maxwell’s equations ∇ × E = ikµ(x)H, ∇ × H = −ikε(x)E + J R(ν, E, H )|∂ N = b

on M1 ,

∇ × E = ikµ(x)H, ∇ × H = −ikε(x)E + J

on M2 .

and

Proof. First we prove that Maxwell’s equations on M imply Maxwell equations on N . Assume now that the 1-forms E and H are classical solutions of Maxwell’s equations on M = M1 ∪ M2 , ∇ × E = ikµ(x)H, ∇ × H = −ikε(x)E + J R(ν, E, H )|∂ N = b.

on M = M1 ∪ M2 , (87)

Since J vanishes near γ , ellipticity implies that E and H are smooth near γ .
= (F −1 )∗ H, and J
= (F −1 )∗ J. Then
= (F −1 )∗ E, H Define on N \Σ the forms E
satisfies the Maxwell’s equations on N \Σ, E
= ik

, ∇ × H
= −ik

+ J
on N \Σ. ∇×E µ(x) H ε(x) E

(88)

Again, let Σ(t) be the t-neighborhood of Σ with respect to the metric
g and γ (t) the t-neighborhood of γ with respect to g. Let It : ∂γ (t) → M be the identity embedding. Denote by ν be the unit normal vector of ∂Σ(t) and ∂γ (t) in Euclidean metric.

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A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann

Now, writing E = E j (x)d x j on M, we see using the transformation rule for
= (F −1 )∗ E in local coordinates is differential 1-forms that the form E
= E
j (
E x )d
x j = (D F −1 )kj (
x ) E k (F −1 (
x ))d
x j,
x ∈ N \Σ, and, using Ft = F ◦ It : ∂γ (t) → ∂Σ(t), we have −1

j (x)d x j ) = (D Ft−1 )k (
I ∗( E x )) d
x j,
x = F(x). j x ) E k (F (


(89)

Let us now do computations in the Euclidean coordinates. In the Euclidean metric ge , the matrix D Ft−1 satisfies D Ft−1 (T ∂Σ(t),ge )→(T ∂γ (t),ge ) ≤ Ct,

(90)

and since E is smooth near γ we see
|ν × E(y)| R3 ≤ Ct,

y ∈ ∂Σ(t).

Thus using (88) we see that for
h ∈ C0∞ (Ω 1 N ), 
− ik

) d V0 (x) ((∇ ×
h) · E h ·
µ(x) H N 
− ik

) d V0 (x) ((∇ ×
h) · E h ·
µ(x) H = lim t→0 N\Σ(t) 
·
= − lim (ν × E) h d S(x) = 0. t→0 ∂Σ(t)

(91)

Thus, we have shown that
= ik

∇×E µ(x) H

in N

(92)

in the sense of Definition 4. Similarly, we see that
= −ik

+ J
in N ∇×H ε(x) E

(93)

in the same finite energy sense. Next we show that Maxwell’s equations on N implies Maxwell’s equations on M. Let U ⊂ M be a bounded neighborhood of γ and W = F(U\γ ) ∪ Σ be a neighborhood of Σ such that supp ( J
) ∩ W = ∅.
and H
form a finite energy solution of Maxwell’s equations (86) on Assume that E (N , g) in the finite energy sense with a source J
supported away from Σ, implying in particular-that
j E
k ∈ L 1 (W, d x),

j H
k ∈ L 1 (W, d x).
ε jk E µ jk H
H = F∗ H
and J = F ∗ J
on M \γ . We have Define E = F ∗ E, ∇ × E = ikµ(x)H, ∇ × H = −ikε(x)E + J in M \γ and ε jk E j E k ∈ L 1 (U \γ , d V0 (x)), µ jk H j Hk ∈ L 1 (U \γ , d V0 (x)).

Full-Wave Invisibility of Active Devices at All Frequencies

779

As ε and µ on M are bounded from above and below, these imply that ∇ × E ∈ L 2 (U \γ , d V0 (x)), ∇ × H ∈ L 2 (U \γ , d V0 (x)), ∇· (εE) = 0, ∇· (µH ) = 0 in U \γ . Let E e , H e ∈ L 2 (U, d V0 (x)) be measurable extensions of E and H to γ . Then ∇ × E e − ikµ(x)H e = 0 in U \γ , ∇ × E e − ikµ(x)H e ∈ H −1 (U, d V0 (x)), ∇ × H e + ikε(x)E e = 0 in U \γ , ∇ × H e + ikε(x)E e ∈ H −1 (U, d V0 (x)). Since γ is a subset with (Hausdorff) dimension 1 of the 3-dimensional domain U , it has zero capacitance. Thus, the Lipschitz functions on U that vanish on γ are dense in H 1 (U ), see [KKM, Thm. 4.8 and Remark 4.2(4)], or [AF, Thm. 3.28]. Thus there are no non-zero distributions in H −1 (U ) supported on γ . Hence we see that ∇ × E e − ikµ(x)H e = 0, ∇ × H e + ikε(x)E e = 0 in U. This also implies that ∇· (εE e ) = 0, ∇· (µH e ) = 0 in U. These imply that E e and H e are in C ∞ smooth in U . Summarizing, E and H have unique continuous extensions to γ , and the extensions are classical solutions to Maxwell’s equations. 6. Cauchy Data for the Single Coating Must Vanish In this section (M, N , F, γ , Σ, g) denotes a single coating construction. The following gives the counterpart for Maxwell’s equations of the hidden Neumann boundary condition on ∂ M2 that appeared for the Helmholtz equation.
and Theorem 5. Let E and H be 1-forms with measurable coefficients on M \γ and E
be 1-forms with measurable coefficients on N \Σ such that E = F ∗ E,
H = F∗ H
. H Let J and J
be 2-forms with smooth coefficients on M\γ and N \Σ, that are supported away from γ and Σ. Then the following are equivalent:
and H
on N satisfy Maxwell’s equations 1. The 1-forms E
= ik

, ∇ × H
= −ik

+ J
on N , ∇×E µ(x) H ε(x) E
∂N = f ν × E|

(94)

in the sense of Definition 4. 2. The forms E and H satisfy Maxwell’s equations on M, ∇ × E = ikµ(x)H, ∇ × H = −ikε(x)E + J ν × E|∂ M1 = f

on M1 ,

(95)

∇ × E = ikµ(x)H, ∇ × H = −ikε(x)E + J

on M2

(96)

and

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A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann

with Cauchy data ν × E|∂ M2 = be , ν × H |∂ M2 = bh

(97)

that satisfies be = bh = 0. Moreover, if E and H solve (95), (96), and (97) with non-zero be or bh , then the fields

are not solutions of Maxwell equations on N in the sense of Definition 4. E and H Proof. Assume first that the 1-forms E and H are classical solutions of Maxwell’s equations in M. Moreover, assume that both E and H satisfy the homogeneous boundary condition ν × E|∂ M2 = 0, ν × H |∂ M2 = 0,

(98)

that is, for the field in M2 the Cauchy data on ∂ M2 vanishes. (Here, ν again denotes the Euclidean unit normal of these surfaces.)
= (F −1 )∗ E, H
= (F −1 )∗ H, and J
= (F −1 )∗ J . Again, define on N \Σ forms E
satisfies Maxwell’s equations on N \Σ, Then E
= ik

, ∇ × H
= −ik

+ J
in N \Σ. ∇×E µ(x) H ε(x) E

(99)

Again, let Σ(t) be the t-neighborhood of Σ in the
g -metric and γ (t) be the t-neighborhood of γ in the g-metric. Arguing as in (90) and below, we see that
|ν × E(y)| R3 ≤ Ct,

y ∈ ∂Σ(t) ∩ N2 .

(100)

Recall that Σ1 (ε) = N1 ∩ Σ(ε). Then, using (88) we see that for
h ∈ C0∞ (Ω 1 N ), 
− ik

) d V0 (x) ((∇ ×
h) · E h ·
µ(x) H N 
− ik

) d V0 (x) ((∇ ×
h) · E h ·
µ(x) H = lim t→0 (N\Σ1 (t)  
·

·
= − lim (ν × E) h d S(x) − (ν × E) h d S(x) = 0, (101) t→0 ∂Σ1 (t)

∂ M2

where we used (100) and (98). Thus, we have shown that
= ik

∇×E µ(x) H

on N

(102)

in the sense of Definition 4. Similarly, we see that
= −ik

+ J
on N , ∇×H ε(x) E

(103)

also in the sense of Definition 4. Next we show that Maxwell’s equations on N imply Maxwell’s equations on M.
and H
form a finite energy solution of Maxwell’s equations (94) on Assume that E
H = F∗ H
, and J = F ∗ J
. (N , g). Again, define on M \γ forms E = F ∗ E, As before, we see that E and H satisfy Maxwell’s equations on M1 \γ1 and the E and H are in L 2 (M1 , d V0 (x)). Using the removable of singularity arguments as in the

Full-Wave Invisibility of Active Devices at All Frequencies

781

case of double coating, we see that E and H have extensions E e and H e in M1 that are classical solutions of ∇ × E e − ikµ(x)H e = 0 on M1 , ∇ × H e + ikε(x)E e = J on M1 .

(104) (105)


Note that (104) implies that, for the original field E,  lim

t→0 ∂Σ(t)∩N1


·
(ν × E) h d S(x) = lim



t→0 ∂γ (t)∩M1

(ν × E) · h d S(x) = 0,

(106)

where h = F ∗
h. Moreover, Maxwell’s equations hold in the interior of M2 : ∇ × E − ikµ(x)H = 0, ∇ × H + ikε(x)E = J

on M2 .


− ik

= 0 on N Let us start to analyze what the validity of the equation ∇ × E µ(x) H in the sense of Definition 4 implies about the boundary values on ∂ M2 . Using (106), we see that for
h ∈ C0∞ (Ω 1 N ), 


− ik

) d V0 (x) ((∇ ×
h) · E h ·
µ(x) H 
− ik

) d V0 (x) ((∇ ×
h) · E h ·
µ(x) H = lim t→0 (N\Σ1 (t)

  



= − lim (ν × E) · h d S(x) + (ν × E) · h d S(x) t→0 ∂Σ1 (t) ∂ N2  (ν × E) ·
h d S(x). = 0−

0=

(107)

N

∂ N2

(108)


+ ik

= J
holding on This shows ν × E|∂ M2 = 0. Similarly, the equation ∇ × H ε(x) E N in the finite energy sense implies that ν × H |∂ M2 = 0.   Assume that E and H satisfy the time-harmonic Maxwell’s equations on M2 ⊂ R3 such that the Cauchy data (ν × E|∂ M2 , ν × H |∂ M2 ) vanishes. By continuing E and H by zero to R3 \ M2 we obtain solutions of Maxwell’s equation in R3 . Thus J must be a current for which there exist solutions of Maxwell’s equations in R3 both satisfying the Sommerfeld radiation condition and vanishing outside N2 . Such currents are nowhere dense in L 2 (N2 ), as then the fields E and H corresponding to J satisfy the Sommerfeld radiation condition and, using Stokes’ theorem, we see that the source J is orthogonal to all (vector-valued) Green’s functions G e (· , y, k; a) with y ∈ R3 \M 2 and a ∈ R3 . Here, the Green’s function (G e (· , y, k; a), G h (· , y, k; a)) satisfies Maxwell’s equations in R3 with current aδ y and the Sommerfeld radiation condition. We thus conclude that finite energy solutions to Maxwell’s equations on N with the single coating exist only if the Cauchy data (ν × E|∂ M2 , ν × H |∂ M2 ) vanishes on the inner surface of the cloaked region. Thus, finite energy solutions do not exist for generic sources, i.e., internal currents J , in the cloaked region.

782

A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann

7. Cloaking an Infinite Cylindrical Domain We now consider an infinite cylindrical domain, N = B2 (0, 2) × R for simplicity, with the double coating. Here, B2 (0, r ) ⊂ R2 is a Euclidean disc with center 0 and radius r . Numerics for cloaking an infinite cylinder have been presented in [CPSSP]. This may also provide a picture of the cloaking that was physically implemented with a “sliced cylinder” geometry in [SMJCPSS], although precise modelling has not been carried out. With this limitation in mind, the physical interpretation of Theorems 6 and 7 below is that the cloaking would be more effective with the insertion of a liner to implement the SHS boundary conditions which are necessary for the existence of finite energy solutions. Here, we modify the treatment from §2 to the noncompact setting, blowing up a line and trying to obtain an infinitely long, invisible cable. Let M1 = B2 (0, 2) × R, γ1 = {(0, 0)} × R ⊂ M1 , M2 = S 2 × R, γ2 = {N P} × R ⊂ M2 . Let M = M1 ∪ M2 , γ = γ1 ∪ γ2 , N1 = B2 (0, 2) × R\(B 2 (0, 1) × R), N2 = B2 (0, 1) × R, Σ = ∂ B2 (0, 1) × R, and N = B2 (0, 2) × R = N1 ∪ N2 ∪ Σ. Let F = (F1 , F2 ) : M \γ → N \Σ be such that F1 : M1 \γ1 → N1 , F2 : M2 \γ2 → N2 . are diffeomorphisms. Let X : B2 (0, 2) × R\{(0, 0)} × R) → (r, θ, z) be the standard cylindrical coordinates on M1 . We assume that F is stretching only in radial direction, that is, X (F(X −1 (r, θ, z))) = (F1 (r ), θ, z).

(109)

Similarly, on M2 we have variables (r, θ, z), where r = dist(x, S P) and we assume that F has a form analogous to (109) in M2 . For simplicity, let g1 be the Euclidean metric on M1 and g2 the product of standard metric on S 2 and standard metric of R on M2 . Let
g = F∗ g on N \Σ, so that (M, N , F, γ , Σ, g) is a double coating construction in this context. On M and N \Σ we define permittivity and permeability by setting ε jk = µ jk = |g|1/2 g jk , on M1 ∪ M2 ,
ε jk =
µ jk = |
g |1/2
g jk , on N \Σ.
By finite energy solutions of Maxwell’s equations on N we will mean one-forms E
and H satisfying the conditions of Definition 4.1, where we emphasize the assumption
:=

and

are in L 1 (N \Σ, d x), making the integrals at the end of that D εE B :=
µH Definition 4.1 well defined.

Full-Wave Invisibility of Active Devices at All Frequencies

783

To formulate the results, we need to define the restrictions of fields on the lines γ1 ⊂ M1 and γ2 ⊂ M2 . First, assume that the 1-forms E and H on M are classical solutions to Maxwell’s equations on M, ∇ × E = ikµ(x)H, in M = M1 ∪ M2 , ∇ × H = −ikε(x)E + J, in M = M1 ∪ M2 , ν × E|∂ M1 = f,

(110)

where J is supported away from γ = γ1 ∪ γ2 . Note that then E and H are C ∞ near γ , and thus we can define the restrictions of the vertical components of the fields on γ1 ⊂ M 1 , ζ · E|γ1 = b1e , ζ · H |γ1 = b1h ,

(111)

∂ , z := x 3 . where ζ := (0, 0, 1) = ∂z Similarly, we can define b2e and b2h to be the restrictions on γ2 ⊂ M2 ,

ζ · E|γ2 = b2e , ζ · H |γ2 = b2h .

(112)

Note that bej = bej (z) and bhj = bhj (z), j = 1, 2, depend only on z.
and Theorem 6. Let E and H be 1-forms with measurable coefficients on M \γ and E ∗

.
H be 1-forms with measurable coefficients on N \Σ such that E = F E , H = F ∗ H
Let J and J be 2-forms with smooth coefficients on M\γ and N \Σ, that are supported away from γ and Σ, respectively Then the following are equivalent:
and H
satisfy Maxwell’s equations 1. On N , the 1-forms E
= ik

, ∇ × H
= −ik

+ J
in N , ∇×E µ(x) H ε(x) E
ν × E|∂ N = f

(113)


and H
are finite energy solutions. and E 2. On M, the forms E and H are classical solutions to Maxwell’s equations (110) on M, with data b1e = ζ · E|γ1 , b2e = ζ · E|γ2 , b1h = ζ · H |γ1 , b2h = ζ · H |γ2 ,

(114)

that satisfy b1e (z) = b2e (z) and b1h (z) = b2h (z), z ∈ R.

(115)

Moreover, if E and H solve (110) with restrictions (114) that do not satisfy (115), then
and H
are not finite energy solutions of Maxwell equations on N . the fields E Proof. First we show that the equations on M imply that the equations hold on N . Assume that the forms E and H satisfy Maxwell’s equations (110) in M in the classical sense, with traces (114) that satisfy (115). Then E and H are C ∞ smooth near γ .
H
and the 2-form J
on N\Σ by E
= (F −1 )∗ E, H
= (F −1 )∗ H , Define the 1-forms E,
satisfies Maxwell’s equations on N \Σ, and J
= ((F −1 )∗ J. Then E
= ik

, ∇ × H
= −ik

+ J
in N \Σ. ∇×E µ(x) H ε(x) E

(116)

784

A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann


H
, D
=

and

are forms on N with A simple computation shows that E, ε E, B=
µH L 1 (N , d x) coefficients. Again, let Σ(t) be the t-neighborhood of Σ in
g -metric and γ (t) be the t-neighborhood of γ in g-metric. Let It : ∂γ (t) → M be the identity embedding. Denote by ν be the unit normal vector of ∂Σ(t) and ∂γ (t) in Euclidean metric. Now, writing E = E j (x)d x j on M, we see as above using Ft = F ◦ It : ∂γ (t) → ∂Σ(t), we have in local coordinates formula (89). Let us next do computations in the Euclidean coordinates. Using (109), the angular direction η := ∂θ , and vertical direction ζ = ∂z , we see that the matrix D Ft−1 (x) satisfies |η · (D Ft−1 (x)η)|R3 ≤ Ct, x ∈ ∂Σ(t),

|ζ · (D Ft−1 (x)ζ )|R3 = 1, x ∈ ∂Σ(t), ζ · (D Ft−1 (x)η) = 0, x ∈ ∂Σ(t),

η · (D Ft−1 (x)ζ ) = 0, x ∈ ∂Σ(t).
vanish on Σ, and we have This implies that only angular components of E
R3 ≤ Ct, x ∈ ∂Σ(t), |η · E|
∂Σ(t)∩N j =
lim ζ · E| bej , j = 1, 2,

(117)

t→0


|∂Σ(t)∩N j =
lim ζ · H bhj ,

t→0

j = 1, 2,

where, for (x 1 , x 2 , x 3 ) ∈ Σ ⊂ N , we denote
bhj (x 1 , x 2 , x 3 ) = bhj (x 3 ), bej (x 1 , x 2 , x 3 ) = bej (x 3 ),


j = 1, 2.

Thus, using (116) we see that for
h ∈ C0∞ (Ω 1 N ) 
− ik

) d V0 (x) ((∇ ×
h) · E h ·
µ(x) H N 
− ik

) d V0 (x) ((∇ ×
h) · E h ·
µ(x) H = lim t→0 N\Σ(t) 
·
= − lim (ν × E) h d S(x), t→0 ∂Σ(t)  = − (ν × (
b1e −
b2e )ζ ) ·
h d S(x) = 0,

Σ

(118)

where ν is the Euclidean unit normal of ∂ N2 = Σ. This shows that Maxwell’s equations are satisfied on N . Observe that if
b1e =
b2e , there exists a test function
h such that the last integral is nonzero, precluding the existence of a finite energy solution. Similar
= −ik

+ J
. considerations are valid for the equation ∇ × H εE
and H
satisfy on N Maxwell’s equations On the other hand, assume that 1-forms E (113) in the finite energy sense. Then, as E and H are forms with L 2 (M)-valued coefficients that satisfy Maxwell’s equations in M1 \γ1 and M2 \γ2 , we see that they have to satisfy Maxwell’s equations in M1 and M2 , and thus they are C ∞ -smooth forms near
and H
are finite energy solutions on N , the above arguments show that γ1 and γ2 . As E b1e = b2e and b1h = b2h . This finishes the proof of Theorem 6.  

Full-Wave Invisibility of Active Devices at All Frequencies

785

Remark. If E, H , and J on M are solutions of Maxwell’s equations as in Proposition 6 (1) such that conditions (114) are not satisfied, then the proof of Proposition 7.1 shows
= F∗ E, H
= F∗ H , and J
= F∗ J extend to forms with L 1 (N , d x) that the fields E coefficients that satisfy
= ik

new on N , ∇×E B+K
= −ik D
+ J
+ J
new on N ∇×H

(119)


and D
=

are 2-forms with measurable in the sense of distributions. Here,
B=
µH εE

coefficients and K new = se δΣ and Jnew = sh δΣ , where δΣ is a measure supported on Σ and se and sh are smooth 2-forms. Similarly, if E, H , and J on M are solutions of Maxwell’s equations as in Theo
H
, and J
on N rem 6 (2) with non-vanishing Cauchy data (97), we see that that E,

satisfy Eqs. (119) with distributional sources K new and Jnew defined as above. 8. Cloaking a Cylinder with the SHS Boundary Condition Next, we consider N2 as an obstacle, while the domain N1 is equipped with a metric corresponding to the single coating. Motivated by the conditions at Σ in the previous section, we impose the soft-and-hard boundary condition on the boundary of the obstacle. To this end, let us give still one more definition of weak solutions, appropriate for this construction. We consider only solutions on the set N1 ; nevertheless, we continue to denote ∂ N = ∂ N1 \Σ. Definition 5. Let (M1 , N1 , F, γ1 , Σ, g1 ) be a single coating construction. We say that
U and H
are finite energy solutions of Maxwell’s equations on N1 with the 1-forms E the soft-and-hard (SHS) boundary conditions on Σ,
= ik

, ∇ × H
= −ik

+ J
on N1 , ∇×E µ(x) H ε(x) E
Σ = 0, η · H
|Σ = 0, η · E|
ν × E|∂ N = f,

(120) (121)


and

are 2-forms with measurable coefficients
and H
are 1-forms on N1 and
εE µH if E satisfying 
22
j E
k d V0 (x) < ∞,  E =
ε jk E (122) 1/2 L (N1 ,|
g | d V0 ) N1 
2 2
j H
k d V0 (x) < ∞; =
µ jk H (123) H L (N ,|
g |1/2 d V ) 1

0

N1

Maxwell’s equation are valid in the classical sense in a neighborhood U of ∂ N :
= ik

, ∇ × H
= −ikε(x) E
+ J
in U, ∇×E µ(x) H
ν × E|∂ N = f ; and finally,

 


− ik

) d V0 (x) = 0, ((∇ ×
h) · E h ·
µ(x) H N1


+

− J
)) d V0 (x) = 0, ((∇ ×
e) · H e · (ik
ε(x) E N

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A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann

for all
e,
h ∈ C0∞ (Ω 1 N1 ) satisfying η ·
e|Σ = 0, η ·
h|Σ = 0,

(124)

where η = ∂θ is the angular vector field that is tangential to Σ. We have the following invisibility result. In this section (M1 , N1 , F, γ1 , Σ) is a coating configuration corresponding to single coating of a cylindrical obstacle B2 (0, 1) × R.
and Theorem 7. Let E and H be 1-forms with measurable coefficients on M1\γ1 and E ∗ ∗


H be 1-forms with measurable coefficients on N1 such that E = F E, H = F H . Let J and J
be 2-forms with smooth coefficients on M1 \γ1 and N1 \Σ, that are supported away from γ1 and Σ. Then the following are equivalent:
and H
satisfy Maxwell’s equations (120) with SHS boundary 1. On N1 , the 1-forms E conditions (121) in the sense of Definition 5. 2. On M1 , the forms E and H are classical solutions of Maxwell’s equations, ∇ × E = ikµ(x)H, in M1 , ∇ × H = −ikε(x)E + J, in M1 , ν × E|∂ M1 = f.

(125)

Proof. First, assume that the forms E and H satisfy Maxwell’s equations (125) in M1 . Then E satisfies identities (117). Considerations similar to those yielding formula (118) imply that 
− ik

) d V0 (x) ((∇ ×
h) · E h ·
µ(x) H N1 
− ik

) d V0 (x) ((∇ ×
h) · E h ·
µ(x) H = lim t→0 N1\Σ(t) 
·
= − lim (ν × E) h d S(x), t→0 ∂Σ(t)∩N1 
+ (ζ · E)ζ
) ·
= − lim (ν × ((η · E)η h d S(x), =0

t→0 ∂Σ(t)∩N1

(126)

for a test function
h satisfying (124).
shows that 1-forms E
and H
satisfy Maxwell’s equations with Similar analysis for H SHS boundary conditions in the sense of Definition 5.
Next, we show that equations on N1 imply equations on M1 . Assume that 1-forms E
satisfy Maxwell’s equations with SHS boundary conditions, and internal current and H J
, in the sense of Definition 5. Then E and H are classical solutions of Maxwell’s equation in M1 \γ1 . Let U ⊂ M1 be a neighborhood of γ1 and W = F(U \γ1 ) ∪ Σ be a neighborhood of Σ in N1 such that supp ( J
) ∩ W = ∅. Then we have
j E
k ∈ L 1 (W, d V0 (x)),

j H
k ∈ L 1 (W, d V0 (x)).
ε jk E µ jk H
H = F∗ H
and J = F ∗ J
on M1 \ γ1 . Again, we see that E, H , Define E = F ∗ E, and J satisfy Maxwell’s equations on U \γ , and as above we see that E and H have

Full-Wave Invisibility of Active Devices at All Frequencies

787

measurable extensions on γ , E e , H e ∈ L 2 (U, d V0 (x)), such that ∇ × E e − ikµ(x)H e and ∇ × H e + ikε(x)E e are distributions in H −1 (U, d V0 ) supported on γ1 . As before, we obtain ∇ × E e − ikµ(x)H e = 0, ∇ × H e + ikε(x)E e = 0 in U. This shows that E and H are classical solutions of Maxwell’s equations on M1 .

 

Similar analysis can be done in the case when we have a physical surface Σ = S 1 × R dividing R3 into two regions, having the SHS boundary conditions on both sides, and we define the material parameters according to double coating construction, i.e., on both sides of the surface. 9. Appendix: Single and Double Coating for Arbitrary Domains and Metrics The constructions of §2 and the results that follow easily extend to general domains and metrics. Let us assume that Ω ⊂ R3 now is an arbitrary domain with smooth boundary, equipped with an arbitrary smooth Riemannian metric, g = gi j (x). This defines the Laplace operator ∆g with, say Dirichlet boundary condition, cf. Remark 3.6. Choose a point O ∈ Ω to be blown up, and assume that the injectivity radius of (Ω, g) at O is larger than 3a for some a > 0. Let B(O, r ) denote a metric ball of (M, g) with center O and radius r . Introduce Riemannian normal coordinates in B(O, 3a) ⊂ Ω: x = (x 1 , x 2 , x 3 ) → (τ, ω), τ > 0, ω ∈ S2 ⊂ TO Ω, so that x = exp O (τ ω). Let f (τ ) : [0, 3a] → [a, 3a] be a smooth strictly increasing function coinciding with τ/2 + a near τ = 0 and with τ for τ > 2a. Define, in these coordinates, F : B(O, 3a)\{O} → B(O, 3a)\ B(O, a), (τ, ω) → ( f (τ ), ω). We extend F by the identity to Ω \ B(O, 3a) and obtain a diffeomorphism F1 : Ω \{O} → N1 = Ω \ B(O, a). Consider the metric
g = F1∗ g in N1 . Observe that surfaces lying at distance τ from ∂ B(O, a) with respect to the metric
g coincide with surfaces lying at distance f (τ ) − a from ∂ B(O, a) with respect to the metric g. Therefore, the directions normal to these surfaces are the same with respect to the metrics g and
g . In particular, the direction of these normals, in the metric
g , is transversal to ∂ B(O, a). Thus, Eqs. (5) remain valid if we use τ − a instead of r − 1. Similarly, we again have the estimate |
g |1/2 ≤ C1 (τ − a)2 . One may also extend the double coating construction as follows. Let (D, g D ) be a compact Riemannian manifold without boundary, and choose a point N P ∈ D. Using Riemannian normal coordinates centered at N P, introduce, similar to the above, a diffeomorphism F2 : D\{N P} → N2 = D\ B(N P, b), where we assume that 3b is smaller than the injectivity radius of D. Pulling back the metric g D , we get a metric g D on D\B(N P, b) with the same properties near ∂ B(N P, b) as
g has near ∂ B(O, a).

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A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann

Observe that, as we are inside the injectivity radii, ∂ B(O, a) and ∂ B(N P, b) are both diffeomorphic to S2 , with diffeomorphisms given by ex p O (aω) and exp N P (bω). Thus, ∂ B(O, a) and ∂ B(N P, b) are diffeomorphic to each other. Gluing these boundaries, we obtain a smooth manifold N = N1 ∪ N2 ∪ Σ with a Riemannian metric singular on Σ which, as one approaches Σ, satisfies conditions (5). This makes it possible to carry out all of the preceding analysis for the double coating. Note that if D is diffeomorphic to S 3 (as earlier), then N is diffeomorphic to Ω M1 . If however D has a non-trivial topology, N may have topology different from that of Ω. However, due to the full-wave invisibility, one is unable to observe this change of topology from observations made at ∂Ω. Note that this is in contrast to the uniqueness result that holds for C ω Riemannian manifolds [LTU]. Similar generalizations of the single coating construction are possible when ∂ D is diffeomorphic to S2 . Acknowledgement. We would like to thank Bob Kohn for bringing the papers [Le, PSS1] to our attention, and Ismo Lindell for discussions concerning the SHS boundary condition. A.G. was partially supported by US NSF, M.L. by CoE-program 213476 of Academy of Finland, and G.U. was partially supported by US NSF and a Walker Family Endowed Professorship.

References [AF] [AE] [AP] [ALP] [BeK] [C] [CPSSP] [GLU1] [GLU2] [GLU3] [HLS] [KK] [KKL] [Ki1] [Ki2] [KSVW] [KV]

Adams, R., Fournier, J.: Sobolev Spaces. Pure and Applied Mathematics 140, New York-London: Academic Press, 2003 Alu, A., Engheta, N.: Achieving transparency with plasmonic and metamaterial coatings. Phys. Rev. E 72, 016623 (2005) Astala, K., Päivärinta, L.: Calderón’s inverse conductivity problem in the plane. Annals of Math. 163, 265–299 (2006) Astala, K., Lassas, M., Päiväirinta, L.: Calderón’s inverse problem for anisotropic conductivity in the plane. Comm. Partial Diff. Eqs. 30, 207–224 (2005) Belishev, M., Kurylev, Y.: To the reconstruction of a riemannian manifold via its spectral data (B-method). Comm. Part. Diff. Eq. 17, 767–804 (1992) Calderón, A.P.: On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), Rio de Janeiro: Soc. Brasil. Mat., 1980, pp. 65–73 Cummer, S., Popa, B.-I., Schurig, D., Smith, D., Pendry, J.: Full-wave simulations of electromagnetic cloaking structures. Phys. Rev. E 74, 036621 (2006) Greenleaf, A., Lassas, M., Uhlmann, G.: The calderón problem for conormal potentials, I: global uniqueness and reconstruction. Comm. Pure Appl. Math. 56(3), 328–352 (2003) Greenleaf, A., Lassas, M., Uhlmann, G.: Anisotropic conductivities that cannot detected in EIT, Physiological Measurement (special issue on Impedance Tomography), 24, 413–420 (2003) Greenleaf, A., Lassas, M., Uhlmann, G.: On nonuniqueness for calderón’s inverse problem. Math. Res. Let. 10(5-6), 685–693 (2003) Hänninen, I., Lindell, I., Sihvola, A.: Realization of Generalized Soft-and-Hard Boundary. Progress In Electromagnetics Research, PIER 64, 2006, pp. 317–333 Kachalov, A., Kurylev, Y.: Multidimensional inverse problem with incomplete boundary spectral data. Comm. Part. Diff. Eq. 23, 55–95 (1998) Kachalov, A., Kurylev, Y., Lassas, M.: Inverse Boundary Spectral Problems, Chapman and Hall/CRC Monogr. and Surv. in Pure and Appl. Math. 123. Boca Raton, FL: Chapman and Hall/CRC, 2001 Kildal, P.-S.: Definition of artificially soft and hard surfaces for electromagnetic waves. Electron. Let. 24, 168–170 (1988) Kildal, P.-S.: Artificially soft and hard surfaces in electromagnetics. IEEE Transactions on Antennas and Propagation 38, 10, 1537–1544 (1990) Kohn, R., Shen, H., Vogelius, M., Weinstein, M.: In preparation Kohn, R., Vogelius, M.: Identification of an unknown conductivity by means of measurements at the boundary. In: Inverse Problems, SIAM-AMS Proceedings., 14 (1984)

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Kato, T.: Perturbation theory for linear operators. Classics in Mathematics. Berlin: Springer-Verlag, 1995 [KKM] Kilpeläinen, T., Kinnunen, J., Martio, O.: Sobolev spaces with zero boundary values on metric spaces. Potential Anal. 12(3), 233–247 (2000) [Ku] Kurylev, Y.: Multidimensional inverse boundary problems by the BC-method: groups of transformations and uniqueness results. Math. Comput. Modelling 18, 33–46 (1993) [KLS] Kurylev, Y., Lassas, M., Somersalo, E.: Maxwell’s equations with a polarization independent wave velocity: direct and inverse problems. J. Math. Pures Et Appl. 86, 237–270 (2006) [LaU] Lassas, M., Uhlmann, G.: Determining Riemannian manifold from boundary measurements. Ann. Sci. École Norm. Sup. 34(5), 771–787 (2001) [LTU] Lassas, M., Taylor, M., Uhlmann, G.: The dirichlet-to-neumann map for complete Riemannian manifolds with boundary. Comm. Geom. Anal. 11, 207–222 (2003) [LeU] Lee, J., Uhlmann, G.: Determining anisotropic real-analytic conductivities by boundary measurements. Comm. Pure Appl. Math. 42, 1097–1112 (1989) [Le] Leonhardt, U.: Optical Conformal Mapping. Science 312, 1777–1780, 23 June, 2006 [LeP] Leonhardt, U., Philbin, T.: General relativity in electrical engineering. New J. Phys. 8, 247 (2006) [Li] Lindell, I.: Generalized soft-and-hard surface. IEEE Tran. Ant. and Propag. 50, 926–929 (2002) [Ma] Maz‘ja, V.: Sobolev Spaces. Berlin: Springer-Verlag, 1985 [M] Melrose, R.: Geometric scattering theory. Cambridge: Cambridge Univ. Press, 1995 [MBW] Milton, G., Briane, M., Willis, J.: On cloaking for elasticity and physical equations with a transformation invariant form. New J. Phys. 8, 248 (2006) [MN] Milton, G., Nicorovici, N.-A.: On the cloaking effects associated with anomalous localized resonance. Proc. Royal Soc. A 462, 3027–3059 (2006) [N] Nachman, A.: Reconstructions from boundary measurements. Ann. of Math. (2) 128, 531–576 (1988) [N1] Nachman, A.: Global uniqueness for a two-dimensional inverse boundary value problem. Ann. of Math. 143, 71–96 (1996) [PSS1] Pendry, J.B., Schurig, D., Smith, D.R.: Controlling electromagnetic fields. Science 312, 1780–1782 (2006) [PSS2] Pendry, J.B., Schurig, D., Smith, D.R.: Calculation of material properties and ray tracing in transformation media. Opt. Exp. 14, 9794 (2006) [SMJCPSS] Schurig, D., Mock, J., Justice, B., Cummer, S., Pendry, J., Starr, A., Smith, D.: Metamaterial electromagnetic cloak at microwave frequencies. Science Online, 10.1126/science.1133628, Oct. 19, 2006 [Se] Serrin, J.: Local behavior of solutions of quasi-linear equations. Acta Math. 111, 247–302 (1964) [SuU] Sun, Z., Uhlmann, G.: Anisotropic inverse problems in two dimensions. Inverse Problems 19, 1001–1010 (2003) [S] Sylvester, J.: An anisotropic inverse boundary value problem. Comm. Pure Appl. Math. 43, 201–232 (1990) [SyU] Sylvester, J., Uhlmann, G.: A global uniqueness theorem for an inverse boundary value problem. Ann. of Math. 125, 153–169 (1987) [U] Uhlmann, G.: Scattering by a metric. In: Encyclopedia on Scattering, R. Pike and P. Sabatier, eds. Chap. 6.1.5, London-New York: Academic Press, 2002, pp. 1668–1677 [V] Vogelius, M.: Lecture, Workshop on Inverse Problems and Applications, BIRS, August, 2006 Communicated by P. Constantin

Commun. Math. Phys. 275, 791–803 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0315-2

Communications in

Mathematical Physics

The Boundary Control Approach to the Titchmarsh-Weyl m−Function. I. The Response Operator and the A−Amplitude Sergei Avdonin, Victor Mikhaylov, Alexei Rybkin Department of Mathematics and Statistics, University of Alaska Fairbanks, PO Box 756660, Fairbanks, AK 99775, USA. E-mail: [email protected]; [email protected]; [email protected] Received: 6 December 2006 / Accepted: 12 February 2007 Published online: 16 August 2007 – © Springer-Verlag 2007

Dedicated with great pleasure to B.S. Pavlov on the occasion of his 70 th birthday Abstract: We link the Boundary Control Theory and the Titchmarsh-Weyl Theory. This provides a natural interpretation of the A−amplitude due to Simon and yields a new efficient method to evaluate the Titchmarsh-Weyl m−function associated with the Schrödinger operator H = −∂x2 + q(x) on L 2 (0, ∞) with Dirichlet boundary condition at x = 0. 1. Introduction Consider the Schrödinger operator H = −∂x2 + q(x)

(1.1)

on L 2 (R+ ) , R+ := [0, ∞), with a real-valued locally integrable potential q. We assume that (1.1) is limit point case at ∞, that is, for each z ∈ C+ := {z ∈ C : Im z > 0} the equation −u  + q(x)u = zu (1.2) has a unique, up to a multiplicative constant, solution u + which is in L 2 at ∞:
|u + (x, z)|2 d x < ∞, z ∈ C+ . R+

(1.3)

Such solution u + is called a Weyl solution and its existence for a very broad class of real potentials q is the central point of the Titchmarsh-Weyl theory. The (principal or Dirichlet) Titchmarsh-Weyl m-function, m(z), is defined for z ∈ C+ as u  (0, z) . (1.4) m(z) := + u + (0, z)

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S. Avdonin, V. Mikhaylov, A. Rybkin

The function m(z) is analytic in C+ and satisfies the Herglotz property: m : C+ → C+ , so m satisfies a Herglotz representation theorem, 
 1 t dµ(t), − m(z) = c + 1 + t2 R t −z where c = Re m(i) and µ is a positive measure subject to
dµ(t) < ∞, 2 R 1+t 1 dµ(t) = w- lim Im m(t + iε) dt. ε→+0 π

(1.5)

(1.6)

(1.7) (1.8)

It is a fundamental fact of the spectral theory of ordinary differential operators that the measure µ is the spectral measure of the Schrödinger operator (1.1) with a Dirichlet boundary condition at x = 0. Another fundamental fact is the Borg-Marchenko uniqueness theorem [4, 15] stating m 1 = m 2 =⇒ q1 = q2 .

(1.9)

There is no explicit formula realizing (1.9) but there are some Gelfand-Levitan-Marchenko type procedures to recover the potential q by the given m−function. The Titchmarsh-Weyl m−function is a central object of the spectral theory of linear ordinary differential operators but its actual computation is problematic. In fact, (1.6) is suitable if the spectral measure µ of (1.1) with Dirichlet boundary condition at 0 is available, which is not usually the case. Instead, (1.6) is used to find µ by (1.8) but not the other way around. The definition (1.4) is not always practical either since finding m(z) by (1.4) is essentially equivalent to solving  −u + q(x)u = zu (1.10) ∞ u(0, z) = 1, 0 |u(x, z)|2 d x < ∞ for all z ∈ C+ . The analysis of the asymptotic behavior of m(z) for large |z| has received enormous attention and the picture is now quite clear (see, e.g. [7, 21] and the literature cited therein). If x = 0 is a (right) Lebesgue point of q(x) then   √ 1 q(0) m(z) = i z + √ + o √ , z → ∞, ε ≤ arg z ≤ π − ε, ε > 0, (1.11) 2i z z which means that the m−functions for all q coinciding on [0, a] with arbitrarily small a > 0 have the same asymptotic behavior. Due to (1.9), it is therefore m(z) for finite z that is of particular interest, which requires a very accurate control of the solution to (1.10) at x → ∞. In other words, the main issue here is the asymptotic behavior of u(x, z) as x → ∞ for finite z. Typically, such asymptotics are derived by transforming (1.10) to a suitable linear Volterra type integral equation. This can efficiently be done when, e.g., q decays at ∞ fast enough (q ∈ L 1 (R+ ) is sufficient). Equation (1.10) can then be transformed to
∞ √ K (x, s, z)y(s, z)ds, y(x, z) := e−i zx u(x, z), y(x, z) = 1 + x

Boundary Control Approach to the Titchmarsh-Weyl m−Function. I.

where

e−2i

K (x, s, z) :=

√

−1

z(s−x)

√ 2i z

793

q(x),

which can be solved by iteration. Another well-known transformation of (1.2) is the Green-Liouville transformation (see, e.g. [23])   q 2 (x) 1 q  (x) 5 y(x, z) = 0, (1.12) + y  (x, z) + y(x, z) + 4 {z − q(x)}2 16 {z − q(x)}3 where y(x, z) = {z − q(x)}1/4 u(x, z). Equation (1.12) is a crucial ingredient in the WKB-analysis and can be reduced to a linear Volterra integral equation for a wide range of potentials (even growing at infinity) but requires that q be twice differentiable. Even for smooth potentials like q(x) = x −α sin x β , 0 < α ≤ β ≤ 1, the transformation (1.12) is not of much help since q  and q  unboundedly oscillate at ∞. Note that, as it was shown by Buslaev-Matveev [5], the Green-Liouville transformation (1.12) works well for slowly decaying potentials subject to  (l)  q (x) ≤ C x −α−l , α > 0, l = 0, 1, 2. (1.13) One of the authors [20] has recently put forward yet another transformation that allows one to obtain and analyze the asymptotics for the solution to (1.10) for general non-smooth potentials

∞ mild decay at ∞. Namely, if a potential q is such  q with a very j+1 that the sequence j |q(x)| d x is from l p , p = 2n with some n ∈ N then (1.10) j=1

can be transformed to




∞

y(x, z) = 1 +

K n (x, s, z)y(s, z)ds,

(1.14)

x

where y(x, λ) := −1 n (x, λ)u(x, λ) and



√




s+x

n (s, z) := n (0, s, z), n (x, s, z) := exp i zs + x




s

K n (x, s, λ) := (qn n )2 (s, λ) x

n

qm (t, z)dt ,

m=1

−2 n (t, λ)dt.

The functions qm are, in turn, defined by the following recursion formulas:
∞ √ e2i zs q(s + x)ds, q1 (x, z) := − 0
∞ qm+1 (x, z) := 2m (x, s, z)qm2 (s + x, z)ds, m ∈ N.

(1.15)

0

Formulas (1.15) can be viewed as “energy dependent” transformations of the original potential q improving its rate of decay at infinity. For n ≥ 2 these transformations are highly nonlinear and were previously considered by many authors (see, e.g. [10, 11, 13]) in connection with a variety of improvements of asymptotics (1.11). The main feature of the transformations (1.14)–(1.15) is that they yield higher order WKB type asymptotics of the Weyl solution u + (x, z) as x → ∞ at fixed finite z (see [20]).

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Our list of transformations of the original equation (1.2) is of course incomplete and given here to demonstrate how drastically computational complexity of solving (1.10) (and hence the m−function) tends to increase when one relaxes decay conditions. It should be particularly emphasized that in order to get m(z) one has to solve the integral equations for each z separately. In addition, q’s with no decay at ∞ should be considered on an ad hoc basis. In the present note we put forward a different approach to evaluate the TitchmarshWeyl m−function which is based on the Boundary Control Theory. The main idea is that to study the (dynamic) Dirichlet-to-Neumann map u(0, t) → u x (0, t) for the wave equation associated with (1.2): u tt − u x x + q(x)u = 0, x > 0, t > 0, with zero initial conditions. The Dirichlet-to-Neumann map defined this way turns out to be the so-called response operator, an important object of the Boundary Control method in inverse problems [2, 3]. In the frequency domain the latter becomes the operator of multiplication by the Titchmarsh-Weyl m–function1 associated with the operator −∂x2 + q(x) with Dirichlet boundary condition at x = 0. (Pavlov [17] has noticed that the m-function can be interpreted as a one-dimensional (spectral) Dirichlet-to-Neumann map). This approach allows one to apply powerful techniques developed for the wave equation to the study of the Titchmarsh-Weyl m−function. In this paper we concentrate on the direct problem only. That is, given potential q, we evaluate the m−function in terms of the response operator (response function, to be precise) which is exactly Simon’s representation of the m−function via his A−amplitude (see [22] and [9]). Our approach however provides a clear physical interpretation of the A−amplitude and gives a new procedure to compute it. The latter can potentially be used for numerical analysis of the m−function. We emphasize that all the ingredients we use in the present paper are already known in different inverse problems communities ([6, 8, 9, 12, 14] to name just five) but it is the new way to combine them that makes our main contribution to this well developed area. However, we do not utilize here the full power of the Boundary Control approach, which is in inverse methods. We plan to address this important issue in our sequel on this topic. The paper is organized as follows. In Sect. 2 we introduce the main ingredient of our approach, the response operator R, and give its connection with the Titchmarsh-Weyl m−function. We also show that its kernel is closely related to the A−amplitude. In Sect. 3 we derive a linear Volterra type integral equation for a function A(x, y) which diagonal value is the A−amplitude (Theorem 1). Section 4 is devoted to the analysis of the integral equation for the kernel A(x, y) producing an important bound for the A−amplitude (Theorem 2) which answers an open question by Gesztesy-Simon [9]. In the short Sect. 5 we present our algorithm of practical evaluation of the m−function and make some concluding remarks. 2. The Response Operator and the A–Amplitude Let us associate with the Schrödinger equation (1.1) the axillary wave equation  u tt (x, t) − u x x (x, t) + q(x)u(x, t) = 0, x > 0, t > 0, u(x, 0) = u t (x, 0) = 0, u(0, t) = f (t), 1 Usually referred to as the Dirichlet (or principal) Titchmarsh-Weyl m−function

(2.1)

Boundary Control Approach to the Titchmarsh-Weyl m−Function. I.

795

where f is an arbitrary L 2 (R+ ) function referred to as a boundary control. It can be verified by a direct computation that the weak solution u f (x, t) to the initial-boundary value problem (2.1) admits the representation  t f (t − x) + x w(x, s) f (t − s) ds, x ≤ t, u f (x, t) = (2.2) 0, x > t, in terms of the solution w(x, s) to the Goursat problem:  wss (x, s) − wx x (x, s) + q(x)w(x, s)= 0, 0 < x < s, x w(0, s) = 0, w(x, x) = − 21 0 q(t)dt.

(2.3)

We introduce now the response operator R: (R f )(t) = u x (0, t),

(2.4)

so it transforms u(0, t) → u x (0, t). By this reason it can also be called the (dynamic) Dirichlet-to-Neumann map. From (2.2) we easily get the representation
t d (R f )(t) = − f (t) + r (t − s) f (s) ds, (2.5) dt 0 r (·) := wx (0, ·). (2.6) In other words, the response operator is the operator of differentiation plus the convolution. The kernel r of the convolution part of (2.5) is called the response function which plays an important role in the Boundary Control method (see, e.g. [2, 3]). In fact, besides the Boundary Control method some close analogs of the response function have independently been discovered in the half-line short-range scattering [18] and more recently in the connection with inverse spectral problem for the half-line Schrodinger operator [9, 19, 22]. We now demonstrate the connection between the response function r (s) and the (Dirichlet) Titchmarsh-Weyl m-function. An interplay between spectral and time-domain data is widely used in inverse problems, see, e.g., [12] where the equivalence of several types of boundary inverse problems is discussed for smooth coefficients; notice, 1 potentials. however, that we consider the case of not smooth but just L loc ∞ Let f ∈ C0 (0, ∞) and
∞  f (t) e−kt dt f (k) := 0

be its Laplace transform. Function  f (k) is well defined for k ∈ C and, if Re k > 0, | f (k)| ≤ Cα (1 + |k|)−α

(2.7)

for any α > 0. Going in (2.1) and (2.4) over to the Laplace transforms, one has

and respectively.

u (x, k) = −k 2 u (x, k), − u x x (x, k) + q(x)  u (0, k) =  f (k),

(2.8) (2.9)


(R f )(k) =  u x (0, k),

(2.10)

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Estimate (2.7) implies that | u (x, k)| decreases rapidly when |k| → ∞ , Re k ≥  > 0. The values of the function  u (0, k) and its first derivative at the origin,  u x (0, k), are related through the Titchmarsh-Weyl m-function  u x (0, k) = m(−k 2 )  f (k).

(2.11)

Therefore,


(R f )(k) = m(−k 2 )  f (k), (2.12) and thus the spectral and dynamic Dirichlet-to-Neumann maps are in one-to-one correspondence. Taking the Laplace transform of (2.5) we get
(R f )(k) = −k  f (k) +  r (k)  f (k).

(2.13)

In Sect. 4 we show that, under some mild conditions on the potential q, (2.12) and (2.13) imply
∞ 2 m(−k ) = −k + e−kα r (α) dα, (2.14) 0

where the integral is absolutely convergent in a proper domain of k. Representation (2.14) is not new. In the form
∞ 2 m(−k ) = −k − A(α)e−2αk dα

(2.15)

0

(with the absolutely convergent integral) it was proven for q ∈ L 1 (R+ ) and q ∈ L ∞ (R+ ) by Gesztesy-Simon [9] who call the function A in (2.15) the A−amplitude. Clearly, one has A(α) = −2r (2α). (2.16) Remark 1. The fundamental role of representation (2.15) and the A−amplitude was emphasized in [22] and [9]. However no direct interpretation of (2.15) and A is given in [9, 22] . On the other hand, (2.14) says that the Titchmarsh-Weyl m–function is the Laplace transform of the kernel of the response operator R (see (2.5)). Or, equivalently, the matrix (one by one in our case) of the response operator R in the spectral representation of H0 = −∂x2 , u(0) = 0, coincides with the Titchmarsh-Weyl m−function associated with H = H0 + q. The response operator R, in turn, describes the reaction of the system. In particular, for the string the operator R connects the displacement and tension at the endpoint x = 0. For electric circuits it relates the current and voltage (see, e.g. [12, 17] and references therein for additional information about the physical meaning of the Dirichlet-to-Neumann map). In the theory of linear dynamical systems the response operator is the input-output map and the Laplace transform of its kernel is the transfer function of a system. Remark 2. In fact, (2.12) can be regarded as a definition of the Titchmarsh-Weyl m− function which could be effortlessly extended to matrix valued and complex potentials since the Boundary Control method is readily available in these situations [2, 3]. We hope to return to this important point elsewhere. Also, since the Dirichlet-to-Neumann map can be viewed as a 3D analog of the m−function, (2.12 ) could hopefully yield a canonical way to define (operator valued) m−functions for certain partial differential operators. It is worth mentioning that Amrein-Pearson [1] have recently generalized (using quite different methods) the theory of the Weyl-Titchmarsh m−function for second-order ordinary differential operators to partial differential operators of the form − + q(x) acting in three space dimensions.

Boundary Control Approach to the Titchmarsh-Weyl m−Function. I.

797

Despite the clear physical interpretation of the response function r some formulas in Sect. 3 look slightly prettier in terms of the A−amplitude. Since our interest to the topic was originally influenced by [9, 22] we therefore are going to deal with A related to r by (2.16). In Sect. 4 we prove the absolute convergence of the integral in (2.15) (and, therefore, of the integral in (2.14)) for 
n+1   1   ∞ ∞ |q(x)| d x ∈ l q ∈ l L R+ := q : . n

3. An Integral Equation for the A-Amplitude In this section we derive a linear Volterra type integral equation closely related to the A−amplitude. Theorem 1. Let q ∈ L 1loc (R+ ). Then for a. e. α > 0, A(α) = A(α, α), where A(x, y) is the solution to the integral equation 
y 
x A(u, v)du q(x − v)dv; A(x, y) = q(x) − 0

v

(3.1)

x, y > 0.

(3.2)

Proof. We go through a chain of standard transformations of the Goursat problem (2.3). By setting u = s + x, v = s − x and   u−v u+v , , (3.3) V (u, v) = w 2 2 Eq. (2.3) reduces to

  ⎧ u−v ⎪ ⎪ V =0 ⎨ Vuv + 4q 2 V (u, u) = 0 ⎪ ⎪  u/2 ⎩ V (u, 0) = − 21 0 q(x) d x

which can be easily transformed into
u

1 u/2 1 v u 1 − v1 )V(u 1 , v1 ). V (u, v) = − q(x) d x − dv1 du 1 q( 2 v/2 4 0 2 v Doubling the variables in (3.4) yields
v
u
1 u V (2u, 2v) = − q(x) d x − dv1 du 1 q(u 1 − v1 ) V (2u 1 , 2v1 ) . 2 v 0 v Introduce a new function




U (x, y) :=

y

dvq(x − v)V (2x, 2v).

(3.4)

(3.5)

(3.6)

0

It follows from (3.5) that U (x, y) satisfies the integral equation
x
y
x
1 y U (x, y) = − dv q(x − v) du q(u) − dv q(x − v) du U (u, v). 2 0 v 0 v

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The function

A(x, y) = q(x) + 2U (x, y)

(3.7)

then obeys Eq. (3.2). It is left to show (3.1). By (2.16) and (2.6), A(α) = −2r (2α) = −2wx (0, 2α).

(3.8)

But it follows from (3.3) that wx (x, s) = (Vu − Vv ) (s + x, s − x) and hence

wx (0, 2α) = (Vu − Vv ) (2α, 2α).

(3.9)

Differentiating (3.5) with respect to u and v and setting u = v = 2α, we have
 1 2α 1 v1  V(2α, v1 ), dv1 q α − Vu (2α, 2α) = − q(α) − 4 4 0 2
2α  1 1 v1  V (2α, v1 ). Vv (2α, 2α) = q(α) + dv1 q α − 4 4 0 2 Inserting these formulas into (3.9) we get
 1 v1  1 2α V (2α, v1 ). dv1 q α − wx (0, 2α) = − q(α) − 2 2 0 2 Setting in (3.10) v1 = 2v and plugging it then in (3.8), yields
α A(α) = q(α) + 2 dv q(α − v)V (2α, 2v).

(3.10)

(3.11)

0

It is left to notice that by (3.6) the right-hand side of (3.11) is q(α) + 2U (2α, 2α) which by (3.7) is equal to A(α, α) and (3.1) is proven.

 The kernel A(x, y) in Theorem 1 is not related to A(α, x) appearing in [9, 22] where A(α, x) is the A−amplitude corresponding to the m−function associated with the interval (x, ∞). 4. Analysis of Iterations In this section we demonstrate that integral equation (3.2 ) is quite easy to analyze. We need the following technical Lemma 1. Let f (x) be a non-negative function and
x+1 || f || := sup f (s) ds < ∞.

(4.1)

x≥0 x

Then for any a, b ≥ 0 and natural n,
a (a + b + 1)n+1 || f ||. (x + b)n f (x)d x ≤ n+1 0

(4.2)

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Proof. We may assume || f || = 1. Integrating the left-hand side of (4.2 ) by parts yields 
a 
a
a n n (x + b) f (x)d x = − (x + b) d f (s)ds 0 0 x 
a 
a
a n f (x)d x + f (s)ds d(x + b)n . (4.3) =b 0

0

x

Due to the trivial inequality


β α

f (x)d x < β − α + 1,

(4.3) can be estimated above as follows:
a
a (x + b)n f (x)d x < bn (a + 1) + (a − x + 1)d(x + b)n 0

0

= bn (a + 1) + (a + b)n +

(a + b)n+1 bn+1 − bn (a + 1) − n+1 n+1

(a + b)n+1 ≤ (a + b)n + n+1

1 = (a + b)n+1 + (n + 1)(a + b)n n+1 (a + b + 1)n+1 . ≤ n+1 At the last step we used the obvious inequality (x ≥ 0) (x + 1)n+1 ≥ x n+1 + (n + 1)x n . 

The following theorem is the main result of this section. Theorem 2. Let q be subject to
||q|| := sup

x+1

|q(s)| ds < ∞ .

(4.4)

x≥0 x

Then for α ≥ 0, |A(α) − q(α)| ≤

1 2




α

|q(x)| d x

0

 2   √   1 exp 2 2 ||q||α + √ exp (2e||q||α) . 2π (4.5)

Proof. Rewriting (3.2) as A = q − K A,


where

y

(K f )(x, y) := 0


dv q(x − v)

x v

du f (u, v),

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and formally solving it by iteration, we get   A(α) = q(α) + (−1)n An (α), An (α) := K n q (α, α), n≥1

and hence



|A(α) − q(α)| ≤

|An (α)| ≤

n≥1



In (α),

(4.6)

n≥1

where

  (4.7) In (α) := |K |n |q| (α, α) |q| with the agreement that |K |is the integral operator K with in place of q. We now  need a suitable estimate for |K |n |q| (x, y). For n = 1, 
y 
x |q(u)| du |q(x − v)| dv (|K | |q|) (x, y) =


v

0 x

≤




ds |q(s)|

x−y

Similarly,
  2 |K | |q| (x, y) =

y

≤




sup

2 |q(s)| ds

=: Q 2 (x).

0



x v

x

dt |q(t)| ≤

0




0

x

(|K | |q|) (u, v)du |q(x − v)| dv
y 
x  du |q(x − v)| dv (|K | |q|) (u, v)

0≤v≤u≤x

0




y

≤ Q 2 (x)

v

(4.8)

(x − v) |q(x − v)| dv

0




y

≤ Q (x) x 2




y

|q(x − v)| dv = Q (x) x 2

0

|q(v + (x − y))| dv

0

≤ Q 2 (x) x(y + 1) ||q||. Here the supremum in (4.8) was estimated by (4.2). We are now able to make the induction assumption 

 x n−1 (y + n − 1)n−1 |K |n |q| (x, y) ≤ Q 2 (x) ||q||n−1 . (n − 1)! (n − 1)!

Then
  |K |n+1 |q| (x, y) ≤

y




x

v

0




(4.9)

  n  |K | |q| (u, v) du |q(x − v)| dv

 u n−1 (v+n−1)n−1 n−1 ≤ ||q|| du |q(x−v)| dv Q (u) (n−1)! (n−1)! 0 v  
x u n−1 du ≤ sup Q 2 (u) 0≤u≤x 0 (n − 1)!  
y (v + n − 1)n−1 |q(x − v)| dv ||q||n−1 × (n − 1)! 0


y

≤ Q 2 (x)

x

2

x n (y + n)n ||q||n . n! n!

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801

At the last step we estimated the second integral by Lemma 1. One now concludes that (4.9) holds for all natural n. Combining (4.7) and (4.9), we get   α n−1 (α + n − 1)n−1 ||q||n−1 , In (α) = |K |n |q| (α, α) ≤ Q 2 (α) (n − 1)! (n − 1)! and hence for (4.6) we have |A(α) − q(α)| ≤ Q 2 (α)

α n (α + n)n ||q||n . n! n!

(4.10)

n≥0

By the inequality (a + b)n ≤ 2n−1 (a n + bn ), estimate (4.10) continues ⎧ ⎨

⎫ ⎬ n nn α |A(α) − q(α)| ≤ Q 2 (α) ||q||n 2n−1 ||q||n + 2n−1 ⎩ ⎭ n! n! n≥0 n≥1 ⎧ ⎫ ⎨  √2||q||α 2n (2α)n n n ⎬ 1 ||q||n . = Q 2 (α) + ⎩ ⎭ 2 n! n! n! 

αn n!

2

n≥0



(4.11)

n≥1

For the first series on the right hand side of (4.11) one has  √2||q||α 2n n≥0

n!

≤ exp2

     2||q||α = exp 2 2||q||α .

(4.12)

Evaluate the other one. It follows from the Stirling formula that n! ≥

√ √  n n 2π n e

and hence (2α)n n n 1 (2α)n en ||q||n ≤ √ √ ||q||n n! n! n! n 2π n≥1 n≥1 1 1 (2e||q||α)n < √ exp (2e||q||α). = √ n! 2π n≥1 2π It follows now from (4.11)–(4.13) that |A(α) − q(α)| < and (4.5) is proven.



     1 1 2 Q (α) exp 2 2||q||α + √ exp (2e||q||α) 2 2π

(4.13)

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S. Avdonin, V. Mikhaylov, A. Rybkin

Remark 3. The exponential bounds (4.5) on A(α) in Theorem 2 can be easily improved for the cases q ∈ L 1 (0, ∞) and q ∈ L ∞ (0, ∞):
α |q(x)| d x, (4.14) |A(α) − q(α)|
Q 2 (α)eα Q(α) , Q(α) = 0

||q||n α 2n ∞ |A(α) − q(α)|
||q||∞ , ||q||∞ := sup |q(x)| , n!(n + 1)! 0≤x≤∞

(4.15)

n≥1

respectively. Bounds (4.14) and (4.15 ) were found in [9]. In [9] (Sect. 10) GesztesySimon also conjectured that A has exponential bound for potentials obeying (4.4). Theorem 2 gives an affirmative answer to their conjecture. 5. A New Procedure for Evaluating the Titchmarsh-Weyl m−Function In this short section we present an algorithm to evaluate the m−function.  x+1 Algorithm 1. Given real valued potential q subject to ||q|| = sup x |q(s)| ds < ∞ , the m−function can be computed as follows: 1. Solve integral Eq. (3.2) for A(x, y) and evaluate A(α) by (3.1). 2 2. Evaluate m(z) by (2.15). √ The integral in (2.15) is absolutely convergent for z = −k , where Re k > 2 max{ 2 ||q||, e ||q||}. Remark 4. Our algorithm yields an absolutely convergent series representation of the m−function,
∞ 2 n m(−k ) = −k − (−1) An (α)e−2αk dα, A0 := q. (5.1) n≥0

0

Under weaker conditions on q representation (5.1) was obtained in [9] by completely different methods which do not imply the linear Volterra integral equation (3.2). Some other series representations can be found in [10, 11, 13, 20, 21]. It should be pointed out though that those series are quite unwieldy and, in addition, should be computed for each z separately. Our procedure has the advantage that once A(α) is found one only needs to compute its Laplace transform for different k. Remark 5. It can be easily seen that if q(x) ≥ 0 then An (α) ≥ 0 which improves the rate of convergence of A(α) = n≥0 (−1)n An (α) making our algorithm more efficient. Acknowledgements. Research of Sergei Avdonin was partially supported by the National Science Foundation, grant # OPP-0414128. Victor Mikhaylov was partially supported by the University of Alaska Fairbanks Graduate Fellowship.

References 1. Amrein, W.O., Pearson, D.B.: M operators: a generalization of Weyl-Titchmarsh theory. J. Comput. Appl. Math. 171(1–2), 1–26 (2004) 2. Avdonin, S.A., Belishev, M.I.: Boundary control and dynamic inverse problem for nonselfadjoint Sturm–Liouville operator. Control and Cybernetics 25, 429–440 (1996) 3. Avdonin, S.A., Belishev, M.I., Ivanov, S.A.: Matrix inverse problem for the equation u tt −u x x + Q(x)u = 0. Math. USSR Sbornik 7, 287–310 (1992)

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4. Borg, G.: Uniqueness theorems in the spectral theory of y  +(λ−q(x))y = 0. In: Proc. 11th Scandinavian Congress of Mathematicians. Oslo: Johan Grundt Tanums Forlag, 1952, pp. 276–287 5. Buslaev, V.S., Matveev, V.B.: Wave operators for the Schrödinger equation with slowly decreasing potential. Theoret. and Math. Phys. 2(3), 266–274 (1970) 6. Chadan, K., Sabatier, P.C.: Inverse problems in quantum scattering theory, Second edition. Texts and Monographs in Physics. New York: Springer-Verlag, 1989 7. Clark, S., Gesztesy, F.: Weyl-Titchmarsh M-function asymptotics, local uniqueness results, trace formulas, and Borg-type theorems for Dirac operators. Trans. Amer. Math. Soc. 354(9), 3475–3534 (2002) 8. Freiling, G., Yurko, V.: Inverse Sturm-Liouville problems and their applications. Huntington, NY: Nova Science Publishers, Inc., 2001 9. Gesztesy, F., Simon, B.: A new approach to inverse spectral theory. II. General real potentials and the connection to the spectral measure. Ann. of Math. 152(2), 593–643 (2000) 10. Harris, B.J.: An exact method for the calculation of certain Titchmarsh-Weyl m-functions. Proc. Roy. Soc. Edinburgh Sect. A 106(1–2), 137–142 (1987) 11. Hinton, D.B., Klaus, M., Shaw, J.K.: Series representation and asymptotics for Titchmarsh-Weyl m-functions. Differ. Int. Eq. 2(4), 419–429 (1989) 12. Kachalov, A., Kurylev, Y., Lassas, M., Mandache, N.: Equivalence of time-domain inverse problems and boundary spectral problems. Inverse Problems 20, 436–491 (2004) 13. Kaper, H.G., Kwong, M.K.: Asymptotics of the Titchmarsh-Weyl m-coefficient for integrable potentials. II. Differential equations and mathematical physics (Birmingham, Ala., 1986), Lecture Notes in Math. 1285, Berlin: Springer, 1987, pp. 222–229 14. Levitan, B.M., Sargsjan, I.S.: Introduction to spectral theory: selfadjoint ordinary differential operators. Translations of Mathematical Monographs, Vol. 39. Providence, RI: Amer. Math. Soc. 1975 15. Marchenko, V.A.: Certain problems in the theory of second-order differential operators. Doklady Akad. Nauk SSSR 72, 457–460 (1950) (Russian) 16. Matveev, V.B., Skriganov, M.M.: Scattering problem for radial Schrödinger equation with slowly decreasing potential. Teor. Mat. Fiz. 10(2), 238–248 (1972) 17. Pavlov, B.: S-Matrix and Dirichlet-to-Neumann Operators. In: Encyclopedia of Scattering, ed. R. Pike, P. Sabatier, London Newyork-San Diego: Academic Press (Harcourt Science and Tech. Company), 2001, pp. 1678–1688 18. Ramm, A.G.: Recovery of the potential from I − function. C. R. Math. Rep. Acad. Sci. Canada 9(4), 177–182 (1987) 19. Ramm, A., Simon, B.: A new approach to inverse spectral theory. III. Short-range potentials. J. Anal. Math. 80, 319–334 (2000) 20. Rybkin, A.: On a transformation of the Sturm-Liouville equation with slowly decaying potentials and the Titchmarsh-Weyl m−function. In: Spectral methods for operators of mathematical physics, Oper. Theory Adv. Appl. 154, Basel: Birkhäuser, 2004, pp. 185–201 21. Rybkin, A.: Some new and old asymptotic representations of the Jost solution and the Weyl m-function for Schrödinger operators on the line. Bull. London Math. Soc. 34(1), 61–72 (2002) 22. Simon, B.: A new approach to inverse spectral theory. I. Fundamental formalism. Ann. of Math. 150(2), 1029–1057 (1999) 23. Titchmarsh, E.C.: Eigenfunction expansions associated with second-order differential equations. Part I. Second Edition, Oxford: Clarendon Press, 1962 Communicated by B. Simon

Commun. Math. Phys. 275, 805–826 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0316-1

Communications in

Mathematical Physics

On the Spectra of Carbon Nano-Structures Peter Kuchment1 , Olaf Post2 1 Mathematics Department, Texas A&M University, College Station, TX 77843-3368, USA.

E-mail: [email protected]

2 Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25,

12489 Berlin, Germany. E-mail: [email protected] Received: 11 December 2006 / Accepted: 18 January 2007 Published online: 15 August 2007 – © Springer-Verlag 2007

Abstract: An explicit derivation of dispersion relations and spectra for periodic Schrödinger operators on carbon nano-structures (including graphene and all types of single-wall nano-tubes) is provided. 1. Introduction Carbon nano-structures, in particular fullerenes (buckyballs), carbon nano-tubes, and graphene have attracted a lot of attention recently, due to their peculiar properties and existing or expected applications (e.g., [21, 25, 55]). Such structures have in particular been modelled by quantum networks (e.g., [2, 3, 26, 27]), also called quantum graphs, which goes back to quantum graph models in chemistry [51, 54] and physics [1, 5, 11, 48, 45] (see also [7, 32, 33] and references therein). A direct and inverse spectral study of Schrödinger operators on zig-zag carbon nano-tubes was conducted in [26, 27]. In this paper, we take an approach different from [26, 27] to such a study. Namely, we provide a simple explicit derivation of the dispersion relations for Schrödinger operators on the graphene structure, which in turn implies the structure of the spectrum and density of states. This derivation was triggered by the one done in [36] for the photonic crystal case, as well by [2, 3], albeit the presented computation is simpler and more convenient for our purpose than the one in [36]. It reflects the known idea (e.g., [1, 5, 34, 35, 50]) that spectral analysis of quantum graph Hamiltonians (at least on graphs with all edges of equal lengths) splits into two essentially unrelated parts: analysis on a single edge, and then spectral analysis on the combinatorial graph, the former being independent on the graph structure, and the latter independent on the potential. The results are formulated in terms of the monodromy matrix (or rather its trace, also called the Hill discriminant [17]) of the 1D potential on one edge of the graphene lattice. Then this dispersion relation, just by a simple restriction procedure, gives answers for any carbon nano-tube: zig-zag, armchair, or chiral. We would like to emphasize that relations of properties of the discriminant to the properties of the one-dimensional periodic

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potential have been studied for a long time and are understood by now extremely well (e.g., [12, 17–19, 24, 41, 43, 44, 53, 58] and references therein). Thus, one can extract all spectral information that might be needed from our explicit description of dispersion relations that involve the discriminant. The paper is structured as follows: in the next Sect. 2, the main geometries and operators are introduced. Section 3 is devoted to derivation of the dispersion relation and the spectral structure for graphene, the main results provided in Theorem 3.6. The following Sect. 4 deals with nano-tubes. The main results accumulated in Theorem 4.3 provide dispersion relations and all parts of the spectra of the nano-tube operators. The last sections contain additional remarks and results, and acknowledgments. 2. Schrödinger Operators on Carbon Nano-Structures All structures studied in this text can be introduced through the honeycomb graphene structure [21, 25, 55]. So, we start with discussing the latter. 2.1. Graphen. It is assumed that in graphene, the carbon atoms are situated at the vertices of a hexagonal 2D structure G shown in Fig. 1 below. We will assume that all edges of G have length 1. It will be crucial for us to consider the following action of 2 the group Z2 on G: it acts √ by the shifts by √ vectors p1 e1 + p2 e2 , where ( p1 , p2 ) ∈ Z and vectors e1 = (3/2, 3/2), e2 = (0, 3) are shown in Fig. 1. We choose as a fundamental domain (Wigner-Seitz cell) of this action the parallelogram region W shown in the picture. Here two vertices V (W ) = {a, b} are assumed to belong to W , while the vertices b , b and b belong to other shifted copies of the fundamental domain. Three edges f, g, h belong to W . Although the graph G does not have to be directed, it will be convenient for us to assign directions to the edges in W as shown in the picture. We will now equip the graph G with the structure of a quantum graph (quantum network) [7, 28, 29, 33, 34]. This requires introduction of a metric structure and of a differential Hamiltonian. We assume that G is naturally embedded into the Euclidean plane, which induces the arc length metric on G, as well as the identification of each

Fig. 1. The hexagonal lattice G and a fundamental domain W together with its set of vertices V (W ) = {a, b} and set of edges E(W ) = { f, g, h}

On the Spectra of Carbon Nano-Structures

807

edge e in G with the segment [0, 1]. Under this identification, the end points of an edge correspond to the points 0 and 1 (this identification is unique up to a symmetry with respect to the center of the edge, i.e., up to the direction of the edge). We also obtain a measure (that we will call d x), and the ability to differentiate functions along
edges and to integrate functions on G. In particular, the Hilbert space L 2 (G) := e∈E(G) L 2 (e) consisting of all square integrable functions on G can be naturally defined. Here E(G) denotes the set of edges in G. We now describe the graphene Hamiltonian H in L 2 (G) that will be studied in our paper (it has also been considered for some special potentials in [2]). Such operators are used for approximating the band structure of carbon nano-structures, as well of other compounds (e.g., [2, 3, 54], and references therein). Let q0 (x) be an L 2 -function on the segment [0, 1]. We will assume that it is even: q0 (x) = q0 (1 − x).

(2.1)

The evenness assumption is made not just for mathematical convenience. As the proposition below shows, this condition is required if one considers operators invariant with respect to all symmetries of the graphene lattice. One could consider a directed honeycomb graph, and thus avoid the evenness condition (hence losing invariance of the operator), but the authors did not see any compelling physical reason for doing so. Using the fixed identification of the edges e ∈ E(G) with [0, 1], we can pullback the function q0 (x) to a function (potential) q(x) on G. Notice that due to the evenness condition imposed on q0 (x), the definition of the potential q does not depend on the orientations chosen along the edges. It is also easy to see that the following claim holds: Proposition 2.1. The potential q defined as above, is invariant with respect to the full symmetry group of the honeycomb lattice G. Moreover, all invariant potentials from L 2,loc (G) are obtainable by this procedure. We skip the immediate proof of this statement. We can now define our Hamiltonian H . It acts along each edge e as H u(x) = −

d 2 u(x) + q(x)u(x), dx2

(2.2)

where we use the shorthand notation x for the coordinate xe along the edge e. The domain dom H of the operator H consists of the functions u that belong to the Sobolev space H 2 (e) on each edge e in G and satisfy the inequality  u2H 2 (e) < ∞. (2.3) e∈E(G)

They also must satisfy the so-called Neumann vertex conditions (also somewhat misleadingly called Kirchhoff vertex conditions) at vertices. These conditions require continuity of the functions at each vertex v (and thus on all graphs G) and vanishing of the total flux, i.e., u e1 (v) = u e2 (v)  e∈E v (G)

if e1 , e2 ∈ E v (G), u e (v)

=0

(2.4a) (2.4b)

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P. Kuchment, O. Post

at each vertex v ∈ V (G). Here E v (G) := { e ∈ E(G) | v ∈ e } is the set of edges incident to the vertex v, u e is the restriction of a function u to the edge e, and u e (v) denotes the derivative of u e along e in the direction away from the vertex v (outgoing direction). Thus defined operator H is unbounded and self-adjoint in the Hilbert space L 2 (G) [28, 34]. Due to the condition on the potential and Proposition 2.1, the Hamiltonian H is invariant with respect to all symmetries of the hexagonal lattice G, in particular with respect to the Z2 -shifts, which will play a crucial role in our considerations.

2.2. Nano-tubes. We provide here a very brief introduction to carbon nano-tubes. One can find a more detailed discussion and classification of nano-tubes in [21, 55]. We also emphasize that only single-wall nano-tubes are considered. Let p ∈ R2 \{0} be a vector that belongs to the lattice of translation symmetries of the honeycomb structure G. In other words, G + p = G. Let us denote by ι p the equivalence relation that identifies vectors z 1 , z 2 ∈ G such that z 2 − z 1 is an integer multiple of the vector p. A nano-tube T p is the graph obtained as the quotient of G with respect to this equivalence relation: T p := G/ι p .

(2.5)

This graph is naturally isometrically embedded into the cylinder R2 /ι p . If p = p1 e1 + p2 e2 , we will abuse notations denoting T p by T( p1 , p2 ) . For example, T(0,N ) is the so-called zig-zag nano-tube, while T(N ,N ) is the so-called armchair nano-tube. The names come from the shape of the boundary of a fundamental domain (cf. Fig. 2). There are many other types of nano-tubes, besides the zig-zag and armchair ones. They are usually called chiral. A degenerate example is given by the zig-zag nano-tube T(0,1) , which consists of a sequence of loops (“beads”) joined by edges into a 1D-periodic necklace structure (see Fig. 2). One can notice that due to existence of rotational and mirror symmetries of the hexagonal structure G, different vectors p can produce the same nano-tubes T p . For instance, T(m,n) = T(n,m) . Also, zig-zag tubes T(0,N ) , T(N ,0) , and T(N ,−N ) are the same (see [21] and references therein for a more detailed classification of nano-tubes).

Fig. 2. A zig-zag (left) and armchair (right) nano-tube T(0,2) and T(1,1) , respectively. The vectors show the translation vector p. The name-giving fundamental domain of each of the nano-tubes is shaded in dark grey. The dashed lines have to be identified. Below, the (degenerate) zig-zag nano-tube T(0,1) is shown

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The Hamiltonian H p on T = T p is defined exactly as for the graphene lattice G. Alternatively, one can think of H acting on functions on G that are periodic with the period vector p (this definition requires some precision, since such functions do not belong to L 2 (G)). 3. Spectra of Graphene Operators Here we calculate the dispersion relation and thus all parts of the spectrum of the graphene Hamiltonian H (see also [2, 3] for related considerations). One can notice that the density of states is determined by the dispersion relation, and thus when the latter is known, the former can be determined as well [4, 53]. We apply now the standard Floquet-Bloch theory (e.g., [17, 31, 53]) with respect to the Z2 -action that we specified before. This theory also holds in the quantum graph case, (see, e.g., [14, 15, 30, 35, 49] and references therein). This reduces the study of the Hamiltonian H to the study of the family of Bloch Hamiltonians H θ acting in L 2 (W ) for the values of the quasimomentum θ in the Brillouin zone [−π, π ]2 . Here the Bloch Hamiltonian H θ acts the same way H does, but it is applied to a different space of functions. Each function u = {u e } in the domain of H θ must belong to the Sobolev space u e ∈ H 2 (e) on each edge e and satisfy the vertex conditions (2.4), as well as the cyclic conditions (Floquet-Bloch conditions) u(x + p1 e1 + p2 e2 ) = ei p·θ u(x) = ei( p1 θ1 + p2 θ2 ) u(x)

(3.1)

for any vector p = ( p1 , p2 ) ∈ Z2 and any x ∈ G. Due to the conditions (3.1), functions u are uniquely determined by their restrictions to the fundamental domain W . Then conditions (2.4) and (3.1) reduce to ⎧ u f (0) = u g (0) = u h (0) =: A ⎪ ⎪ ⎪ ⎨u  (0) + u  (0) + u  (0) = 0 g f h . (3.2) ⎪ u f (1) = eiθ1 u g (1) = eiθ2 u h (1) =: B ⎪ ⎪ ⎩u  (1) + eiθ1 u  (1) + eiθ1 u  (1) = 0. g f h By standard arguments (see e.g. [34, Theorem 18]), H θ has purely discrete spectrum σ (H θ ) = {λ j (θ )}. The graph of the multiple valued function θ → {λ j (θ )} is known as the dispersion relation, or Bloch variety of the operator H . It is known [17, 24, 31, 53] that the range of this function is the spectrum of H :  σ (H ) = σ (H θ ). (3.3) θ∈[−π,π ]2

It is also well known that the dispersion relation determines not only the spectrum, but the density of states of H as well [4, 53]. So, our goal now is the determination of the spectrum of H θ and thus the dispersion relation of H . In order to determine this spectrum, we have to solve the eigenvalue problem H θ u = λu

(3.4)

for λ ∈ R and a non-trivial function u ∈ L 2 (W ) satisfying the above boundary conditions.

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Let us denote by  D the spectrum of the Dirichlet Hamiltonian H D acting as in (2.2) on (0, 1) with boundary conditions u(0) = u(1) = 0. If λ ∈ /  D , there exist two linearly independent solutions ϕ0 , ϕ1 (depending on λ) of the equation −ϕ  + q0 ϕ = λϕ

(3.5)

on (0, 1), such that ϕ0 (0) = 1, ϕ0 (1) = 0 and ϕ1 (0) = 0, ϕ√ 1 (1) = 1. For example, if D /  if and only if µ := λ ∈ / π Z and q0 = 0 and λ > 0 then we have λ ∈ ϕ0 (t) =

sin µ(1 − t) sin µ

ϕ1 (t) =

and

sin µt . sin µ

(3.6)

We will often use the notation ϕ0,λ , ϕ1,λ to emphasize dependence on the spectral parameter. We will assume that the functions ϕ j are lifted to each of the edges in W , using the identifications of these edges with the segment [0, 1] described before. Abusing notations, we will use the same names ϕ j for the lifted functions. Then, for λ ∈ /  D we can use (3.2) to represent any solution u of (3.4) from the domain of H θ on each edge in W as follows: ⎧ ⎪ ⎨u f = Aϕ0 + Bϕ1 (3.7) u g = Aϕ0 + e−iθ1 Bϕ1 ⎪ ⎩u = Aϕ + e−iθ2 Bϕ . h 0 1 With this choice, the continuity conditions in (3.2) and the eigenvalue equation on each edge are automatically satisfied. The remaining two equations that guarantee zero fluxes at the vertices, lead to the system   3ϕ0 (0)A + (1 + e−iθ1 + e−iθ2 )ϕ1 (0)B = 0, (3.8) (1 + eiθ1 + eiθ2 )ϕ0 (1)A + 3ϕ1 (1)B = 0. Using the symmetry (2.1) of the potential q0 , we obtain ϕ1 (1) = −ϕ0 (0)

and

ϕ0 (1) = −ϕ1 (0).

Thus, the system (3.8) reduces to  −3ϕ1 (1)A + (1 + e−iθ1 + e−iθ2 )ϕ1 (0)B = 0 (1 + eiθ1 + eiθ2 )ϕ1 (0)A − 3ϕ1 (1)B = 0.

(3.9)

(3.10)

The quotient η(λ) :=

 (1) ϕ1,λ ϕ1 (1) =  (0) ϕ1 (0) ϕ1,λ

is well defined, since ϕ1 (0) = 0. Thus, the system (3.10) can be rewritten as  −3η(λ)A + (1 + e−iθ1 + e−iθ2 )B = 0 (1 + eiθ1 + eiθ2 )A − 3η(λ)B = 0.

(3.11)

(3.12)

The determinant of this system is clearly equal to |1 + eiθ1 + eiθ2 |2 − 9η2 (λ) = 3 + 2 cos θ1 +2 cos θ2 + 2 cos (θ1 − θ2 )−9η2 (λ) θ1 θ2 2 2 = 1 + 8 cos θ1 −θ 2 cos 2 cos 2 − 9η (λ). Thus, we have proven the following lemma:

(3.13)

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Lemma 3.1. If λ ∈ /  D , then λ is in the spectrum of the graphene Hamiltonian H if and only if there is θ ∈ [−π, π ]2 such that 9η2 (λ) = |1 + eiθ1 + eiθ2 |2 , or 9η2 (λ) = 1 + 8 cos

θ1 θ2 θ1 − θ2 cos cos . 2 2 2

(3.14)

Remark 3.2. Note that this lemma gives a nice relation between the spectrum of the metric graph Hamiltonian H and the discrete graph Laplacian [9, 10] U := deg1 v w∼v U (w). Indeed, the Bloch Laplacian θ on 2 ({a, b}) ∼ = C2 has the matrix

 1 0 1 + e−iθ1 + e−iθ2 θ ∼ . = 0 3 1 + eiθ1 + eiθ2 Thus, the lemma is just the statement that for λ ∈ /  D , we have λ ∈ σ(H θ ) if and only θ if η(λ) ∈ σ ( ), and hence λ ∈ σ(H ) if and only if η(λ) ∈ σ ( ). This relation between quantum and combinatorial graph operators is well known and has been exploited many times (e.g., [1, 5, 8, 34, 35, 39, 50]). Lemma 3.1, in particular, says that in order to find the spectrum of H , we need to calculate the range of the following function on [−π, π ]2 : θ1 θ2 θ1 − θ2 cos cos . (3.15) F(θ1 , θ2 ) = 1 + 8 cos 2 2 2 Lemma 3.3. The function F has range [0, 3]; its maximum is attained at (0, 0) and minimum at (2π/3, −2π/3) and (−2π/3, 2π/3). The proof of the lemma is straightforward, after noticing that the function F(θ1 , θ2 )2 = 1 + 8 cos

θ1 θ2 θ1 − θ2 cos cos = |1 + eiθ1 + eiθ2 |2 2 2 2

ranges from 0 to 9. Next, we want to interpret the function η(λ) in terms of the original potential q0 (x) on [0, 1]. To this end, let us extend q0 (x) periodically to the whole real axis R and consider the Hill operator H per on R given as in (2.2) with the resulting periodic potential: H per u(x) = −

d 2 u(x) + q0 (x)u(x). dx2

(3.16)

(We maintain the notation q0 (x) for the extended potential.) We will be interested in the well studied spectral problem H per u = λu.

(3.17)

As it is usually done in the theory of periodic Hill operators (e.g., [17, 24, 41, 12, 53]), we consider the so called monodromy matrix M(λ) of H per . It is defined as follows: 



ϕ(0) ϕ(1) = M(λ) , ϕ  (0) ϕ  (1)

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where ϕ is any solution of (3.17). In other words, the monodromy matrix shifts by the period along the solutions of (3.17). The matrix valued function λ → M(λ) is entire. It is well known (see the references above) and easy to see that the monodromy matrix has determinant equal to 1. Its trace plays the major role in the spectral theory of the Hill operator. Namely, let D(λ) = tr M(λ) be the so called discriminant (or Lyapunov function) of the Hill operator H per . The next proposition collects well known results concerning the spectra of Hill operators [17, 24, 41, 12, 53]: Proposition 3.4. per per (i) The spectrum

σ (H  ) of H ispurely absolutely continuous. per  (ii) σ (H ) = λ ∈ R |D(λ)| ≤ 2 . (iii) σ(H per ) consists of the union of closed non-overlapping and non-zero length finite intervals (bands) B2k := [a2k , b2k ], B2k+1 := [b2k+1 , a2k+1 ] such that

a0 < b0 ≤ b1 < a1 ≤ a2 < b2 ≤ · · · and limk→∞ ak = ∞. The (possibly empty) segments (b2k , b2k+1 ) and (a2k+1 , a2k+2 ) are called the spectral gaps. Here {ak } and {bk } are the spectra of the operator with periodic and anti-periodic conditions on [0, 1] correspondingly. D th (iv) Let λD k ∈  be the k Dirichlet eigenvalue, labelled in increasing order. Then D λk belongs to (the closure of) the k th gap1 . When q0 is symmetric, as in our case, th 2 λD k coincides with an edge of the k gap . (v) If λ is inside the k th band Bk , then D  (λ) = 0, and D(λ) is a homeomorphism of the band Bk onto [−2, 2]. Moreover, D(λ) is decaying on (−∞, b0 ) and (a2k , b2k ) and is increasing on (b2k+1 , a2k+1 ). It has a simple extremum in each spectral gap [ak , ak+1 ] and [bk , bk+1 ]. (vi) The dispersion relation for H per is given by D(λ) = 2 cos θ,

(3.18)

where θ is the one-dimensional quasimomentum. There are many other important direct and inverse spectral results concerning the well studied (in particular, due to the inverse scattering transform research) operator H per (see, e.g., [17–19, 24, 44, 41, 43, 53, 58]). We will now see the relation between the function η(λ) that was introduced for the graphene operator H and the discriminant of H per . In order to do so, we introduce another basis of solutions of (3.17), namely cλ and sλ with cλ (0) = 1, cλ (0) = 0 and sλ (0) = 0, sλ (0) = 1 (the notations are chosen to recall the cosine and sine functions in the case of zero potential). Using this basis of the solution space, we obtain M(λ) =



cλ (1) sλ (1) . cλ (1) sλ (1)

1 If the gap closes, this boils down to a single point. 2 The same comment applies here.

(3.19)

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A simple calculation (assuming the symmetry (2.1)) relates the new basis with the one of ϕ0 and ϕ1 (which we now denote ϕ0,λ , ϕ1,λ to emphasize dependence on the spectral parameter): cλ = ϕ0,λ + η(λ)ϕ1,λ

and

sλ =

1

 (0) ϕ1,λ . ϕ1,λ

In particular, cλ (1) = sλ (1) = η(λ). Thus, 1 D(λ). (3.20) 2 √ For example, if q0 = 0, then η(λ) = cos( λ). So far, we have been avoiding points of the Dirichlet spectrum  D of a single edge (i.e, the Dirichlet spectrum of the potential q0 (x) on [0, 1]). We will now deal with exactly these points. η(λ) =

Lemma 3.5. Each point λ ∈  D is an eigenvalue of infinite multiplicity of the graphene Hamiltonian H . The corresponding eigenspace is generated by simple loop states, i.e., by eigenfunctions which live on a single hexagon and vanish at the vertices (see Fig. 3 below). Proof. We need to guarantee first that each λ ∈  D is an eigenvalue. Indeed, an eigenfunction is provided by a simple loop state of the type shown for zero potential in Fig. 3 below. It is constructed as follows. Let ψλ (x) be an eigenfunction of the operator −d 2 /d x 2 + q0 (x) with the eigenvalue λ and Dirichlet boundary conditions on [0, 1]. Then, due to the symmetry (evenness) of the potential, we can assume the eigenfunction to be either even, or odd. For an odd eigenfunction ψλ (x), repeating it on each of the six edges of a hexagon, we clearly get an eigenfunction of the operator H . If it is an even eigenfunction, then repeating it around the hexagon with an alternating sign does the same trick. Thus λ ∈ σpp (H ). It is well known then (e.g., [17]) that the multiplicity of the eigenvalue must be infinite. For completeness, we repeat here this simple argument. Let Mλ ⊂ L 2 (G) be the eigenspace. Consider a vector γ that is a period of the lattice G and the operator Sγ of shift by γ in L 2 (G). Then Sγ acts in Mλ as a unitary operator. If dim Mλ were finite, Sγ would have had an eigenvector f ∈ Mλ ⊂ L 2 (G) with an eigenvalue µ, |µ| = 1. However, such a function f obviously cannot belong to L 2 (G), since it is quasi-periodic in the direction of the vector γ . This proves infinite multiplicity.

Fig. 3. A simple loop state constructed from an odd eigenfunction on [0, 1]

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We note now that due to [30] (see also [35, Thm. 11]), linear combinations of compactly supported eigenfunctions are dense in the whole eigenspace Mλ . Thus, we only need to show that the simple loop states just described generate all compactly supported eigenfunctions in the space Mλ . Suppose that ϕ is a compactly supported eigenfunction of H corresponding to the eigenvalue λ ∈  D . First, note that ϕ vanishes at each vertex. Indeed, if this were not true, due to compactness of support, there would have been an edge such that at its one end v0 (corresponding to x = 0), ϕ(v0 ) = 0, while at the other v1 (corresponding to x = 1), ϕ(v1 ) = 0. Expanding into the basis cλ , sλ , we get ϕ(x) = Acλ (x) + Bsλ (x). In particular, ϕ(0) = A = 0. Then ϕ(1) = Acλ (1) = Asλ (1) = 0, since sλ (1) = 0. This leads to a contradiction. Thus, ϕ vanishes at all vertices. In particular, on each edge it constitutes a Dirichlet eigenfunction for the Hill operator on this edge. This also implies that the support of ϕ, as a graph, cannot have vertices of degree 1 and supp ϕ cannot be a tree. Thus, there must be a loop in the support of ϕ. In particular, the outer boundary of the support must be a loop. Take one boundary edge e0 . There is a hexagon inside the boundary loop which contains this edge. Consider a simple loop state ϕ0 coinciding with the eigenfunction ϕ on e0 and extended to the hexagon the way it was described before. Subtracting ϕ0 from ϕ, we obtain a new eigenfunction  ϕ . The number of hexagons inside the boundary loop of the support of  ϕ is less than inside the support of ϕ. Thus, continuing this procedure, we eventually represent ϕ as a combination of simple loop eigenstates. Figure 4 below illustrates this process. We can now completely describe the spectral structure of the graphene operator H . Theorem 3.6. (i) The singular continuous spectrum σsc (H ) is empty. (ii) The absolutely continuous spectrum σac (H ) has band-gap structure and coincides as a set with the spectrum σ(H per ) of the Hill operator H per with potential q0 periodically extended from [0, 1]. In particular,  

σac (H ) = λ ∈ R  |D(λ)| ≤ 2 , where D(λ) is the discriminant of H per . (iii) The pure point spectrum σpp (H ) coincides with  D and thus, due to the evenness of the potential, belongs to the union of the edges of spectral gaps of σ(H per ) = σac (H ).

Fig. 4. Deleting simple loop states (dark grey) from the support of an eigenfunction (light grey)

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(iv) The dispersion relation consists of the variety θ1 θ2 θ1 − θ2 2 cos cos 1 + 8 cos D(λ) = ± 3 2 2 2

(3.21)

and the collection of flat branches λ ∈  D . (v) Eigenvalues λ ∈  D of the pure point spectrum are of infinite multiplicity and the corresponding eigenspaces are generated by simple loop eigenstates. In particular, σ(H ) has gaps if and only if σ(H per ) has gaps. Proof. The claim (i) about absence of the singular continuous spectrum is a general fact about periodic “elliptic” operators. For instance, the standard proof applied for the case of periodic Schrödinger operators in [53, 56] or [31, Theorem 4.5.9] works in our situation. Alternatively, in [20] one can find this statement proven for a general case of analytically fibered operators, which covers our situation as well. Statement (iv) is a combination of Lemmas 3.1 and 3.5 and formula (3.20). The statement (ii) about absolute continuity of the spectrum outside the points of  D follows from (iv) and the standard Thomas’ analytic continuation argument [31, 53, 56]. We remind the reader that according to this argument, eigenvalues correspond to constant branches of the dispersion relation. It is clear that the dispersion curves (3.21) have no constant branches outside  D . The claim (iii) is just a combination of Lemma 3.5 and of the Thomas’ argument again, which shows that there are no eigenvalues outside  D . Finally, (v) is a combination of (iii) and Lemma 3.5. It is clear that the function F 2 has non-degenerate minima F = 0 at the points θ = ±(2π/3, −2π/3). Thus, the function ±F has conical singularities at these points. Since, according to Proposition 3.4, the discriminant D is monotonic with non-degenerate derivative near each point where D(λ) = 0, one obtains the following conclusion: Corollary 3.7. The dispersion curve of the graphene operator H has conical singularities at all spectral values λ such that D(λ) = 0. These singularities (sometimes described as “linear spectra”) represent one of the most interesting features of graphene structures (cf. Fig. 5). These singularities resemble Dirac spectra for massless fermions and thus lead to unusual physical properties of graphene (e.g., [25]). We see that quantum graph models with arbitrary periodic potentials preserve this feature. 4. Spectra of Nano-Tube Operators We use here the notations concerning nano-tubes that were introduced in Sect. 2.2. Consider a vector p = ( p1 , p2 ) ∈ Z2 that belongs to the lattice of translation symmetries of the graphene structure G, i.e., it shifts the structure by p1 e1 + p2 e2 (see Fig. 1). We will use, as before, the corresponding nano-tube T p = T( p1 , p2 ) and the Hamiltonian H p = H( p1 , p2 ) on T p (see Sect. 2.2). Let B = [−π, π ]2 be the Brillouin zone of graphene. Then, as we discussed in the previous section, the Floquet-Bloch theory provides the direct integral expansion ⊕ H= B

H θ dθ.

(4.1)

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√ Fig. 5. The dispersion relation cos λ = ±F(θ )/3 for the free (i.e., q0 = 0) case. The cones arising from √ D(λ) = 2 cos( λ) = 0 are at the levels λ = (π(2k + 1))2 . They are located inside a band of the corresponding Hill operator. Note that the cones at D(λ) = ±2 (i.e., λ = (π k)2 in the free case) are located at the band edges of the Hill operator

In the case of the nano-tube T p , only the values of quasimomenta θ enter that satisfy the condition p · θ = p1 θ1 + p2 θ2 ∈ 2π Z

(4.2)

since a function on T p lifts to a p-periodic function u on G, i.e., u(x + p1 e1 + p2 e2 ) = u(x) (cf. (3.1)). Thus, let us consider the following subset B p ⊂ B:  

B p := θ ∈ [−π, π ]2  p · θ ∈ 2π Z .

(4.3)

Then, we have the direct integral decomposition ⊕ Hp =

H θ dθ.

(4.4)

Bp

In particular, the spectrum of H p is given by σ(H p ) =



σ(H θ ),

(4.5)

θ∈B p

and the dispersion relation for H p is just the restriction to B p of the dispersion relation for H described in part (iv) of Theorem 3.6.

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This implies for instance that we still have  D ⊂ σpp (H p ) and the rest of the spectrum is determined by the pre-image  1  2   η−1 ± F(B p ) = D −1 ± F(B p ) . 3 3

(4.6)

One should notice that it is conceivable that non-constant branches of the dispersion curves (3.14) might sometimes have constant restrictions to B p , thus providing new eigenvalues for the nano-tube Hamiltonian. This happens if the line (4.2) is a level set of the function F. It is easy to find such lines. Lemma 4.1. The only linear level sets of the function F inside B are the following ones: (i) θ1 = ±π , (ii) θ2 = ±π , (iii) θ1 − θ2 = ±π . On these lines F(θ1 , θ2 ) = 1. The proof of the lemma is immediate from the expression (3.15) for the function F (see also Fig. 6 for the level sets of F, which illustrates this statement). We will see that existence of such lines leads to additional pure point spectrum for some types of nano-tubes. In order to determine the spectra of nano-tubes, we need to know the ranges of the function F (cf. (3.15)) restricted to B p . These are described in the following Lemma 4.2.

Fig. 6. The level curves of F for levels varying from 0 (the two dots at ±(2π/3, −2π/3)) till 3 (the dot at the origin). The level curve associated to F = 1 is dotted, the areas where F < 1 are shaded. The dashed line is the line in B p closest to the minimum point (−2π/3, 2π/3) (in the case p1 − p2 = 3m ± 1)

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(i) The function F restricted to B p , achieves its maximum 3 at (0, 0) ∈ B p for any p. (ii) The location and value of the minimum depends on the vector p. The minimal value α( p) := min F(θ ) θ∈B p

satisfies α( p) ∈ [0, 1]

(4.7)

for any p. (iii) α( p) = 0 if and only if p1 − p2 is divisible by 3. (iv) α( p) = 1 if and only if p = (0, ±1), (±1, 0), (0, ±2), (±2, 0), (1, −1), (−1, 1), (2, −2), or (−2, 2). (All these cases correspond to zig-zag nano-tubes.) (v) In all cases not covered by (iii) and (iv), let p1 − p2 = 3m ± 1. Then α( p) ∈ (0, 1) can be found by minimizing the function F over the line p1 θ1 + p2 θ2 = 2π m. In particular, in the case when p = (0, N ) with N = 3m ± 1 > 2 (m ∈ Z), one has     πm  (4.8) − 1 α (0, N ) = 2 cos N (this formula gives the correct answer α( p) = 0 also when N = 3m). (vi) lim α( p) = 0.

| p|→∞

(4.9)

Proof. (i) The claim about the maximum is straightforward, since the only maximum point (0, 0) of F in B belongs to B p for any p. (ii) The expression (3.15) shows that the set of points θ ∈ B where F = 1, consists of four lines θ j = ±π , as well as two lines θ1 − θ2 = ±π . Since no line p · θ = 0 can miss all these points, we conclude that α( p) ≤ 1 for any p. The inequality α( p) ≥ 0 is obvious, since, as we have already discovered, the expression under the square root in (3.15) has its minimum equal to 0. (iii) As we have already indicated before, the points where F reaches its minimum are (2π/3, −2π/3) and (−2π/3, 2π/3). Thus, for α( p) = 0 to hold, one of the lines p · θ = 2π n must pass through one of these points. Thus, p1 − p2 = ±3n, and the claim is proven. (iv) In order for α( p) to be equal to 1, the lines p · θ ∈ 2π Z should not enter the zones where F < 1 (shaded in Fig. 6). It is clear that when p1 , p2 are of the same sign, this is impossible for the line p · θ = 0, unless one of the coordinates p j is equal to zero. One can assume then, due to symmetries, that p = (0, N ), N > 0. In this case, the line p · θ = N θ2 = 2π enters the shaded region, unless N ≤ 2. If the coordinates p j have opposite signs, then in order for the first “nontrivial” line p · θ = ±2π not to enter the Brillouin zone (and thus in particular the shaded area), one has to satisfy the condition | p1 π − p2 π | ≤ 2π . Due to the signs of p j s being opposite, this means that | p1 | + | p2 | ≤ 2. This restricts the situation to the vectors p = (1, −1) and (−1, 1) only, which do satisfy α( p) = 1. If this line does enter the Brillouin zone, the only case when the shaded area is not entered is when p = (2, −2) or (−2, 2) and thus the line goes along the boundary of the shaded region.

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(v) In order to find α( p), we need to minimize F over the set of lines p · θ = 2π n for all such integers n that the line intersects the Brillouin zone B. This entails first determining the appropriate value of n and then minimizing over the corresponding line. The minima of F are located at the points (−2π/3, 2π/3) and (2π/3, −2π/3) and are the only local minima in the shaded regions shown in Fig. 6. Thus, we need to find the value of n that provides a line closest to a minimum point (see again Fig. 6). Evaluating p · θ at the point (−2π/3, 2π/3), we get 2π( p1 − p2 )/3 = 2π(m ± 1/3). This suggests that the line p · θ = 2π m is the right one. When p = (0, N ) with N = 3m ± 1 > 2, we conclude that we need to minimize F over the line N θ2 = 2π m. Substituting the value θ2 = 2π m/N into the modified expression for F, F(θ1 , θ2 ) =

1 + 4 cos

 θ2 θ2  θ2  cos + cos θ1 − , 2 2 2

(4.10)

leads to a simple minimization with respect to θ1 and thus to the formula (4.8). (vi) This claim is clear, since when | p| → ∞, the lines p · θ ∈ 2π Z form a dense set in B, and thus the minimum of F over B p approaches zero. Let us now concentrate for a moment on the additional pure point spectrum that arises due to the linear level sets described in Lemma 4.1. Let us assume that p = (0, 2N ), N ∈ Z. Then, according to that lemma, the line p · θ = 2N π is a level 1 set of F(θ ). Consider λ such that η(λ) = 1/3 (or η(λ) = −1/3). We will now construct a compactly supported eigenfunction for H(0,2N ) . In order to do so, let us notice that η(λ) = 1/3  (0) = 3ϕ  (1). Let us construct a function ϕ on the boldface structure means that ϕ1,λ + 1,λ Z in Fig. 7 below. It is constructed as follows: on the two “horns” directed toward the vertex a, we define the function to be equal to ϕ1,λ (x). It is similarly defined on the “horns” leading towards b. On the “bridge” between a and b, we define the function  (0) = 3ϕ  (1) implies that as ϕ0,λ + ϕ1,λ . It is easy to conclude that the equality ϕ1,λ 1,λ the function satisfies Neumann conditions at both vertices (and certainly the equation H ϕ = λϕ on the edges of Z ). The graph of the function ϕ+ is visualized in the middle of Fig. 7.

Fig. 7. The extra eigenstates outside the Dirichlet spectrum for the zig-zag nano-tubes with even number of “zig-zags”. On the left, √ the support is shown; on the right, the eigenfunctions ϕ± corresponding to the smallest solution of η(λ) = cos λ = ±1/3 are plotted (in the case of zero potential q0 = 0)

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Analogously, if η(λ) = −1/3, one creates a function ϕ− , changing the value on the bridge to ϕ0,λ − ϕ1,λ (see the right graph in Fig. 7). The functions ϕ± are defined on Z only, but can be extended to the whole structure G as follows: one repeats the functions up and down (to dashed-line hexagons in Fig. 7), alternating the sign. Outside this column of hexagons, we define the functions to be equal to zero. These functions are periodic with respect to the vector e2 with period 2, and thus define compactly supported eigenfunctions for any even zig-zag nano-tube T(0,2N ) . We will call such eigenfunctions the three-leaf eigenfunctions (the name suggested by the right graph in Fig. 7). We are now ready to establish the main result about the spectra of carbon nano-tubes. First of all, let us collect all the notions we need here. As before, p = ( p1 , p2 ) ∈ Z2 is a translation vector that determines the nano-tube T p . The Hamiltonian H p on L 2 (T p ) is defined as before, using the pull-back of a potential q0 (x) on [0, 1] symmetric with respect to the point 1/2. We also denote by D(λ) the Hill discriminant (trace of the monodromy matrix) of the Hill operator H per on R with periodized potential q0 . The subset B p of the Brillouin zone B is defined in (4.3). Finally, the function F(θ ) of the quasimomentum θ is defined in (3.15), and α( p) is described in Lemma 4.2. In order to avoid lengthy formulation, in the theorem and two corollaries below, when dispersion relations are described, the flat branches corresponding to the pure point spectrum are omitted. The pure point spectra (and thus the flat branches) are described in separate statements. Theorem 4.3. (i) The non-constant part of the dispersion relation for H p is provided by 2 D(λ) = ± F(θ ), 3

θ ∈ Bp.

(ii) The singular continuous spectrum σsc (H p ) is empty. (iii) The absolutely continuous spectrum is given by   2  2 σac (H p ) = D −1 −2, − α( p) ∪ α( p), 2 3 3

(4.11)

(4.12)

and    2   2  D −1 −2, − ∪ , 2 ⊆ σac (H p ) ⊆ σ(H per ) = D −1 [−2, 2] . (4.13) 3 3 (iv) σac (H p ) = σ(H per ) if and only if p1 − p2 is divisible by 3.   (v) σac (H p ) = D −1 [−2, − 23 ]∪[ 23 , 2] if and only if T p is either a (0, 1)-, or a (0, 2)zig-zag nano-tube (or equivalent, e.g. T(1,−1) ). (vi) Unless T p is a zig-zag nano-tube with an even number of zig-zags (i.e., T(0,2N ) ), one has σpp (H p ) =  D .

(4.14)

This spectrum consists of one edge of each spectral gap (including the closed ones) of σ(H per ). All these eigenvalues are of infinite multiplicity and the corresponding eigenspaces are spanned by simple loop eigenfunctions (supported on a single hexagon) and tube loop eigenfunctions (supported on a loop around the tube).

On the Spectra of Carbon Nano-Structures

821

(vii) If T p is a zig-zag nano-tube with an even number of zig-zags (i.e., T(0,2N ) ), one has

where

σpp (H p ) =  D ∪ ,

(4.15)

 2  = D −1 ± . 3

(4.16)

The eigenvalues from  are of infinite multiplicity, are embedded into σac (H p ), and are located two per each band of σ(H per ). The corresponding eigenspaces are generated by the compactly supported three-leaf functions. The eigenvalues from  D are of infinite multiplicity and the corresponding eigenspaces are spanned by simple loop eigenfunctions and tube loop eigenfunctions. (viii) If p1 − p2 is divisible by 3, the ac-spectrum of H p has exactly the same gaps as   σ(H per ). Otherwise, there is an additional gap D −1 (− 23 α( p), 23 α( p)) inside each band of the spectrum of H per . Proof. The first statement coincides with (4.6). The claim (ii) is proven exactly as the corresponding statement in Theorem 3.6. Statements (iii) through (v) follow from (i) and Lemma 4.2. The statement (vi) is almost completely proven, except the description of the eigenfunctions. The proof of this description works exactly like in Theorem 3.6, except that the procedure of eliminating hexagons does not have to end with an empty set. One can end up with a loop of edges around the tube, which thus does not encircle any hexagons. This would provide a loop eigenfunction that runs around the tube, rather than around a hexagon. The similar claim in (vii) concerning the eigenvalues from  D is proven exactly the same way. What remains to be proven in (vii), is the structure of the eigenfunctions corresponding to λ ∈ . It is proven similarly to the structure of eigenfunctions corresponding to  D . Indeed, again according to [35], the eigenspace is spanned by compactly supported eigenfunctions. Consider the outer boundary of the support and start eliminating hexagons inside as follows. There must be a vertex (like the ends of horns in Fig. 7) that borders zero values. Then, on the corresponding horn the function must be proportional to ϕ1,λ . Let us now extend it to a three-leaf eigenfunction and subtract from the original one. Continuing this process, we eventually eliminate all hexagons. Notice that in this case we cannot end up with a loop around the tube, since this would force all the vertex values to be equal to zero, which is impossible, when λ does not belong to  D . Thus, only the three-leaf states enter the eigenfunction. The statement (viii) follows from the previous ones. We will specify this result for the cases of zig-zag ( p = (0, N )) and armchair ( p = (N , N )) nano-tubes. The zig-zag case was also considered in [26]. Corollary 4.4. Let T(0,N ) be a zig-zag nano-tube with N zig-zags. (i) The non-constant part of the dispersion relation for H(0,N ) is given by  πn 2 πn  π n  (cos , D(λ) = ± 1 + 4 cos + cos θ1 − 3 N N N where 0 ≤ n < N .

(4.17)

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(ii) The singular continuous spectrum is empty. (iii) The absolutely continuous spectrum is given by   

 2 2 −1 α, 2 , −2, − α ∪ σac (H(0,N ) ) = D 3 3

(4.18)

where α = α((0, N )) ∈ [0, 1] is defined in (4.8). In particular, α = 0 (i.e., σac (H(0,N ) ) = σ(H per )) if and only if N is divisible by 3. Furthermore, α = 1 if and only if N = 1 or N = 2. (iv) If N is odd, then the pure point spectrum is given by σpp (H(0,N ) ) =  D . The corresponding eigenspaces are infinite-dimensional and generated by simple loop eigenfunctions (supported on a single hexagon) and tube loop eigenfunctions (supported on a loop around the tube). If N is even, then σpp (H(0,N ) ) =  D ∪ , where  is defined in (4.16). In particular, if N = 2 then the embedded eigenvalues from  are located at the band edges of σac (H p ). If N > 2 is even, this new point spectrum is located inside the bands. The eigenspaces corresponding to  are infinite-dimensional and generated by the compactly supported three-leaf functions. (v) σ(H(0,N ) ) has additional gaps (other than the gaps of σ(H per )) if and only if N is not divisible by 3. Remark 4.5. In order to avoid confusion, we need to specify what a simple loop eigenstate is for the case of the necklace tube T(0,1) . In this case, the image of a hexagon in the tube is a “dumbbell” consisting of two beads of the necklace connected with a link (see Fig. 8 below). A simple loop eigenfunction in this case can concentrate either on a single bead, or on the whole dumbbell. Corollary 4.6. Let T(N ,N ) be an armchair nano-tube. (i) The non-constant part of the dispersion relation for H(N ,N ) is given by θ  2 πn  π n   θ1  1 D(λ) = ± cos cos − , 1 + 8 cos θ1 − 3 N 2 2 N where 0 ≤ n < N . (ii) The singular continuous spectrum is empty. (iii) The absolutely continuous spectrum is given by   σac (H(N ,N ) ) = D −1 [−2, 2] = σ(H per ).

(4.19)

(4.20)

(iv) The pure point spectrum is given by σpp (H(N ,N ) ) =  D and is located at an edge of each gap in σac (H(N ,N ) ). The eigenvalues are of infinite multiplicity and the eigenspaces are generated by simple loop eigenfunctions (either on a single hexagon or a loop around the tube). (v) σ(H(N ,N ) ) has exactly the same gaps as σ(H per ).

Fig. 8. A dumbbell image of a hexagon

On the Spectra of Carbon Nano-Structures

823

5. Final Remarks • The zig-zag nano-tube case has been thoroughly studied by a different method in [26, 27]. The methods employed in this paper seem to be significantly simpler than the ones in [26, 27] and also apply to all 2D carbon nano-structures such as graphene and any single-wall nano-tube. After the paper was accepted for publication, the authors received the preprint [6], where the case of armchair nano-tubes is considered by methods analogous to the ones of [26, 27]. • Other spectral properties of the operators H and H p , e.g. asymptotics of gaps lengths or properties of (and formulas for) the density of states can be easily derived from the explicit dispersion relations that we obtained and the well studied properties of the Hill discriminant. As an example, we provide a theorem below that describes the smoothness of the potential in terms of the gap decay. In order to do so, we call the gaps arising as D −1 ([− 23 α( p), 23 α( p)]) the odd gaps G 2k−1 , k = 1, 2, ... and the gaps of the Hill operator the even gaps G 2k , k = 1, 2, .... Notice that we count the gaps even when they close (e.g., all odd gaps close for graphene and for nano-tubes with integer ( p1 − p2 )/3). Let also γk be the lengths of the gap G k . In the theorem below, the operator is either the graphene operator H , or the nano-tube operator H p . Theorem 5.1. (i) The periodized 1D potential q0 is infinitely differentiable if and only if γ2k decays faster than any power of k when k → ∞. (ii) The periodized 1D potential q0 is analytic if and only if γ2k decays exponentially with k. Since the even gaps are exactly the spectral gaps for the Hill operator with the periodized q0 , this is an immediate corollary of the results of this text and known results of the same nature for the Hill operator [12, 22, 23, 40, 57]. Statements similar to this theorem can be derived as easily for other functional classes of potentials, using the corresponding results for the Hill operator in [12]. • Albeit for graphene (as well as for the nano-tubes with p1 − p2 divisible by 3) the absolutely continuous spectrum coincides with the one of the periodic Hill operator as a set, the structure of the spectrum is different, due to the appearance of the conical singularities inside of each band of the Hill spectrum. Such singularities can also appear when the even gaps close, but this situation is non-generic with respect to the potential q0 . However, as we discussed above, closing the odd gaps and thus the appearance of conical singularities there is mandatory for any potential in the graphene case, as well for nano-tubes T p with p1 − p2 divisible by 3. • As we have indicated in the beginning, quantum graphs (quantum networks) have been used to model spectra of molecules at least since [51, 54]. However, the validity of such models is still under investigation, see e.g., [13, 16, 33, 46, 47, 52] and references therein. • The graphene operator H provides also an interesting example in terms of Liouville type theorems. As it was established in [38] (see also [37] for related considerations), the Liouville theorem for periodic operators holds if and only if the Fermi surface consists of finitely many points. Albeit this normally occurs at the spectral edges only, it was indicated in [37, 38] that in principle this can happen inside the spectrum. The graphene operator provides just such an example. Namely, a direct corollary of our results and the ones of [38] is the following

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Theorem 5.2. Let H be the graphene operator. Suppose D(λ) = 0 and n > 0. Then the space of solutions u of the equation H u − λu = 0 such that |u(x)| ≤ Cu (1 + |x|)n is finite dimensional. Acknowledgement. The authors express their gratitude to E. Korotyaev, K. Pankrashkin and V. Pokrovsky for information and comments. In particular, it was V. Pokrovsky who attracted our attention to the importance of conical singularities. The authors are also grateful to the reviewer for useful remarks. This research of both authors was partly sponsored by the NSF through the NSF Grant DMS-0406022. The authors thank the NSF for this support. The second author was partly supported by the DFG through the Grant Po 1034/1-1. Part of this work was done while O. P. visited Texas A&M University. The second author thanks the DFG for support and Texas A&M University for hospitality.

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22. Hochstadt, H.: Estimates on the stability intervals for the Hill’s equation. Proc. AMS 14, 930–932 (1963) 23. Hochstadt, H.: On the determination of a Hill’s equation from its spectrum. Arch. Rat. Mech. Anal. 19, 353–362 (1968) 24. Iakubovich, V.A., Starzhinski, V.M.: Linear Differential Equations with Periodic Coefficients. NY: Wiley, 1975 25. Katsnelson, M.I.: Graphene: carbon in two dimensions. Materials Today 10(1–2), 20–27 (2007) 26. Korotyaev, E., Lobanov, I.: Schrödinger operators on zigzag graphs. http://arxiv.org/list/math.SP/ 0604006, 2006 27. Korotyaev, E., Lobanov, I.: Zigzag periodic nanotube in magnetic field. http://arxiv.org/list/math.SP/ 0604007, 2006 28. Kostrykin, V., Schrader, R.: Kirchhoff’s rule for quantum wires. J. Phys. A 32, 595–630 (1999) 29. Kottos, T., Smilansky, U.: Quantum chaos on graphs. Phys. Rev. Lett. 79, 4794–4797 (1997) 30. Kuchment, P.: To the Floquet theory of periodic difference equations. In: Geometrical and Algebraical Aspects in Several Complex Variables, Cetraro (Italy), June 1989, Carouge: EditEl, 1991, pp 203–209 31. Kuchment, P.: Floquet Theory for Partial Differential Equations. Basel: Birkhäuser Verlag, 1993 32. Kuchment, P. (ed).: Quantum graphs and their applications. Special issue, Waves in Random Media, 14(1), S107–S128 (2004) 33. Kuchment, P.: Graph models of wave propagation in thin structures. Waves in Random Media 12(4), R1–R24 (2002) 34. Kuchment, P.: Quantum graphs: I. Some basic structures. Waves Random Media 14, S107–S128 (2004) 35. Kuchment, P.: Quantum graphs. II. Some spectral properties of quantum and combinatorial graphs. J. Phys. A 38(22), 4887–4900 (2005) 36. Kuchment, P., Kunyansky, L.: Spectral Properties of High Contrast Band-Gap Materials and Operators on Graphs. Exp. Math. 8(1), 1–28 (1999) 37. Kuchment, P., Pinchover, Y.: Integral representations and Liouville theorems for solutions of periodic elliptic equations. J. Funct. Anal. 181, 402–446 (2001) 38. Kuchment, P., Pinchover, Y.: Liouville theorems and spectral edge behavior on abelian coverings of compact manifolds. http://arxiv.org/list/math-ph/0503010, 2005 to appear in Trans. Amer. Math. Soc. 39. Kuchment, P., Vainberg, B.: On the structure of eigenfunctions corresponding to embedded eigenvalues of locally perturbed periodic graph operators. Commun. Math. Phys. 268, 673–686 (2006) 40. Lazutkin, V.F., Pankratova, T.F.: Asymptotics of the width of gaps in the spectrum of the Sturm-Liouville operators with periodic potential. Soviet Math. Dokl. 15, 649–653 (1974) 41. Magnus, W., Winkler, S.: Hill’s Equation. NY: Wiley, 1966 42. Marchenko, V.A., Ostrovskii, I.V.: A characterization of the spectrum of Hill’s operator. Matem. Sborn. 97, 540–606 (1975); English transl. in Math. USSR-Sb. 26, 493–554 (1975) 43. Marchenko, V.A., Ostrovskii, I.V.: Approximation of periodic potentials by finite zone potentials. (Russian) Vestnik Kharkov. Gos. Univ. No. 205, 4–40, 139 (1980) 44. McKean, H.P., Trubowitz, E.: Hill’s surfaces and their theta functions. Bull. Amer. Math. Soc. 84(6), 1042–1085 (1978) 45. Mills, R.G.J., Montroll, E.W.: Quantum theory on a network. II. A solvable model which may have several bound states per node point. J. Math. Phys. 11(8), 2525–2538 (1970) 46. Molchanov, S., Vainberg, B.: Transition from a network of thin fibers to the quantum graph: an explicitly solvable model. Cont. Math. 415, Providence, RI: Amer. Math. Soc., 2006, pp 227–240 47. Molchanov, S., Vainberg, B.: Scattering solutions in a network of thin fibers: small diameter asymptotics. http://arixiv.org/list/math-ph/0609021, 2006 48. Montroll, E.: Quantum theory on a network. I. A solvable model whose wavefunctions are elementary functions. J. Math. Phys. 11(2), 635–648 (1970) 49. Oleinik, V.L., Pavlov, B.S., Sibirev, N.V.: Analysis of the dispersion equation for the Schrödinger operator on periodic metric graphs. Waves in Random Media 14, 157–183 (2004) 50. Pankrashkin, K.: Spectra of Schrödinger operators on equilateral quantum graphs. Lett. Math. Phys. 77, 139–154 (2006) 51. Pauling, L.: The diamagnetic anisotropy of aromatic molecules. J. Chem. Phys. 4, 673–677 (1936) 52. Post, O.: Branched quantum wave guides with Dirichlet boundary conditions: the decoupling case. J. Phys. A 38(22), 4917–4931 (2005) 53. Reed, M., Simon, B.: Methods of modern mathematical physics IV: Analysis of operators. New York: Academic Press, 1978 54. Ruedenberg, K., Scherr, C.W.: Free-electron network model for conjugated systems. I. Theory. J. Chem. Phys., 21(9), 1565–1581 (1953) 55. Saito, R., Dresselhaus, G., Dresselhaus, M.S.: Physical Properties of Carbon Nanotubes. London: Imperial College Press, 1998 56. Thomas, L.E.: Time dependent approach to scattering from impurities in a crystal. Comm. Math. Phys. 33, 335–343 (1973)

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57. Trubowitz, E.: The inverse problem for periodic potentials, Comm. Pure and Appl. Math. 30, 321–342 (1977) 58. Zakharov, V.E., Manakov, S.V., Novikov, S.P., Pitaevskii, L.P.: Theory of Solitons: The Inverse Scattering Method. London: Plenum, 1984 Communicated by B. Simon

Commun. Math. Phys. 275, 827–838 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0318-z

Communications in

Mathematical Physics

Spectral Estimates for Two-Dimensional Schrödinger Operators with Application to Quantum Layers Hynek Kovaˇrík, Semjon Vugalter, Timo Weidl Institute of Analysis, Dynamics and Modeling, Universität Stuttgart, PF-80 11 40, D-70569 Stuttgart, Germany. E-mail: [email protected] Received: 21 December 2006 / Accepted: 11 January 2007 Published online: 16 August 2007 – © Springer-Verlag 2007

Abstract: A logarithmic type Lieb-Thirring inequality for two-dimensional Schrödinger operators is established. The result is applied to prove spectral estimates on trapped modes in quantum layers.

1. Introduction It is well known that the sum of the moments of negative eigenvalues −λ j of a 2 one-dimensional Schrödinger operator − ddx 2 − V can be estimated by


γ

λ j ≤ L γ ,1

j



1

R

V+ (x)γ + 2 d x, γ ≥

1 , 2

(1)

where L γ ,1 is a constant independent of V , see [10, 16]. For γ = 21 this bound has the correct weak coupling behavior, see [13], and it also shows the correct Weyl-type asymptotics in the semi-classical limit. Moreover, (1) fails to hold whenever γ < 21 . The case γ = 21 therefore represents a certain borderline inequality in dimension one. The situation is much less satisfactory in dimension two. The corresponding two-dimensional Lieb-Thirring bound
j

γ λj

= tr (− −

γ V )−

 ≤ L γ ,2

R2

V+ (x)γ +1 d x

(2)

holds for all γ > 0, [10]. Dimensional analysis shows that here the borderline should be γ = 0. However, (2)  fails for γ = 0, because − − V has at least one negative eigenvalue whenever V ≥ 0, see [13]. In addition, it was shown in [13] that if V

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decays fast enough, the operator − − αV has for small α only one eigenvalue which goes to zero exponentially fast: λ1 ∼ e−4π(α



V )−1

, α → 0.

(3)

It follows from (3) that the optimal behavior for α → 0 cannot be reached in the power-like scale (2), no matter how small γ is, since the l.h.s. decays faster than any power of α. This means that in order to obtain a Lieb-Thirring type inequality with the optimal behavior in the weak coupling limit, one should introduce a different scale on the l.h.s. of (2). In the present paper we want to find a two-dimensional analog of the one-dimensional borderline inequality, which corresponds to γ = 21 in (1). In other words, we want to establish an inequality with the r.h.s. proportional to V and with the correct order of asymptotics in the weak and strong coupling regime. Obviously, we have to replace the power function on the l.h.s. of (2) by a new function F(λ), which will approximate identity as close as possible. On the other hand, since − − V has always at least one eigenvalue, it is necessary that F(0) = 0. Moreover, Eq. (3) shows that F should grow from zero faster than any power of λ, namely as | ln λ|−1 . This leads us to define the family of functions Fs : (0, ∞) → (0, 1] by ⎧ 0 < t ≤ e−1 s −2 , ⎨ | ln ts 2 |−1 ∀s > 0 Fs (t) := (4) ⎩ 1 t > e−1 s −2 . Notice that each Fs is non-decreasing and continuous and that Fs (t) → 1 point-wise as s → ∞. Hence our  goal is to establish an appropriate estimate on the regularized counting functionn j Fs (λ j ) for large values of the parameter s.  Our main results are formulated in the next section. It turns out that j Fs (λ j ) can be estimated by a sum of two integrals, one of which includes a local logarithmic weight, see Theorem 1. The inequality (8) established in Theorem 1 has the correct behavior for weak as well as for strong potentials, see Remark 1. We also show that the logarithmic weight in (8) cannot be removed, see Remark 2. Moreover, in Corollary 1 we obtain individual estimates on eigenvalues of Schrödinger operators with slowly decaying potentials. The proof of the main result, including two auxiliary lemmata, is then given in Sect. 3. In Remark 4 we give some numerical estimates on the constants in the inequality (9). In the closing Sect. 4 we apply Theorem 1 to analyze discrete spectrum of a Schrödinger operator corresponding to quantum layers. The result established in Sect. 4 may be regarded as a two-dimensional analog of Lieb-Thirring inequalities on trapped modes in quantum waveguides obtained in [6]. 2. Main Results For a given V we define the Schrödinger operator − − V in L 2 (R2 ) as the Friedrich extension of the operator associated with the quadratic form   Q V [u] = |∇u|2 − V |u|2 d x on C0∞ (R2 ), R2

(5)

(6)

provided Q V is bounded from below. Throughout the paper we will suppose that V satisfies

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Assumption A. The function V (x) is such that σess (− − V ) = [0, ∞). The following notation will be used in the text. Given a self-adjoint operator T , the number of negative eigenvalues, counting their multiplicity, of T to the left of a point −ν is denoted by N (ν, T ). The symbol R+ stands for the set (0, ∞). Moreover, as in [7] we define the space L 1 (R+ , L p (S1 )) in polar coordinates (r, θ ) in R2 , as the space of functions f such that 1/ p  ∞  2π p  f  L 1 (R+ ,L p (S1 )) := | f (r, θ )| dθ r dr < ∞. (7) 0

0

Finally, given s > 0 we denote B(s) := {x ∈ R2 : |x| < s}. We then have 1 (R2 , | ln |x|| d x). Assume that V ∈ L 1 (R , L p (S1 )) Theorem 1. Let V ≥ 0 and V ∈ L loc + for some p > 1. Then the quadratic form (6) is bounded from below and closable. The negative eigenvalues −λ j of the operator associated with its closure satisfy the inequality
Fs (λ j ) ≤ c1 V ln(|x|/s) L 1 (B(s)) + c p V  L 1 (R+ ,L p (S1 )) (8) j

for all s ∈ R+ . The constants c1 and c p are independent of s and V . In particular, if V (x) = V (|x|), then there exists a constant c4 , such that
Fs (λ j ) ≤ c1 V ln(|x|/s) L 1 (B(s)) + c4 V  L 1 (R2 ) (9) j

holds true for all s ∈ R+ . Remark 1. Notice that the r.h.s. of (8) has the right order of asymptotics in both weak and strong coupling limits. Indeed, replacing V by αV and assuming that V ∈ L 1 (R2 , (| ln |x|| + 1) d x) it can be seen from the definition of Fs that
Fs (λ j ) ∼ α, α → 0 ∨ α → ∞. j

 For α → 0 this follows from (3). For α → ∞ the behavior of j Fs is governed by the Weyl asymptotics for the counting function:
N (e−1 s −2 , − − αV ) ≤ Fs (λ j ) ≤ N (0, − − αV ). (10) j

The latter is linear in α when α → ∞ provided V ∈ L 1 (R2 , (| ln |x|| + 1) d x), see also Remark 5.  Remark 2. We would like to emphasize that j Fs (λ j ) cannot be estimated only in terms of V  L 1 (R2 ) . In particular, the logarithmic term in (8) and (9) cannot be removed. This is due to the fact that there exist potentials V ∈ L 1 (R2 ) with a strong local singularity, such that the semi-classical asymptotics of N (ν, − − V ) is non-Weyl for any ν > 0, [2]. Namely if we define Vσ (x) = r −2 | ln r |−2 | ln | ln r ||−1/σ , r < e−2 , σ > 1, Vσ (x) = 0,

r ≥ e−2 ,

(11)

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H. Kovaˇrík, S. Vugalter, T. Weidl

where r = |x|, then Vσ ∈ L 1 (R2 ) for all σ > 1, but N (ν, − − αVσ ) ∼ α σ

α → ∞, ∀ ν > 0,

(12)

see [2, Sect. 6.5]. If (9) were true with the logarithmic factor removed, it would be in obvious contradiction with (10) and (12). Moreover, the asymptotics (12) remain valid also if the singularity of V is not placed at zero, but at some other point. This shows that the condition p > 1 in Theorem 1 is necessary. Remark 3. The non-Weyl asymptotics of N (0, − − αV ) can also occur for potentials which have no singularities, but which decay at infinity too slowly, so that the associated eigenvalues accumulate at zero. For example, if Vσ (x) = (θ ) r −2 (ln r )−2 (ln ln r )−1/σ , r > e2 , σ > 1, Vσ (x) = 0,

r ≤ e2 ,

(13)

then N (0, − − αVσ ) ∼ α σ , see [2]. In this case, however, Theorem 1 says that  the eigenvalues accumulating at zero are small enough so that their total contribution to j Fs (λ j ) grows at most linearly in α. More exactly, inequality (8) gives the following estimate: Corollary 1. Let ∈ L p (0, 2π ) for some p > 1. Let V satisfy the assumptions of Theorem 1 and suppose that  V (x) − Vσ (x) = o Vσ| | (x) , |x| → ∞, where Vσ (x) is defined by (13). Denote n(α) = N (0, − − αV ) and let −λn(α) be the largest eigenvalue of − − αV . Then, for any fixed s > 0 there exists a constant cs > 0 such that for α large enough we have λn(α) ≤ s −2 exp(−cs α σ −1 ). Proof. Inequality (8) shows that implies

 j

(14)

Fs (λ j ) ≤ cs α for some cs . In particular, this

j Fs (λ j ) ≤ cs α, ∀ j.

(15)

On the other hand, from [2, Prop. 6.1] it follows that n(α) ≥ c˜ α σ for some c˜ and α large enough. An application of the inequality (15) with j = n(α) then yields (14). Analogous estimates for λn(α)−k , k ∈ N can be obtained by an obvious modification.

Spectral Estimates for 2-D Schrödinger Operators

831

3. Proof of Theorem 1 We prove the inequality (8) for continuous potentials with compact support. The general case then follows by approximating V by a sequence of continuous compactly supported functions and using a standard limiting argument in (8). As usual in the borderline situations, the method of [10] cannot be directly applied and a different strategy is needed. We shall treat the operator − − V separately on the space of spherically symmetric functions in L 2 (R2 ) and on its orthogonal complement. To this end we define the corresponding projection operators:  2π 1 (Pu)(r ) = u(r, θ ) dθ, Qu = u − Pu, u ∈ L 2 (R2 ). 2π 0 Since P and Q commute with −, the variational principle says that for each a > 1 the operator inequality − − V ≥ P (− − (1 + a −1 ) V ) P + Q (− − (1 + a) V ) Q

(16)

holds. Let us denote by −λ Pj and −λ Qj the non decreasing sequences of negative eigenvalues of the operators P (− − (1 + a −1 ) V ) P and Q (− − (1 + a V ) Q respectively. Clearly we have


Q Fs (λ j ) ≤ Fs (λ Pj ) + Fs (λ j ). (17) j

j

j

We are going to find appropriate bounds on the two terms on the r.h.s. of (17) separately. First we note that P (− − (1 + a −1 ) V ) P is unitarily equivalent to the operator h=−

d2 1 − 2 − W (r ) = h 0 − W (r ) in L 2 (R+ ) dr 2 4r

with the Dirichlet boundary condition at zero and with the potential  1 + a 2π W (r ) = V (r, θ ) dθ. 2πa 0 More precisely, h is associated with the closure of the quadratic form   q[ϕ] = |ϕ  |2 − W |ϕ|2 r dr on C0∞ (R+ ). R+

(18)

(19)

(20)

We start with the estimate on the lowest eigenvalue of h. Lemma 1. Let V be continuous and compactly supported and let W be given by (19). Denote by −λ1P the lowest eigenvalue of the operator h. Then there exists a constant c2 , independent of s, such that  ∞

Fs (λ1P ) ≤ c2 W (r ) r 1 + χ(0,s) (r ) | ln r/s| dr (21) 0

holds true for all s ∈ R+ .

832

H. Kovaˇrík, S. Vugalter, T. Weidl

Proof. From the Sturm-Liouville theory we find the Green function of the operator h 0 at the point −κ 2 : ⎧√ 0 ≤ r ≤ r  < ∞, ⎨ rr  I0 (κr ) K 0 (κr  )  G 0 (r, r , κ) := √ ⎩ rr  I0 (κr  ) K 0 (κr ) 0 ≤ r  < r < ∞, where I0 , K 0 are the modified Bessel functions, see [1]. The Birman-Schwinger principle tells us that if for a certain value of κ the trace of the operator √ √ K (κ) := W (h 0 + κ 2 )−1 W is less than or equal to 1, then the inequality λ1P ≤ κ 2 holds. Taking into account the continuity of W , this implies  



  ∞ P P r I0 λ1 r K 0 λ1 r W (r ) dr ≥ 1. (22) 0

 Now we introduce the substitutions τ = s λ1P , t = s −1r and recall that I0 (0) = 1 while K 0 has a logarithmic singularity at zero, see [1, Chap. 9]. We thus find out that 

F1 τ 2 I0 (τ t) K 0 (τ t) ≤ c2 1 + χ(0,1) (t) | ln t| , ∀τ ≥ 0, where c2 is a suitable constant independent of τ . Here we have used the fact that |I0 (z) K 0 (z)| ≤ const ∀ z ≥ 1,

(23)

see [1]. Numerical analysis gives c2 ∼ = 0.844. Finally, we multiply both sides of inequality (22) by Fs (λ1P ) and note that   Fs (λ1P ) = Fs τ 2 /s 2 = F1 τ 2 .



The proof is complete.

Next we estimate the higher eigenvalues of h. Lemma 2. Under the assumptions of Lemma 1 there exists a constant c3 such that  s  ∞
P Fs (λ j ) ≤ W (r ) r |ln r/s| dr + c3 W (r ) r dr, ∀ s ∈ R+ . j≥2

0

s

Proof. Let us introduce the auxiliary operator hd = −

d2 1 − 2 − W (r ) in L 2 (R+ ) 2 dr 4r

(24)

subject to the Dirichlet boundary conditions at zero and at the point s. Let −µ j be the non-decreasing sequence of negative eigenvalues of h d . Since imposing the Dirichlet boundary condition at s is a rank one perturbation, it follows from the variational principle that

Fs (λ Pj ) ≤ Fs (µ j ). (25) j≥2

j≥1

Spectral Estimates for 2-D Schrödinger Operators

833

Moreover, h d is unitarily equivalent to the orthogonal sum h 1 ⊕ h 2 , where d2 1 − 2 − W (r ) in L 2 (0, s), dr 2 4r d2 1 h 2 = h 2,0 − W (r ) = − 2 − 2 − W (r ) in L 2 (s, ∞) dr 4r h 1 = h 1,0 − W (r ) = −

with Dirichlet boundary conditions at 0 and s. Keeping in mind that Fs ≤ 1 we will estimate (25) as follows:

Fs (µ j ) ≤ N (0, h 1 ) + Fs (µj ), (26) j

j

where −µj are the negative eigenvalues of h 2 . To continue we calculate the diagonal elements of the Green functions of the free operators h 1,0 and h 2,0 . Similarly as in the proof of Lemma 1 we get  G 1 (r, r, κ) = r I0 (κr ) K 0 (κr ) + βs−1 (κ)I0 (κr ) 0 ≤ r ≤ s, (27) s ≤ r < ∞, G 2 (r, r, κ) = r K 0 (κr ) (I0 (κr ) + βs (κ)K 0 (κr )) where βs (κ) = −

I0 (κs) . K 0 (κs)

The Birman-Schwinger principle thus gives us the following estimates on the number of eigenvalues of h 1 and h 2 to the left of the point −κ 2 :  s  ∞ N (κ 2 , h 1 ) ≤ G 1 (r, r, κ) W (r ) dr, N (κ 2 , h 2 ) ≤ G 2 (r, r, κ) W (r ) dr. (28) 0

s

Passing to the limit κ → 0 and using the asymptotic behavior of the Bessel functions I0 and K 0 , [1], we find out that for any fixed r holds the identity lim G 1 (r, r, κ) = lim G 2 (r, r, κ) = r |ln r/s| .

κ→0

κ→0

(29)

The assumption on W and the dominated convergence theorem then allow us to interchange the limit κ → 0 with the integration in (28) to obtain  s N (0, h 1 ) ≤ r |ln r/s| W (r ) dr. (30) 0

This estimates the first term in (26). In order to find an upper bound on the second term in (26), we employ the formula  ∞
 Fs (µ j ) = Fs (t) N (t, h 2 ) dt, (31) j

0

see [10]. Using (28), the substitution t → t 2 and the Fubini theorem we get  e−1/2 s −1 
G 2 (r, r, t) 1 ∞  Fs (µ j ) ≤ W (r ) dt dr. 2 s t (ln ts)2 0 j

834

H. Kovaˇrík, S. Vugalter, T. Weidl

In view of (27) it suffices to show that the integral 

e−1/2 s −1

0

K 0 (tr ) (I0 (tr ) + βs (t)K 0 (tr )) dt t (ln ts)2

(32)

is uniformly bounded for all s > 0 and r ≥ s. The substitutions r = sy, t = τ/s transform (32) into  g(y) :=

e−1/2

0

K 0 (τ y) (I0 (τ y) + β1 (τ )K 0 (τ y)) dτ, τ (ln τ )2

y ∈ [1, ∞).

(33)

Since g is continuous, due to the continuity of Bessel functions, and g(1) = 0, it is enough to check that g(y) remains bounded as y → ∞. Moreover, the inequality (u, (h 2,0 + t1 )−1 u) ≤ (u, (h 2,0 + t2 )−1 u) ∀ 0 ≤ t2 ≤ t1 , ∀ u ∈ L 2 (s, ∞) shows that G 2 (r, r, t), the diagonal element of the integral kernel of (h 2,0 + t 2 )−1 , is non increasing in t for each r ≥ s. Equations (27) and (29) then imply  0

y −1

K 0 (τ y) (I0 (τ y) + β1 (τ )K 0 (τ y)) dτ ≤ ln y τ (ln τ )2



y −1

0

dτ = 1. τ (ln τ )2

On the other hand, when τ ∈ [y −1 , e−1/2 ], it can be seen from (23) and from the behavior of I0 , K 0 in the vicinity of zero, see [1], that |K 0 (τ y) (I0 (τ y) + β1 (τ )K 0 (τ y))| ≤ const uniformly in y. Equation (31) thus yields  ∞
 Fs (µ j ) ≤ c3 W (r ) r dr ∀ s ∈ R+ , s

j

where c3 is independent of s. Numerical analysis shows that c3 ∼ = 0.7. Together with (25), (26) and (30) this completes the proof.

From Eq. (19), Lemma 1 and Lemma 2 we conclude that


1+a 1+a c2 + 1 V ln(|x|/s) L 1 (B(s)) + c3 V  L 1 (R2 ) . Fs (λ Pj ) ≤ 2πa 2πa j

Let us now turn to the second term on the r.h.s. of (17). The key ingredient in estimating this contribution will be the result of Laptev and Netrusov obtained in [7]. We make use of the estimate
Q Fs (λ j ) ≤ N (0, Q(− − (1 + a) V )Q) j

and of the Hardy-type inequality Q (−) Q ≥ Q

1 Q, |x|2

(34)

Spectral Estimates for 2-D Schrödinger Operators

835

which holds in the sense of quadratic forms on C0∞ (R2 ), see [2]. For any ε ∈ (0, 1) we thus get the lower bound 

1 ε 1+a V Q, (35) Q (− − (1 + a) V ) Q ≥ (1 − ε) Q − + − 1 − ε |x|2 1−ε which implies

N (0, Q (− − (1 + a) V ) Q) ≤ N 0, − +

 1 ε 1+a V . − 1 − ε |x|2 1−ε

The last quantity can be estimated using [7, Thm.1.2], which says that 

1 ε 1+a V ≤ c˜ p V  L 1 (R+ ,L p (S1 )) N 0, − + − 1 − ε |x|2 1−ε

(36)

(37)

for some constant c˜ p that also depends on ε and a. In order to conclude the proof of (8) we note that by the Hölder inequality V  L 1 (R2 ) ≤ const V  L 1 (R+ ,L p (S1 )) . To show that the quadratic form (6) is semi-bounded from below we note that inequality (8) says that there are only finitely many eigenvalues of − − V below −e−1 s −2 . Let −V be the minimum of those. Then Q V [u] ≥ −V u L 2 (R2 ) ∀ u ∈ C0∞ (R2 ). The proof of Theorem 1 is now complete. Remark 4. The constant c p in Theorem 1 depends on p and generically goes to infinity as p → 1, see Remark 2. However, for spherically symmetric potentials we have c p = c4 , which is independent of p, see (9). In this case we can use the result of [8], see also [3], to get an upper bound on c4 . Taking into account the numerical values of c2 and c3 we set a = 1 and optimize w.r.t. ε. This gives c1 ∼ = 1.27 and c4 ∼ = 1.08. Remark 5. As a corollary of the proof of Theorem 1 we immediately obtain

N (0, − − V ) ≤ 1 + const V ln |x/s| L 1 (R2 ) + V  L 1 (R+ ,L p (S1 )) ,

(38)

which agrees with [14, Thm.3]. Estimates on N (0, − − V ), different from (38), including logarithmic weights have been obtained earlier in [11, 15]. For spherically symmetric potentials (38) reduces to the inequality established, with explicit constants, already in [3]. Remark 6. Lieb-Thirring inequalities for the operator h = h 0 − W in the form  γ + 1+a γ tr (h 0 − W )− ≤ Cγ ,a W (r )+ 2 r a dr, γ > 0, a ≥ 1 R+

have been recently established in [5].

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H. Kovaˇrík, S. Vugalter, T. Weidl

4. Application In this section we consider a model of quantum layers. It concerns a conducting plate  = R2 × (0, d) with an electric potential V . We will consider the shifted Hamiltonian HV = − − V −

π2 d2

in L 2 (),

(39)

with Dirichlet boundary conditions at ∂, which is associated with the closed quadratic form  

π2 2 2 2 |∇u| − V |u| − 2 |u| d x on H01 (). (40) d  We assume that for each x3 ∈ (0, d) the function V (·, ·, x3 ) satisfies Assumption A. Without loss of generality we assume that V ≥ 0, otherwise we replace V by its positive part. The essential spectrum of the Operator HV covers the half line [0, ∞). Let us denote by −λ˜ j the non-decreasing sequences of negative eigenvalues of HV . For the sake of brevity we choose s = 1 and prove Theorem 2. Assume that V ∈ L 3/2 () and that  π x 2 d 3 ˜ d x3 V (x1 , x2 ) = V (x1 , x2 , x3 ) sin2 d 0 d satisfies the assumptions of Theorem 1 for some p > 1. Then there exist positive constants C1 , C2 , C3 ( p) such that
F1 (λ˜ j ) ≤ C1 V˜ ln(x12 + x22 ) L 1 (B(1)) + C3 ( p) V˜  L 1 (R+ ,L p (S1 )) j

+ C2 V 3/2  L 1 () .

(41)

Remark 7. Notice that (41) has the right asymptotic behavior in both weak and strong coupling limits. Namely, in the weak coupling limit the r.h.s. is dominated by the term linear in V , while in the strong coupling limit the term proportional to V 3/2 prevails. In this sense our result is similar to the Lieb-Thirring inequalities on trapped modes in quantum wires obtained in [6]. Proof of Theorem 2. Let νk = k 2 π 2 /d 2 , k ∈ N be the eigenvalues of the Dirichlet Laplacian on (0, d) associated with the normalized eigenfunctions 

 k π x3 2 φk (x3 ) = sin . d d Moreover, define R = (φ1 , ·) φ1 , S = I − R. By the same variational argument used in the previous section we obtain the inequality HV ≥ R (− − ν1 − 2V ) R + S (− − ν1 − 2V ) S.

(42)

Spectral Estimates for 2-D Schrödinger Operators

837

The latter implies

F1 (λ˜ j ) ≤ F1 (µ˜ j ) + N (0, S (− − ν1 − 2V ) S), j

(43)

j

where −µ˜ j are the negative eigenvalues of R (− − ν1 − 2V ) R. Since R (− − ν1 − 2V ) R = (−∂x21 − ∂x22 − 2 V˜ ) ⊗ R, the first term on the r.h.s. of (43) can be estimated using (8) as follows:
F1 (µ˜ j ) ≤ C1 V˜1 ln(x12 + x22 ) L 1 (R2 ) + C3 ( p) V˜  L 1 (R+ ,L p (S1 )) .

(44)

j

As for the second term, we note that S (−∂x23

∞ ∞

ν2 − ν1 − ν1 ) S = (νk − ν1 ) (φk , ·) φk ≥ νk (φk , ·) φk ν2 k=2

k=2

3 = S (−∂x23 ) S 4 holds true in the sense of quadratic forms on C0∞ (0, d), which implies the estimate

 3 8 S (− − ν1 − 2V ) S ≥ S − − V S. 4 3 Using the variational principle and the Cwickel-Lieb-Rosenblum inequality, [4, 9, 12], we thus arrive at 

 8 N (0, S (− − ν1 − 2V ) S) ≤ N 0, − − V ≤ C2 V 3/2 . 3  In view of (43) this concludes the proof.



Acknowledgement. We would like to thank Elliott Lieb for useful comments. The support from the DFG grant WE 1964/2 is gratefully acknowledged.

References 1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Washington, DC: National Bureau of Standards, 1964 2. Birman, M.S., Laptev, A.: The negative discrete spectrum of a two-dimensional Schrödinger operator. Comm. Pure and Appl. Math. XLIX, 967–997 (1996) 3. Chadan, K., Khuri, N.N., Martin, A., Wu, T.T.: Bound states in one and two spatial dimensions. J. Math. Phys. 44, 406–422 (2003) 4. Cwikel, M.: Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. of Math. 106, 93–100 (1977) 5. Ekholm, T., Frank, R.L.: Lieb-Thirring inequalities on the half-line with critical exponent. http://arxiv.org/list/math.SP/0611247, 2006 6. Exner, P., Weidl, T.: Lieb-Thirring inequalities on trapped modes in quantum wires. XIIIth International Congress on Mathematical Physics (London, 2000), Boston, MA: Int. Press, 2001, pp. 437–443 7. Laptev, A., Netrusov, Y.: On the negative eigenvalues of a class of Schrödinger operators, In: Diff. operators and spectral theory. Am. Math. Soc. Transl. 2, 189, 173–186 (1999)

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8. Laptev, A.: The negative spectrum of a class of two-dimensional Schrödinger operators with spherically symmetric potentials. Func. Anal. Appl. 34, 305–307 (2000) 9. Lieb, E.: Bound states of the Laplace and Schrödinger operators. Bull. Amer. Math. Soc. 82, 751–753 (1976) 10. Lieb, E., Thirring, W.: Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. Studies in Mathematical Physics. Princeton, NJ: Princeton University Press, 1976, pp. 269–303 11. Newton, R.G.: Bounds on the number of bound states for the Schrödinger equation in one and two dimensions. J. Op. Theory 10, 119–125 (1983) 12. Rosenblum, G.V.: Distribution of the discrete spectrum of singular differential operators (in Russian), Izv. Vassh. Ucheb. Zaved. Matematika 1, 75–86 (1976); English transl. Soviet Math. 20, 63–71 (1976) 13. Simon, B.: The Bound State of Weakly Coupled Schrödinger Operators in One and Two Dimensions. Ann. of Phys. 97, 279–288 (1976) 14. Solomyak, M.: Piecewise-polynomial approximation of functions from H l ((0, 1)d ), 2l = d, and applications to the spectral theory of the Schrödinger operator. Israel J. of Math. 86, 253–275 (1994) 15. Stoiciu, M.: An estimate for the number of bound states of the Schrödinger operator in two dimensions. Proc. of AMS 132, 1143–1151 (2003) 16. Weidl, T.: On the Lieb-Thirring constants L γ ,1 for γ ≥ 1/2. Commun. Math. Phys. 178(1), 135–146 (1996) Communicated by B. Simon

Commun. Math. Phys. 275, 839–859 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0320-5

Communications in

Mathematical Physics

On the Susceptibility Function of Piecewise Expanding Interval Maps Viviane Baladi
CNRS, UMR 7586, Institut de Mathématique de Jussieu, Paris, France. E-mail: [email protected] Received: 15 January 2007 / Accepted: 8 March 2007 Published online: 15 August 2007 – © Springer-Verlag 2007

Abstract: We study the susceptibility function (z) =

∞ 


z n X (y)ρ0 (y)

n=0

∂ ϕ( f n (y)) dy ∂y

associated to the perturbation f t = f + t X ◦ f of a piecewise expanding interval map f , and to  an observable ϕ. (1) is the formal derivative (at t = 0) of the average R(t) = ϕρt d x of ϕ with respect to the SRB measure of f t . Our analysis is based on a spectral description of transfer operators. It gives in particular sufficient conditions on f , X , and ϕ which guarantee that (z) is holomorphic in a disc of larger than one, or which ensure that a number may be associated to the possibly divergent series (1) . We present examples of f , X , and ϕ so that R(t) is not Lipschitz at 0, and we propose a new version of Ruelle’s conjecture. 1. Introduction and Main Results Let us call SRB measure for a dynamical system f : M → M, on a manifold M endowed with Lebesgue measure, an f -invariant ergodic probability measure µ so that   k (x)) = the set {x ∈ M | limn→∞ n1 n−1 ϕ( f ϕ dµ} has positive Lebesgue meak=0 sure, for continuous observables ϕ. (Strictly speaking, this is the definition of a physical measure; we refer to [31] for a discussion of the differences between physical and SRB measures. For the purposes of this introduction, the distinction is not very important.) If f admits a unique SRB measure µ, it is natural to ask how µ varies  when f is changed. More precisely, one considers, for fixed ϕ, the function R(t) = ϕ dµt , where µt is the SRB measure (if it is well-defined) of f t = f +t X ◦ f . Loosely speaking, we say that the SRB measure is differentiable (or Lipschitz) at f for ϕ if R(t) is differentiable at 0. (See
Current address: UMI 2924 CNRS-IMPA, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, Brazil

840

V. Baladi

[9] for the relevance of this issue to nonequilibrium statistical mechanics. Theorems 4 and 5 of [14] show another setting where (Lipschitz) regularity of R(t) is relevant.) If f is a sufficiently smooth uniformly hyperbolic diffeomorphism restricted to a transitive attractor, Ruelle [23] (see also [24]) proved that R(t) is differentiable at t = 0 and gave an explicit formula for R (0). Dolgopyat [7] later showed that R(t) was differentiable for a class of partially hyperbolic diffeomorphisms f . More recently, differentiability, together with a formula for R (0), has been obtained for uniformly hyperbolic continuous-time systems (see [6] and references therein) and infinitedimensional hyperbolic systems (see [10] and references therein). A much more difficult situation consists in studying nonuniformly hyperbolic interval maps f , e.g. within the quadratic family (not to mention higher-dimensional dynamics such as Hénon maps). For quadratic interval maps, one requires in addition that the SRB measure be absolutely continuous with respect to Lebesgue. It is well-known that the SRB measure of f t may exist only for some parameters t, although it is continuous in a nontrivial subset of parameters (see [28, 30]). In this setting, Ruelle ([25, 26]) has outlined a program, replacing differentiability by differentiability in the sense of Whitney’s extension theorem, and proposing (1) with ∞ 
∂ (z) = (1) z n X (y)ρ0 (y) ϕ( f n (y)) dy, ∂y n=0

the “susceptibility function,” 1 as a candidate for the derivative. Beware that (1) needs to be suitably interpreted: It could be simply the value at 1 of a meromorphic extension of (z) such that 1 is not a pole, but also a number associated to the – possibly divergent – series obtained by setting z = 1 in (1), by some (yet undetermined) summability method. Formal arguments (see [22] and Appendix B) justify the choice of (1), which Ruelle [25] calls “the only reasonable formula one can write.” For several nonuniformly hyperbolic interval maps f admitting a finite Markov partition (i.e., the critical point is preperiodic), although (z) has a pole (or several poles) inside the open unit disc, it extends meromorphically to a disc of radius larger than 1 and is holomorphic at z = 1 ([26, 11]). The relation between (1) and (Whitney) differentiability of R(t) for such maps has not been established. The case of nonrecurrent critical points is being investigated [27]. Our goal here is much more modest: We consider unimodal interval maps f which are piecewise uniformly expanding, i.e., | f  | > 1 (except at the critical point). In this case, existence of the SRB measure of all perturbed maps f t is guaranteed, and it is known that R(t) has modulus of continuity |t| ln |t| (we refer to the beginning of Sect. 2 for more details and references). Our intention was to understand the analytic properties of (z) for perturbations f + t X ◦ f of such maps, and to see if they could be related to the differentiability (or lack of differentiability) of R(t). Our results are as follows (the precise setting is described in Sect. 2): We prove (Proposition 3.1) that (z) is always holomorphic in the open unit disc. When the critical point is preperiodic of eventual period n 1 ≥ 1, we show that (Theorems 5.1 and 5.2) (z) extends meromorphically to a disc of radius larger than one, with possible poles at the n 1th roots of unity, and we give sufficient conditions for the residues of the poles to vanish. When the critical point is not periodic, (z) appears to be rarely holomorphic at z = 1. Nevertheless, we have a candidate 1 for the value of 1 Since (eiω ) is the Fourier transform of the “linear response” [22], it is natural to consider the variable ω, but we prefer to work with the variable z = eiω .

On the Susceptibility Function of Piecewise Expanding Interval Maps

841

the possibly divergent series (1), under the condition that the “weighted total jump” J ( f, X ) defined in (16) vanishes (Proposition 4.4). The tools for these results are transfer operators L0 and L1 introduced in Sect. 2 (these operators were also used by Ruelle [26]). A key ingredient is a decomposition (Proposition 3.3) of the invariant density of f into a smooth component and a “jump” component (this was inspired by Ruelle’s work [27] in the nonuniformly hyperbolic case). Finally, we give examples of interval maps and observables for which R(t) is not Lipschitz.2 Applying Theorem 5.1 to these examples we get that (z) has a pole at z = 1. The “weighted total jump” J ( f, X ) associated to these examples is nonzero. In view of our results, we propose to reformulate Ruelle’s conjecture as follows: Conjecture A. Let f be either a mixing, piecewise expanding, piecewise smooth unimodal interval map such that the critical point is not periodic, or a mixing smooth Collet-Eckmann unimodal interval map with nonflat critical point. Let f t = f + X t ◦ f be a smooth perturbation corresponding to a smooth X = ∂t X t |t=0 such that X 0 = 0 and each f t is topologically conjugated to f . Then R(t) is differentiable at 0 for all smooth observables ϕ, and R (0) = (1) (the infinite sum being suitably interpreted). The above conjecture is interesting only if there are examples satisfying the assumptions and for which the conjugacy between f and f t is not smooth. We [5] expect this to be true and that the condition J ( f, X ) = 0 is related to the existence of a topological conjugacy between f and f t (see Remark 4.5). For general perturbations of piecewise expanding maps, our counter-examples show that the (previously known) property that R(t) has modulus of continuity |t| ln |t| cannot be improved. For nonuniformly expanding maps, we propose: Conjecture B. Let f be a mixing smooth Collet-Eckmann unimodal interval map, with nondegenerate critical point c (i.e. f  (c) = 0). Then, for any smooth X , and any smooth observable ϕ, the function R(t) is η-Hölder at 0, in the sense of Whitney over those t for which f t is Collet-Eckmann, for any η < 1/2. For critical points of order p ≥ 3 we expect that the condition η < 1/2 should be replaced by η < 1/ p. We expect Conjectures A and B to be essentially optimal. 2. Setting and Spectral Properties of the Transfer Operators In this work, we consider a continuous f : I → I , where I = [a, b], with: f is strictly increasing on I+ = [a, c], strictly decreasing on I− = [c, b] (a < c < b), (ii) for σ = ±, the map f | Iσ extends to a C 3 map on a neighbourhood of Iσ , and inf | f  | Iσ | > 1; (iii) c is not periodic under f ; (iv) f is topologically mixing on [ f 2 (c), f (c)]. (i)

The point c will be called the critical point of f . We write ck = f k (c) for k ≥ 0. For a function X : R → R, with sup |X | ≤ 1, so that X | f (I ) extends to a C 2 function in a neighbourhood of f (I ) and X  is of bounded variation 3 and supported in [a, b], we shall consider the additive perturbation 4 2 After this paper was written, Carlangelo Liverani mentioned to us that Marco Mazzolena [18] independently constructed examples of families f t such that R(t) is not Lipschitz. 3 A prime denotes derivation, a priori in the sense of distributions. 4 Sometimes we only consider one-sided perturbations, i.e., t ≥ 0 or t ≤ 0.

842

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f t (x) = f (x) + t X ( f (x)), |t| small.

(2)

More precisely, we take > 0 so that (i) and (ii) hold for all f t with |t| < , except that f t | I σ may only extend to a C 2 map. Then we assume that f and X are such that, up to taking perhaps smaller , we have sup|t|< f t (c) ≤ b and inf |t|< min( f t (a), f t (b)) ≥ a, so that each f t maps I into itself. Then each f t admits an absolutely continuous invariant probability measure, with a density ρt which is of bounded variation [16]. There is only one such measure [17] and it is ergodic. In fact, assumptions (iii) and (iv) imply that it is mixing. (We refer to the introduction of [15] for an account of the use of bounded variation spaces, in particular references to the work of Rychlik and Keller. The bibliography there, together with that in Ruelle’s book [20], give a fairly complete picture.) By construction, each ρt is continuous on the complement of the at most countable set Ct = { f tk (c), k ≥ 1}, and it is supported in [ f t2 (c), f t (c)] ⊂ [a, b] (we extend it by zero on R). In addition, denoting by |ϕ|1 the L 1 (R, Lebesgue) norm of ϕ, assumption (iii) implies by [15, Prop. 7] (see (38) below and also [15, Remark 5]) that |ρt − ρ0 |1 = 0(|t| ln |t|).

(3)

We next define the transfer operators L0 and L1 , with L1 the ordinary PerronFrobenius operator, and show that L1 is “the derivative of L0 .” In order to make this precise we need more notation. Recall that a point x is called regular for a function φ if 2φ(x) = lim y↑x φ(y) + lim y↓x φ(y). If φ1 and φ2 are (complex-valued) functions of bounded variation on R having at most regular discontinuities, the Leibniz formula says that (φ1 φ2 ) = φ1 φ2 + φ1 φ2 , where both sides are a priori finite measures. Define J := (−∞, f (c)] and χ : R → {0, 1, 1/2} by ⎧ ⎪ / J ⎨0 x ∈ χ (x) = 1 x ∈ int J ⎪ ⎩ 1 x = f (c). 2 The two inverse branches of f , a priori defined on [ f (a), f (c)] and [ f (b), f (c)], may be extended to C 3 maps ψ+ : J → (−∞, c] and ψ− : J → [c, ∞), with sup |ψσ | < 1 for σ = ±. (In fact there is a C 3 extension of ψ± in a small neighbourhood of J .) The map ψ+ has a single fixed point a0 ≤ a. We can now introduce two linear operators:

and

L0 ϕ(x) := χ (x)ϕ(ψ+ (x)) − χ (x)ϕ(ψ− (x)),

(4)

 L1 ϕ(x) := χ (x)ψ+ (x)ϕ(ψ+ (x)) + χ (x)|ψ− (x)|ϕ(ψ− (x)).

(5)

Note that L1 is the usual (Perron-Frobenius) transfer operator for f , in particular, L1 ρ0 = ρ0 and L∗1 (LebesgueR ) = LebesgueR . The operators L0 and L1 both act boundedly on the Banach space BV = BV (0) := {ϕ : R → C , var(ϕ) < ∞, supp(ϕ) ⊂ [a0 , b]}/ ∼, endowed with the norm ϕ BV = inf φ∼ϕ var(φ), where var(·) denotes the total variation and ϕ1 ∼ ϕ2 if the bounded functions ϕ1 , ϕ2 differ on an at most countable set. The following lemma indicates that BV is the “right space” for L1 , but is not quite good enough for L0 :

On the Susceptibility Function of Piecewise Expanding Interval Maps

843

Lemma 2.1. There is λ < 1 so that the essential spectral radius of L1 on BV is ≤ λ, while 1 is a maximal eigenvalue of L1 , which is simple, for the eigenvector ρ0 . There are no other eigenvalues of L1 of modulus 1 on BV . The essential spectral radius of L0 on BV coincides with its spectral radius, and they are equal to 1. Proof. For the claims on L1 , we refer e.g. to [2, §3.1–3.2] and references therein to works of Hofbauer, Keller, Baladi, Ruelle (see also Appendix A). In fact, we may take any λ ∈ (supx (| f  (x)|−1 ), 1). The essential spectral radius of L0 on BV is equal to 1 (see e.g. [2, §3.2], and in particular the result of [13] for the lower bound). It remains to show that there are no eigenvalues of modulus larger than 1. Now, z is an eigenvalue of modulus > 1 of L0 on BV if and only if (see e.g. [20]) w = 1/z is a pole of ζ (w) = exp


wn n≥1

n




sgn( f n ) (x).

f n (x)=x

 However, since f is continuous, we have that | f n (x)=x sgn( f n ) (x)| ≤ 2 for each n, so that ζ (w) has no poles in the open unit disc.   To get finer information on L0 , we consider the smaller Banach space (see [21] for similar spaces) BV (1) = {ϕ : R → C, supp(ϕ) ⊂ (−∞, b], ϕ  ∈ BV }, for the norm ϕ BV (1) = ϕ   BV . We have the following key lemma: Lemma 2.2. The spectrum of L0 on BV (1) and that of L1 on BV coincide. In particular, the eigenvalues of modulus > λ of the two operators are in bijection. Proof. By construction ϕ → ϕ  is a Banach space isomorphism between BV (1) and BV (0) . The Leibniz formula and the chain rule imply that for any ϕ ∈ BV (1) , (L0 ϕ) = L1 (ϕ  ).

(6)

Indeed, the singular term in the Leibniz formula (corresponding to the derivative of χ , which is a dirac mass at c1 ) vanishes, because ψ+ (c1 ) = ψ− (c1 ) = c and ϕ(c)−ϕ(c)=0. That is, the operators L0 and L1 are conjugated, and L0 on BV (1) inherits the spectral properties of L1 on BV , as claimed.   Lemma 2.2 implies that the spectral radius of L0 on BV (1) is equal to 1. The fixed vector is R0 , where we define for x ∈ R,  x ρ0 (u)du. (7) R0 (x) := −1 + −∞

By construction, R0 is Lipschitz, strictly increasing on [c2 , c1 ], and constant outside of this interval (≡ −1 to the left and ≡ 0 to the right). In addition, R0 coincides with ρ0 on each continuity point of ρ0 , so that R0 ∼ ρ0 . The fixed vector of L∗0 is ν(ϕ) = ϕ(b0 ) − ϕ(a0 ) with b0 = ψ− (a0 ). Indeed, L0 ϕ(b) = 0 and L0 ϕ(a0 ) = ϕ(a0 ) − ϕ(b0 ). Since b0 ≥ b (otherwise we would have ψ− (a0 ) = b0 > b, a contradiction) we have ϕ(b0 ) = ϕ(b).

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3. The Susceptibility Function and the Decomposition ρ0 = ρs + ρ r If K is a compact interval we let C 1 (K ) denote the set of functions on K which extend to C 1 functions in an open neighbourhood of K . The susceptibility function [26] associated to f as above, ϕ ∈ C 1 ([a0 , b]), and the perturbation f t = f + t X , is defined to be the formal power series (z) =

∞ 


z n X (y)ρ0 (y)

n=0

=

∞ 


∂ ϕ( f n (y)) dy ∂y (8)

z

n

Ln0 (ρ0 X )(x)ϕ  (x) d x.

n=0

The expressions (8) evaluated at z = 1 may be obtained by formally differentiating ([22], see also Appendix B below) the map  R : t → ϕ(x)ρt (x) d x (9) at t = 0, when ϕ is at least C 1 . Proposition 3.1. The power series (z) extends to a holomorphic function in the open unit disc, and in this disc we have  (z) = (id − zL0 )−1 (Xρ0 )(y) ϕ  (x) d x. Proof. The spectral properties of L0 on BV (Lemma 2.1) imply that for each δ > 0 there is C so that Ln0  BV ≤ C(1 + δ)n , so that (z) is holomorphic in the open unit disc.   Remark 3.2. Ruelle [26] studied (z) for real-analytic multimodal maps f conjugated to a Chebyshev polynomial (e.g. the “full” quadratic map 2 − x 2 on [−2, 2]). In this nonuniformly expanding analytic setting, the susceptibility function is not holomorphic in the unit disc: It is meromorphic in the complex plane but has poles of modulus < 1. (See also [11] for generalisations to other real-analytic maps with preperiodic critical points, and see [3] for determinants giving the locations of the poles when the dynamics is polynomial.) The study of real analytic non-uniformly hyperbolic interval maps with non-preperiodic, but nonrecurrent, critical point is in progress [27]. In order to analyse further (z), let us next decompose the invariant density ρ0 into a singular and a regular part: Any function ϕ : R → C of bounded variation, with regular discontinuities, can be uniquely decomposed as ϕ = ϕs + ϕr , where the regular term ϕr is continuous and of bounded variation (with var(ϕr ) ≤ ϕ BV ), while the singular (or “saltus”) term ϕs is a sum of jumps
su Hu , ϕs = u∈S

where S is an at most countable set, Hu (x) = −1 if x < u, Hu (x) = 0 if x> u and Hu (u) = −1/2, and the su are nonzero complex numbers so that var(ϕs ) = u |su | ≤ ϕ BV . (See [19], noting that our assumption that the discontinuities of ϕ are regular

On the Susceptibility Function of Piecewise Expanding Interval Maps

845

gives the above formulation.) In the case when ϕ is the invariant density of a piecewise smooth and expanding interval map, we have the following additional smoothness of the regular term (this observation, which was inspired by the analogous statement for nonuniformly expanding maps [27], seems new): Proposition 3.3. Consider the decomposition ρ0 = ρs + ρr of the invariant density ρ0 ∈ BV . Then ρr ∈ BV (1) . Proof. We shall use the following easy remark: If a0 = x0 < x1 < · · · < xm = b for m ≥ 2 and ϕ(x) = ϕi (x) for xi−1 ≤ x ≤ xi , with ϕi extending to a C 1 function in a neighbourhood of [xi−1 , xi ], for i = 1, . . . , m, then if ϕ is supported in [a0 , b] we have ϕ ∈ BV with ϕ BV ≤ (b − a0 ) sup

sup

i=1,...,m [xi−1 ,xi ]

|ϕi | +

m−1


|ϕi (xi ) − ϕi+1 (xi )| + |ϕ1 (x0 )| + |ϕ2 (xm )|.

i=1

(10) In this proof we write ρ instead of ρ0 . We know that if ϕ0 ∈ BV is such that (n) with ρ (n) = Ln (ϕ ), the limit being in the 1 0 a0 ϕ0 d x = 1 then ρ = lim n→∞ ρ BV topology. We can assume in addition that ϕ0 is C 2 and nonnegative. Decomposing ρ (n) = ρs(n) + ρr(n) , we have on the one hand that ρs(n) is a sum of jumps along c j for (n) 1 ≤ j ≤ n. On the other hand, by the remark in the beginning of the proof, ρr is an element of BV (1) . We may estimate the BV norm of n = (ρr(n) ) as follows: First note that n extends to a C 1 function in a neighbourhood of x if x ∈ / {c j , 1 ≤ j ≤ n}. Next, we shall show by an easy distortion estimate that there is C (depending on f and on the C 2 norm of ϕ0 ) so that b

|n (x)| ≤ C, ∀n, ∀x ∈ / {c j , 1 ≤ j ≤ n}.

(11)

/ {c j , 1 ≤ j ≤ n} (so that f k (y) = c for Indeed, note that if x = f n (y) with x ∈ 0 ≤ k ≤ n − 1)
f  ( f k (y)) d 1 1 1 =− . n  d x ( f (y)) f  ( f k (y)) ( f n−k−1 ) ( f k+1 (y)) ( f n (y)) n−1

(12)

k=0

 n−k−1 ) (y )|−1 is bounded by a Since supw=c | f  (w)|/| f  (w)| ≤ C0 and n−1 k k=0 |( f  geometric series, uniformly in {yk | f (yk ) = c, 0 ≤  ≤ n − k − 2}, we get n1 (ϕ0 )(x) + λn Ln1 (|ϕ0 |)(x), |n (x)| ≤ CL where λ ∈ (supx=c | f  (x)|−1 , 1). (We have not detailed the contribution of the terms where ϕ0 has been differentiated.) The claim (11) follows from differentiating the righthand-side of (12) with respect to x, and using that supw=c | f  (w)|/| f  (w)| ≤ C1 and supn supx |Ln1 (φ)(x)| < ∞ for all bounded φ. To conclude our analysis of the BV norm of n , we must consider x ∈ {c j , 1 ≤ j ≤ n} and estimate | limw↑x n (w) − lim z↓x n (z)|. The jump between the left and right limits corresponds to the discrepancy between the sets f −n (w) and f −n (z), i.e., it is of the same type as | limw↑x ρ (n) (w)−lim z↓x ρ (n) (z)|, with the difference that 1/|( f n ) (y)|

846

V. Baladi

or ϕ0 (y) (for f n (y) = x) are replaced by their derivatives with respect to x. We find for all n and all x ∈ {c j , 1 ≤ j ≤ n}, ˜ lim ρ (n) (w) − lim ρ (n) (z)|. | lim (n )(w) − lim(n )(z)| ≤ C| w↑x

w↑x

z↓x

z↓x

(13)

Thus, there is C˜ so that for all n,
n n n  ˜ | lim (n )(w) − lim(n )(z)| ≤ Cvar(L 1 (ϕ0 )) + λ var(L1 (|ϕ0 |)). (14) x∈{c j ,1≤ j≤n}

w↑x

z↓x

By the Lasota-Yorke estimates (see e.g. (38)) on Ln1 , (11) and (14), together with (10)

so that n  BV ≤ C

for all n. Applying Helly’s selection theorem, imply that there is C a subsequence n k converges pointwise and in L1 (Lebesgue) to some  ∈ BV . Similar (n ) (n ) arguments show that ρr k and ρs k converge  to ρˆr and ρˆs , respectively (maybe restricting further the subsequence). It follows that ρˆr ψ  d x = − ψ d x for all C 1 functions ψ, i.e.  = ρˆr . By construction we have ρ = ρˆs + ρˆr , with ρˆr ∈ BV (1) ⊂ BV ∩ C 0 , and ρˆs a sum of jumps along the (at most countable) postcritical orbit. By uniqueness of the decomposition ρ = ρs + ρr , we have proved the lemma.   We may now consider the contribution to (z) of the regular term in the decomposition from Proposition 3.3: Lemma 3.4. If ϕ ∈ C 1 ([a0 , b]) then  (id − zL0 )−1 (Xρr )(x) ϕ  (x) d x extends to a meromorphic function in a disc of radius strictly larger than 1, with only singularity in the closed unit disc an at most simple pole at z = 1. The residue of this  b pole is X (a0 )ρr (a0 ) a0 ϕρ0 d x − ϕ(a0 ) . Proof. The spectral properties of L0 on BV (1) (Lemma 2.2) imply that (id − zL0 )−1 (Xρr ) depends meromorphically on z in a disc of radius strictly larger than 1, where its only possible singularity in the closed unit disc is a simple pole at z = 1, with residue (X (b0 )ρr (b0 ) − X (a0 )ρr (a0 ))R0 (x). Since ρr is continuous and supported in b (−∞, c1 ] ⊂ (−∞, b0 ] we have ρr (b0 ) = 0. To finish, integrate a0 ϕ  R0 d x by parts  and use R0 (b) = 0 and R0 (a0 ) = −1.   ∞ Clearly, (id − zL0 )−1 (Xρs ) = n=0 z n Ln0 (Xρs ) is an element of BV which depends holomorphically on z in the open unit disc. We will be able to say much more about this expression if c is preperiodic, in Sect. 5. If c is not preperiodic, the situation is not as transparent, but some results are collected in Sect. 4. In view of Sects. 4–5, we introduce further notation. If c is preperiodic, i.e. f n 0 (c) has minimal period n 1 ≥ 1 (with n 0 ≥ 2 minimal), we set N = n 0 + n 1 − 1 ≥ 2, otherwise we put N = ∞. By definition of the saltus, we have ρs (x) =

N
n=1

with sn = lim y↓cn ρ(y) − lim x↑cn ρ(x).

sn Hcn (x),

(15)

On the Susceptibility Function of Piecewise Expanding Interval Maps

847

We next define the weighted total jump of f : J ( f, X ) =

N


sn X (cn ).

(16)

n=1

We put J ( f ) = J ( f, 1). Note that J ( f ) = −ρr (b0 ) + ρr (a0 ) = ρr (a0 ). Remark 3.5. If f is a tent-map, i.e. | f  (x)| (for x = c) is constant, then it is easy to see that ρ = ρs is purely a saltus function (for example use ρ = limn→∞ Ln1 (ϕ0 ), with ϕ0 the normalised characteristic function of [c2 , c1 ], the limit being in the variation norm). In particular, we get that J ( f ) = J ( f, 1) = 0 for all tent-maps. 4. The Susceptibility Function in the Non-Markov Case In this section we assume (i)–(iv) and that c is not preperiodic (i.e. for every n ≥ 1, the point f n (c) is not periodic; in other words, there does not exist a finite Markov partition for f ). We can suppose without further restricting generality that f (c) < b and min( f (a), f (b)) > a > a0 . We start with a preparatory lemma: Lemma 4.1. Assume that c is not preperiodic. j  If J ( f ) = 0 then the function ρ˜s = ∞ j=1 Hc j k=1 sk is of bounded variation and satisfies (id − L0 )ρ˜s = ρs . If J ( f, X ) = 0 then, setting δc j to be the dirac mass at c j , the measure j   µs = ∞ j=1 δc j k=1 sk X (ck ) is bounded and satisfies (id − f ∗ )µs = Xρs .  n Remark 4.2. We do not claim that when J ( f ) = 0 the sum ∞ n=0 L0 ρs converges to −1 −1 −1 (id − L0 ) ρs = ρ˜s or that (id − zL0 ) ρs converges to (id − L0 ) ρs as z → 1 (even within [0, 1]), and we do not claim the parallel statements about (id − f ∗ )−1 (Xρs ) = µs when J ( f, X ) = 0. Remark 4.3. For any complex number κ we have (id − L0 )(ρ˜s + κ R0 ) = ρs and (id − f ∗ )(µs + κρ0 ) = Xρs . Our result in this case is: Proposition 4.4. Assume that c is not preperiodic and let ϕ ∈ C 1 ([a0 , b]). For |z| < 1 we have  j ∞

(z) = − ϕ(c j ) z j−k sk X (ck ) − (id − zL1 )−1 (X  ρs + (Xρr ) )(x)ϕ(x) d x. j=1

k=1

(17) The second term above extends to a meromorphic function in a disc of radius strictly larger than 1, with only singularity an at most simple pole at z = 1, with residue b J ( f, X ) a0 ϕρ0 d x. If J ( f, X ) = 0 then the following is a well-defined complex number:  j ∞

ϕ(c j ) sk X (ck ) − (id − L1 )−1 (X  ρs + (Xρr ) )(x)ϕ(x) d x. (18) 1 = − j=1

k=1

848

V. Baladi

Remark 4.5. There exists a unique function α on the postcritical orbit so that X (ck+1 ) = α(ck+1 ) − f  (ck )α(ck ) for k ≥ 1: set
X (ck+1+ j )/( f j+1 ) (ck ). α(ck ) = − j≥0

(See e.g. [29, Proof of Thm 1] for the relevance of this “twisted cohomology equation”, in view of Conjecture A: The possibility to extend α “smoothly” to I is related to the existence of a topological conjugacy between f and f t .) Since sk = f  (ck )sk+1 for all k ≥ 1, and since s1 = 0, our condition J ( f, X ) = 0 is equivalent to requiring that X (c1 ) − α(c1 ) = 0. In view of Lemma 4.1, slightly abusing notation, we may write when c is not preperiodic, and J ( f, X ) = 0,  1 = − (id − L1 )−1 ((Xρ0 ) )(x)ϕ(x) d x. If, in addition, X ≡ 1, we may also write  1 = (id − L0 )−1 (ρ0 )(x)ϕ  (x) d x.

(19)

The orbit of  c is expected to be “generically” dense, so that both conditions “ϕ(ck ) = 0 for all k and ϕρ0 d x = 0” and “X (ck ) = 0 for all k” are very strong 5 . However, we point out that either condition implies that (z) extends holomorphically to a disc of radius larger than 1, with (1) = 1 . The relationship between (z) and 1 (when J ( f, X ) = 0) is unclear for general ϕ and X . (See Remark 4.6. See however Appendix C for an alternative – perhaps artificial– susceptibility function, which can be related to 1 .) If J ( f, X ) = 0, it seems unlikely that a replacement for 1 would exist. (See also Appendix C.) We now prove Lemma 4.1. j

Proof. Note first that if c is not preperiodic then, since ρ = lim j→∞ L1 (ϕ0 ) (for ϕ0 as in the proof of Proposition 3.3) and the convergence is exponentially rapid in the BV norm, there are ξ < 1 and C ≥ 1 so that
|sk | ≤ var(ρ ( j) − ρ) ≤ Cξ j , ∀ j. (20) k≥ j+1

Then apply (20) and the assumption J ( f ) = |

j
k=1

sk | = | −




∞

k=1 sk

= 0, to get

sk | ≤ Cξ j , ∀ j.

(21)

k≥ j+1

Observe next that L0 (Hc j ) = Hc j+1 for all j ≥ 1. Finally, use sup |Hc j | ≤ 1 for all j and (15). 5 They are satisfied for nontrivial X , e.g. if c is not recurrent.

On the Susceptibility Function of Piecewise Expanding Interval Maps

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For the second claim, use also sup |X | ≤ 1, f ∗ (δc j ) = δc j+1 , that J ( f, X ) = 0, implies j
| sk X (ck )| ≤ Cξ j , ∀ j, (22) k=1

and that (ρs is a distribution of order 0 and X is continuous) Xρs =

∞


X (c j )s j δc j .

(23)

j=1

  We next show Proposition 4.4: Proof. Write ρ for ρ0 and consider the decomposition ρ = ρs + ρr . We have ρ(b) = ρ(b0 ) = ρ(a0 ) = 0, ρr (b) = 0, and ρs is continuous at a0 , b0 , and b with ρs (b0 ) = ρs (b) = 0. We may integrate by parts, and get from the Leibniz formula (recall Lemma 2.2, and note that (L0 (ψ)) = f ∗ ψ  , for ψ ∈ BV ) for |z| < 1 that  b   n  n  L0 (ρ X )(x)ϕ (x) d x = − ϕ f ∗ (Xρs ) − Ln1 (X  ρs + (Xρr ) )(x)ϕ(x) d x. (24) a0

(There are no boundary terms in the Stieltjes integration by parts because Ln0 (ρ X ) is continuous and vanishes at b, ρ(b0 ) = 0 and ρ(a0 ) = 0.) It follows that for |z| < 1,

   ∞
zn (25) ϕ f ∗n (Xρs ) + Ln1 (X  ρs + (Xρr ) )(x)ϕ(x) d x . (z) = − n=0

The proof of Lemma 3.4 applies to L1 on BV and allows us to control the terms associated to (Xρr ) and X  ρs . Since J ( f, X ) = R Xρs , the residue of the possible pole at z = 1 is, using Stieltjes integration by parts,    b    b −( X  ρs d x + (Xρr ) d x) ϕρ0 d x = −( X  ρ d x + Xρr d x) ϕρ0 d x R

R

a0

=(

 R

Xρ  − 

= J ( f, X )

R

 R b

Xρr d x)



R

b

a0

ϕρ0 d x

a0

ϕρ0 d x.6

(26)

a0

On the other hand, we get by (20), (23), and since sup |Hc j | ≤ 1 for all j, that for each |z| < 1, ∞


z n f ∗n (Xρs ) =

n=0

∞


zn

n=0

=

∞
j=1

∞


sk X (ck )δck+n

(27)

k=1

δc j

j


z j−k X (ck )sk .

k=1

6 In the case X = 1, recall that J ( f, 1) = ρ (a ), and note that r 0 ϕ(a0 )ρr (a0 ).



 Ln0 (ρs )ϕ  d x = − ϕ f ∗n (ρs ) +

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We have proved (17) in the open disc of radius 1. The fact that 1 is well-defined follows from Lemma 4.1 and our assumption that J ( f, X ) = 0 which implies (22).   Remark 4.6. In spite of Lemma 4.1, we are not  in a position ∞ to apply Fubini’s theorem in (27) at z = 1. It seems unlikely that the sum ∞ n=0 k=1 sk X (ck )δck+n converges in the usual sense to µs , and it is unclear whether µs could be interpreted as a classical (e.g. Norlund or Abelian) limit of this sum.

5. The Susceptibility Function in the Markov Case Assume in this section, in addition to (i)–(iv), that c is preperiodic, i.e. there exist n 0 ≥ 2 and n 1 ≥ 1 so that cn 0 is periodic of minimal period n 1 (we take n 0 minimal for this property). In this Markov case, we have the following result: Theorem 5.1. Assume that c is preperiodic. Let ϕ ∈ C 1 ([a0 , b]). Then (z) admits a meromorphic extension to a disc of radius > 1. The poles of (z) in the closed unit disc are at most simple poles at the n 1th roots of unity.  Assume either (a) ϕ(ck ) = 0 for all k ≥ n 0 and ϕρ0 d x = 0, or (b) X (ck ) = 0 for all k ≥ 1, then the residues of all poles of modulus one of (z) vanish, and (1) = lim

k→∞

k 


Ln0 (ρ0 X )(x)ϕ  (x) d x.

n=0

We next exhibit other sufficient conditions for the residues of the poles of (z) 1,n 0 1,n 0 on the n 0 unit circle to vanish. For this, we introduce Jn 0 = Jn 0 ( f, X ) = J ( f, X ) = k=1 X (ck )sk , and, if n 1 ≥ 2, the following sums of jumps for m = n 0 , . . . , n 0 +n 1 −1:


Jmn 1 ,n 0 = Jmn 1 ,n 0 ( f, X ) =

X (ck )sk .

1≤k≤n 0 +n 1 −1: ∃≥0:k+n 0 −1−n 1 =m

Theorem 5.2. Assume that c is preperiodic. Let ϕ ∈ C 1 ([a0 , b]). If Jmn 1 ,n 0 = 0 for m = n 0 , . . . n 0 + n 1 − 1, then (z) is holomorphic in a disc of radius strictly larger than one with (1) = lim (z) and (1) = lim z→1

k→∞

k 


Ln0 (Xρ0 )(x)ϕ  (x) d x.

n=0

  0 +n 1 −1 The residue of (z) at z = 1 is J ( f, X )( ϕρ0 d x − n11 nj=n ϕ(c j )), in partic0 ular, if J ( f, X ) = 0 then (z) is holomorphic at z = 1 with (1) = We first prove Theorem 5.1:

lim

z∈[0,1),z→1

(z).

On the Susceptibility Function of Piecewise Expanding Interval Maps

851

Proof. Since Ln0 (ρ X ) is continuous and vanishes at b, the term associated to the rightmost boundary in the Stieltjes integration by parts (24) in the proof of Proposition 4.4 vanishes. If c2 = a0 then ρ vanishes and is continuous at a0 and b0 , so that the leftmost boundary term from (24) vanishes. If c2 = a0 , this leftmost boundary term is in fact included in the Stieltjes integral − ϕ f ∗n (Xρs ). We consider X ≡ 1, the general case follows by integration by partsas in (24–26) in the proof of Proposition 4.4 (recall in particular the residue J ( f, X ) ϕρ0 d x), using the remarks in the previous paragraph. By Lemma 2.2, Proposition 3.3, Lemma 3.4 and (15) it suffices to consider L0 acting on the finite-dimensional space generated by Hck , for 1 ≤ k ≤ n 0 + n 1 − 1. We have  L0 Hc j = Hc j+1 , j < n 0 + n 1 − 1, L0 Hcn0 +n1 −1 = Hcn0 . The (n 0 + n 1 − 1) × (n 0 + n 1 − 1) matrix L associated to the above linear operator is such that L n 1 is in lower triangular form, with zeroes in the first n 0 − 1 diagonal elements and with an n 1 × n 1 identity block in the n 1 last rows and columns. It follows that (id − zL0 )−1 (ρ0 ) extends meromorphically to a disc of radius strictly larger than 1, whose singularities on the unit circle are at most simple poles at the n th roots of unity. b1 To show the claim on the vanishing of the residues, we integrate a0 Ln0 (ρs )ϕ  d x by parts: it suffices to consider the boundary terms since our assumption ϕ(c j ) = 0 for all j ≥ n 0 guarantees that the poles corresponding to the eigenvalues of L have zero residue. If a0 = c2 then the boundary term gives a residue −ϕ(a0 )ρs (a0) for the pole at z = 1, which, summed with the residue from Lemma 3.4 gives J ( f, 1) ϕρ0 d x (using ρs (a0 ) + ρr (a0 ) = 0 and J ( f, 1) = ρr (a0 )). If a0 = c2 , the boundary term gives rise to the multiple of ϕ(c2 ) which appears in the contribution of the spectrum of L, and  Lemma 3.4 gives J ( f, 1)( ϕρ0 d x − ϕ(a0 )).   We now prove Theorem 5.2: Proof. Again, we consider X ≡ 1, and the general case follows by integration by parts. If Jmn 1 ,n 0 = 0 for m = n 0 , . . . n 0 + n 1 − 1, then L0n 0 −1 (ρs ) vanishes. It follows (recall Lemma 3.4, the residue there vanishes if X = 1 since ρr (a0 ) = J ( f, 1)) that (z) is holomorphic in a disc of radius strictly larger than one. If J ( f ) = 0 then we claim that the spectral projector  associated to the eigenvalue 1 of the matrix L introduced in the proof of Theorem 5.1 satisfies (ρs ) = 0 (this gives the second statement of the theorem). To show the claim note that the fixed vector for L is v = (v j ) with v j = 0 for j ≤ n 0 − 1 and v j = 1 for n 0 ≤ j ≤ n 0 + n 1 − 1, and that u = (1, . . . , 1) is a left fixed vector for L. The projector  is just (w) = u,w u,v v, and (ρs ) = 0 follows from J ( f, 1) = 0.   6. Non-Differentiability of the SRB Measure In this section, we present examples 7 of perturbations f + t X ◦ f ofmaps f satisfying (i)–(iv), so that f has a preperiodic critical point, and at which t → ϕρt d x fails to be 7 The example in Theorem 6.1 and Remark 6.3 are due to A. Avila. D. Dolgopyat told me several years ago that he believed the SRB measure was not a Lipschitz function of the dynamics in the present setting, and he may have been aware of similar examples. After this paper was written, we learned about [8] which, although mostly nonrigorous, indicated that R(t) should not be expected to be Lipschitz, and C. Liverani brought to our attention Mazzolena’s [18] detailed analysis of families of maps for which R(t) is not Lipschitz.

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Lipschitz at t = 0 for a well-chosen smooth observable ϕ. (In view of (3), we shall see that the examples are “as bad as possible.”) Recall that we call a tent-map a map f satisfying (i)–(iv) and so that | f  (x)| is constant for x = c. For 1 < λ ≤ 2 we let gλ be the tent-map of slopes ±λ on [0, 1], i.e., gλ (x) = λx for x ∈ [0, 1/2], and gλ (x) = λ − λx for x ∈ [1/2, 1]. We put cn (λ) = gλn (1/2) for n ≥ 1. We first present the simplest possible counter-example: Theorem 6.1. There exists a C 1 function ϕ, with ϕ(0) = ϕ(1) = 0, a sequence λk ∈ (1, 2) with limk→∞ λk = 2, so that ck+2 (λk ) is a fixed point of gλk , and a constant C > 0 so that   ϕγλk d x − ϕγ2 d x ≥ Ck(2 − λk ), ∀k, with



ϕγ2 d x = 1, where γλk is the invariant density of gλk .

(In fact we have ϕ(cn (λk )) = 0 for all k ≥ 1 and n ≥ 1 in Theorem 6.1.) The theorem shows that the SRB measure cannot be (one-sided) Lipschitz at g2 for ϕ. Since we can write gλk = g2 + tk X ◦ f , with tk = λk − 2 and X as in §2, with X (0) = 0 (in fact, X (x) = x for x ∈ [0, 1]), and ϕ(0) = ϕ(1) = 0, Theorem 5.2 applies to f = g2 , X , and ϕ, and, since ϕρ0 d x = 0 gives that (z) is meromorphic  in a disc of radius larger than one with a simple pole at z = 1 (the residue is J ( f, X ) ϕρ0 d x with J ( f, X ) = X (c1 )s1 = 0). Note that Ruelle [26] proved that the susceptibility function associated to the full quadratic map and any smooth X and ϕ has a vanishing residue at z = 1. However, (z) has a pole strictly inside the unit disc in the setting of [26]. Of course, the example in Theorem 6.1 is a bit special since g2 is an “extremal” tentmap. But it is not very difficult to provide other examples of tent-maps with preperiodic critical points at which the SRB measure is not a Lipschitz function of the dynamics. Indeed, coding the postcritical orbit by the sequence , with  j = L if c j < 1/2 and  j = R if c j > 1/2, the code of g2 is R L ∞ (that is, 1 = R, and  j = L for all j ≥ 2), while the proof of Theorem 6.1 shows that the code of gλk is 1 = R,  j = L for 2 ≤ j ≤ k + 1 and  j = R for j ≥ k + 2. The following example corresponds to a similar perturbation, starting from  = R L R ∞ (i.e., g√2 ), and considering a sequence gν , for  ≥ 6 and even, where ν is the unique parameter giving the code 1 = R, 2 = L ,  j = R for 3 ≤ j ≤  − 2, −1 = L ,  j = R for j ≥ . (28) (In particular c (ν ) is the fixed point of gν .)

√ √ 1 function ϕ, with ϕ(c ( 2)) = ϕ(c ( 2)) = ϕ Theorem 6.2. There exists a C 1 2 √ √  (c3 ( 2)) = 0, and ϕγ√2 d x = 1, a sequence ν ∈ ( 2, 2), with  even and √ lim→∞ ν = 2, so that c (ν ) is a fixed point of gν , and a constant C > 0 so that   √ ϕγ√2 d x − ϕγν d x ≥ C(ν − 2), ∀. (In fact we have ϕ(cn (ν )) = 0 for all even  ≥ 4 and n ≥ 1 in Theorem 6.2.) Theorem 5.2 applies to the example  in Theorem 6.2 and gives that (z) has a simple pole at z = 1 with residue J ( f, X ) ϕγ√2 d x = 0.

On the Susceptibility Function of Piecewise Expanding Interval Maps

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Remark 6.3. Although the combinatorics will be more complicated, a modification of the proof of Theorems 6.1 and 6.2 should be applicable [1] to all preperiodic tent-maps. 1 This would give a dense countable  set of parameters 0 , and C functions ϕλ , where the SRB measure λ → R(λ) = ϕλ0 γλ d x is not Lipschitz at λ0 if λ0 ∈ 0 , for which (z) is meromorphic at z = 1 (by Theorem 5.2). If this construction is possible, a Baire argument [1] would then imply that there is an uncountable set of parameters 1 where R(λ) is not Lipschitz. This would give rise to counterexamples which are non-Markov tent-maps to which Proposition 4.4 applies (with, presumably, J ( f, X ) = 0). In view of the program sketched in the previous remark, it would seem that the SRB measure of tent-maps is not often Lipschitz. We next prove Theorem 6.1: Proof. The fixed point of gλ is xλ = λ/(1 + λ) > 1/2 and its preimage in [0, 1/2] is yλ = 1/(1 + λ). Let z λ = yλ /λ be the preimage of yλ in [0, 1/2]. The critical value is c1 = λ/2 > 1/2 (in this proof we write c j for c j (λ) whenever the meaning is obvious), which is mapped to c2 = (2 − λ)λ/2 < 1/2. If λ = 2 then c1 = 1, c2 = c3 = 0, and γ2 is constant, equal to 1 on [0, 1]. If 1 < λ < 2, then ck+1 = yλ if λ = λk = 2 − Ck λ−k with Ck = 2/(λ(1 + λ)), and k ≥ 1. The invariant density for such λk is supported in [c2 , c1 ] and constant on each (c j+1 , c j+2 ) for 1 ≤ j ≤ k, with value v j > 0, and on (ck+2 , c1 ) = (xλk , c1 ), with value vk+1 > 0. The fixed point equation for γλk reads vk+1 = λk v1 , v j + vk+1 = λk v j+1 for 1 ≤ j ≤ k − 1, and 2vk = λk vk+1 . This implies that the sequence j → v j is strictly increasing. (Indeed, vk+1 = 2vk /λk > vk , and proceed by decreasing induction, using that v j+1 > (v j +vk+1 )/2 and vk+1 > v j+1 to show that v j+1 > v j for k −1 ≥ j ≥ 1.) We take a nonnegative C 1 function ϕ which is  supported in (2/3,  3/4), and thus in (ck+2 , c1 ) for all large enough k. We assume that ϕ(x)γ2 d x = ϕ(x) d x = 1. We next show that there is D > 0 so that for all k ≥ 1,  ϕγλk d x ≥ 1 + Dkλ−k (29) k , and this will end the proof of the theorem.   c1 To show (29), we use the fact that ϕγλk d x = vk+1 ck+2 ϕ(x) d x = vk+1 . To estimate  vk+1 we exploit γλk d x = 1: This integral is equal to the difference vk+1 (c1 − c2 ) −

k
(c j+2 − c j+1 )(vk+1 − v j ). j=1

We have c j+2 − c j+1 = λk (c3 − c2 ) with (c3 − c2 ) ≥ Aλ−k k with A independent of k, k− j −j and (vk+1 − v j ) ≥ (v j+1 − v j ) = vk+1 λk (2 − λ)/2 ≥ Ck λk /2 for 1 ≤ j ≤ k − 1,8 k so that j=1 (c j+2 − c j+1 )(vk+1 − v j ) ≥ Ekλ−k−1 and k j−1

, 1 ≤ vk+1 (c1 − c2 ) − Ekλ−k−1 k )/(c1 − c2 ). Since c1 − c2 ≤ 1 we proved (29).   which implies vk+1 ≥ (1 + Ekλ−k−1 k 8 To get the equality, first use v + v m k+1 = λk vm+1 at m = j and m = j + 1, repeat this k − j − 1 more times, and end by using that 2vk = λk vk+1 .

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Finally, we show Theorem 6.2:

√ √ Proof. Note that γ√2 is constant equal to u on (c2 ( 2), c3 ( 2), and constant equal to √ √ √ √ 2u on (c3 ( 2), c1 ( 2)), with c3 ( 2) > 1/2 the fixed point. Putting √ √ √ √  √ (30) d = c3 ( 2) − c2 ( 2) + 2(c1 ( 2) − c3 ( 2) , the normalisation condition is

(31) √ du = 1. √ For  ≥ 6 even, we define ν < 2 by (28). Then c (ν ) > c3 ( 2) is a fixed point and the critical orbit of gν is ordered as follows (in the remainder of this proof we write cm for cm (ν ) when the meaning is clear) c2 < c−1 < c0 < c−3 < · · · < c5 < c3 < c < c4 < c6 < · · · < c−2 < c1 . The invariant density of g(ν ) is constant equal to u 1 = u 1 (ν ) on (c2 , c−1 ), constant equal to u 2 on (c−1 , c−3 ), constant equal to u j on (c−(2 j−3) , c−(2 j−1) ) for 3 ≤ j ≤ /2 − 1, constant equal to u /2 on (c3 , c ), constant equal to u /2+1 on (c , c4 ), constant equal to u j on (c2 j− , c2 j+2− ) for /2 + 2 ≤ j ≤ √  − 2, constant equal to (c , c ). As  → ∞ we have that c tends to c ( 2), that c2 and c−1 tend u −1 on −2 1 1 1 √ √ to c2 ( 2), that c tends to c3 ( 2). In particular, (3) implies that u 2 → u. The fixed point equation for γν implies that u −1 = ν u 1 , u −2 = ν u 2 and 2u 2 = √ ν2 ν u −1 (thus, u −1 = ν u 2 /2 tends to 2u). In particular, u −2 = 2 u −1 > u −1 , which implies that s−2 < 0. Now, it is not difficult to see from the fixed point equation for γν that for any 3 ≤ k ≤  we have sk = sk−1 / f  (ck−1 ). It follows that s−1 > 0, and that s2 j < 0 for 4 ≤ 2 j ≤  − 2 and s2 j+1 > 0 for 3 ≤ 2 j + 1 ≤  − 3. In other words, γν is increasing on (c2 , c ) (with minimal value u 1 ) and decreasing on (c , c1 ) (with minimal value u −1 = ν u 1 ). Take anonnegative C 1function which  is supported in (c−1  (ν ), 1/2) for all , and note that ϕγ√2 d x = u ϕ d x. Since ϕγν d x = u 2 (ν ) ϕ d x, it suffices to show that there is a constant K > 0 so that for all large enough , u 2 (ν ) ≤ u − K ν− ,

√ in order to prove the theorem. Note that |c2 (ν ) − c2 ( 2)| = O(ν− ). It follows that √ ν − 2 = O(ν− ) and that u 2 (ν )−u 1 (ν ) = (1−ν2 /2)u −1 /ν = O(ν− ). Therefore, it is enough to prove that u 1 (ν ) ≤ u − K  ν− , (32)  for some K > 0 and all large enough . The rest of the proof is now similar to the argument in Theorem 6.1. Writing (cm( j) , cm  ( j) ) for the interval on which γν is constant equal to u j , we have 1=

−1


u j (cm  ( j) − cm( j) )

(33)

j=1

= u 1 (c − c2 ) +

/2
(u j − u 1 )(cm  ( j) − cm( j) ) j=2

+ ν u 1 (c1 − c ) +

−2


(u j − ν u 1 )(cm  ( j) − cm( j) ).

j=/2+1

On the Susceptibility Function of Piecewise Expanding Interval Maps −+2 j

855 −2 j

If 3 ≤ j ≤ /2 we have u j − u 1 ≥ u j − u j−1 > Dν and cm  ( j) − cm( j) > Dν for D > 0 independent of  and j. The case j > /2 is similar. It follows that there is C  > 0 so that the right-hand-side of (33) is larger than u 1 (c − c2 ) + ν u 1 (c1 − c ) + C  ν− = u 1 d + C  ν−

(34)

for all large enough , where  d = c (ν ) − c2 (ν ) + ν (c1 (ν ) − c3 (ν ) .

(35)

We have thus proved that u1 ≤

1 − C  ν− . d

Combining the above bound with (31), (30), and the easily proved fact that |d − d | = O(ν− ), we get (32).   Appendix A. Uniform Lasota-Yorke Estimates and Spectral Stability We recall how to get uniform Lasota-Yorke estimates. For |t| < , define Jt := (−∞, f t (c)] and χt : R → {0, 1, 1/2} by ⎧ ⎪ / Jt ⎨0 x ∈ χ (x) = 1 x ∈ int Jt ⎪ ⎩ 1 x = f (c). t 2 The two inverse branches of f t , a priori defined on [ f t (a), f t (c)] and [ f t (b), f t (c)], may be extended to C 2 maps ψt,+ : Jt → (−∞, c] and ψt,− : Jt → [c, ∞), with  | < 1 for σ = ±. (In fact there is a C 2 extension of ψ sup |ψt,σ t,± in a small neighbourhood of Jt .) It is no restriction of generality to assume that ψt,+ (a0 ) = a0 for all t. Put   L1,t ϕ(x) := χt (x)ψt,+ (x)ϕ(ψt,+ (x)) + χt (x)|ψt,− (x)|ϕ(ψt,− (x)).

(36)

The first remark is that (see e.g. [12, Lemma 13]) there is D ≥ 1 so that for any ϕ ∈ BV , we have |L1,t ϕ − L1 ϕ|1 ≤ D|t|ϕ BV , ∀|t| < .

(37)

Let λ−1 < inf x=0 | f  (x)|. Now, since c is not periodic, the proof of (3.26) in [2, p. 177] yields D  ≥ 1 so that for all small enough |t|, m  varLm 1,t ϕ ≤ λ varϕ + D |ϕ|1 , ∀m ≥ 1.

(38)

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Appendix B. Formal Relation Between (1) and the Derivative of the SRB Measure We refer to [23] and [6], and references therein, for uniformly hyperbolic instances where the SRB measure is smooth, and where the susceptibility function is related to its derivative. In this appendix, we first recall (in our notation) Ruelle’s formal argument [22] leading to the consideration of (1) as a candidate for the derivative. We then give another (perhaps new) formal argument. For simplicity, we consider only X ≡ 1. The first step is rigorous: By (38) there are C ≥ 1 and ξ ∈ (λ, 1) that for all |t| < and all k ≥ 1,   | ϕ(x)ρt (x) d x − ϕ( f tk (x)) d x| ≤ Cξ k . (39) Now, since ϕ is C 1 , there is s = sk with |s| < |t| so that ϕ( f k (x)) − ϕ( f tk (x)) = −t

k−1
d ϕ( f sn (y))| y= fsk−n (x) . dy

(40)

n=0

Next, 
k−1 d ϕ( f sn (y))| y= fsk−n (x) d x dy n=0

(41)


k−1 d = ϕ( f sn (y))Lk−n 1,s (ϕ0 )(y) dy dy n=0

=


k−1

ϕ  (y)Ln0,s (Lk−n 1,s (ϕ0 ))(y) dy.

n=0

Letting (this is of course a formal step that is not justified here) t → 0 and k → ∞ in the above formula, and using that Lm 1 (ϕ0 ) → ρ0 as m → ∞, we would find 

ϕρt d x − t



ϕρ0 d x

∼

∞ 


ϕ  (y)Ln0 (ρ0 )(y) dy

as t → 0,

(42)

n=0

as announced. Let us give now the second formal argument. Consider   ρt − ρ ϕ d x − ϕ  (id − L0 )−1 ρ0 d x. t Define for x ∈ R and |t| < ,  Rt (x) := −1 +

x −∞

ρt (u)du.

(43)

If t is small, it is tempting (but of course illicit, since there is no continuity of the resolvent on BV ) to replace (id − L0 )−1 by (id − L0,t )−1 where L0,t ϕ(x) := χt (x)ϕ(ψt,+ (x)) − χt (x)ϕ(ψt,− (x)).

(44)

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857

We then get by integration by parts 

 R0 − Rt −1  − (id − L0,t ) R0 d x ϕ t

  R0 − Rt  −1 R0 − Rt  − L0,t ( ) − R0 d x = ϕ (id − L0,t ) t t

   −1 R0 (·) − R0 (· − t)  − R0 (·) d x = ϕ (id − L0,t ) t ∼ 0 as t → 0, 

(45)

where we used L0,t Rt = Rt and L0,t R0 (x) = L0 R0 (x − t) = R0 (x − t), and where we “pretend” again that (id − L0,t )−1 is continuous on BV . Appendix C. A Regularised Susceptibility Function In this section we assume (i)–(iv) and, in addition, that c is not preperiodic and f (c) < b, min( f (a), f (b)) > a > a0 . For simplicity, we only consider the case X ≡ 1. Define a power series ρs (z) =

∞


z k sk Hck .

(46)

k=1

Clearly, (20) implies that z → ρs (z) is holomorphic in a disc of radius larger than 1, with ρs (1) = ρs . Put ρ(z) = ρr +ρs (z). Define next a regularised susceptibility function by ∞ 
˜ z n Ln0 (ρ(z))(x)ϕ  (x) d x. (47) (z) = n=0

˜ Proposition C.1. Let c be non preperiodic. (z) is holomorphic in the open unit disc. 1 ˜ is holomorphic in a disc of radius If J ( f ) = 0 then for every ϕ ∈ C ([a0 , b]), (z) ˜ larger than one, and in addition, (1) = 1 . Proof. The first claim is easily shown. We have for each |z| < 1, ∞


z n Ln0 (ρs (z)) =

n=0

∞
n=0

=

∞
j=1

zn

∞


z k sk Hck+n

(48)

k=1

z j Hc j

j


sk .

k=1

˜ Thus, since sup |Hc j | ≤ 1, (48) and (21) imply that (z) is holomorphic in the disc of radius 1/ξ . The last claim follows easily.   Our second observation follows:

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Proposition C.2. Let c be nonpreperiodic and J ( f ) = 0. For ϕ ∈ C 1 ([a0 , b]) with  ˜ as z → 1 in [0, 1) exists if and only if ϕρ0 d x = 0, the limit of (z) ∞


lim

z→1,z∈[0,1)

z j ϕ(c j )

(49)

j=1

exists. Proof. Replacing ρs by ρs (z) in the proof of Proposition 4.4, it suffices to consider ⎞ ⎛ j ∞
∞ ∞ ∞



sk z k+n Hck+n = z j Hc j ⎝ s k − J ( f )⎠ + J ( f ) z j Hc j . n=1 k=1

j=1

k=1

j=1

The first term in the right-hand-side of the above equality extends holomorphically in the disc of radius 1/ξ . Integrating by parts, this leaves J(f)

∞


z j ϕ(c j ),

j=1

as claimed.   If c j is not recurrent it is easy to find examples of ϕ so that the limit (49) does not exist. This limit may never [1] exist. Acknowledgements. Partially supported by ANR-05-JCJC-0107-01. I am grateful to Dmitry Dolgopyat for important remarks, and to David Ruelle, who explained this problem to me several years ago, and shared ideas on his ongoing work on the nonuniformly hyperbolic case. Artur Avila sketched the counterexample of Theorem 6.1, provided Remark 6.3, and made several useful comments. I thank Gerhard Keller, who found mistakes in previous versions, for helpful suggestions, and Daniel Smania for very useful conversations which helped me formulate Conjecture A and Remark 4.5. Note added in proof: [5] is now available on arxiv.org (Linear response formula for piecewise expanding unimodal maps) and contains a proof of a slight strengthening of Conjecture A for piecewise expanding maps.

References 1. Avila, A.: Personal communication, 2006 2. Baladi, V.: Positive transfer operators and decay of correlations. Advanced Series in Nonlinear Dynamics, Vol. 16, Singapore: World Scientific, 2000 3. Baladi, V., Jiang, Y., Rugh, H.H.: Dynamical determinants via dynamical conjugacies for postcritically finite polynomials. J. Stat. Phys. 108, 973–993 (2002) 4. Baladi, V., Keller, G.: Zeta functions and transfer operators for piecewise monotone transformations. Commun. Math. Phys. 127, 459–479 (1990) 5. Baladi, V., Smania, D.: Work in progress, 2007 6. Butterley, O., Liverani, C.: Smooth Anosov flows: correlation spectra and stability. J. Mod. Dyn. 1, 301–322 (2007) 7. Dolgopyat, D.: On differentiability of SRB states for partially hyperbolic systems. Invent. Math. 155, 389–449 (2004) 8. Ershov, S.V.: Is a perturbation theory for dynamical chaos possible? Phys. Lett. A 177, 180–185 (1993) 9. Gallavotti, G.: Chaotic hypothesis: Onsager reciprocity and fluctuation-dissipation theorem. J. Stat. Phys. 84, 899–926 (1996) 10. Jiang, M., de la Llave, R.: Linear response function for coupled hyperbolic attractors. Commun. Math. Phys. 261, 379–404 (2006)

On the Susceptibility Function of Piecewise Expanding Interval Maps

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11. Jiang, Y., Ruelle, D.: Analyticity of the susceptibility function for unimodal markovian maps of the interval. Nonlinearity 18, 2447–2453 (2005) 12. Keller, G.: Stochastic stability in some chaotic dynamical systems. Monatshefte Math. 94, 313–333 (1982) 13. Keller, G.: On the rate of convergence to equilibrium in one-dimensional systems. Commun. Math. Phys. 96, 181–193 (1984) 14. Keller, G.: An ergodic theoretic approach to mean field coupled maps. Progress in Probability 46, 183–208 (2000) 15. Keller, G., Liverani, C.: Stability of the spectrum for transfer operators. Annali Scuola Normale Superiore di Pisa 28, 141–152 (1999) 16. Lasota, A., Yorke, J.A.: On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186, 481–488 (1973) 17. Li, T.Y., Yorke, J.A.: Ergodic transformations from an interval into itself. Trans. Amer. Math. Soc. 235, 183–192 (1978) 18. Mazzolena, M.: Dinamiche espansive unidimensionali: dipendenza della misura invariante da un parametro. Master’s Thesis, Roma 2, 2007 19. Riesz, F., Sz.-Nagy, B.: Functional analysis. (Reprint of the 1955 original), Dover Books on Advanced Mathematics, New York: Dover, 1990 20. Ruelle, D.: Dynamical zeta functions for piecewise monotone maps of the interval. CRM Monograph Series, 4, Providence, RI: Amer. Math. Soc. 1994 21. Ruelle, D.: Sharp zeta functions for smooth interval maps. In: International Conference on Dynamical Systems (Montevideo 1995), Pitman Research Notes in Math. Vol. 362, Harlow: Longman, 1996, pp. 188–206 22. Ruelle, D.: General linear response formula in statistical mechanics, and the fluctuation-dissipation theorem far from equilibrium. Phys. Lett. A 245, 220–224 (1998) 23. Ruelle, D.: Differentiation of SRB states. Commun. Math. Phys. 187, 227–241 (1997) 24. Ruelle, D.: Differentiation of SRB states: Corrections and complements. Commun. Math. Phys. 234, 185–190 (2003) 25. Ruelle, D.: Application of hyperbolic dynamics to physics: some problems and conjectures. Bull. Amer. Math. Soc. 41, 275–278 (2004) 26. Ruelle, D.: Differentiating the a.c.i.m. of an interval map with respect to f . Commun. Math. Phys. 258, 445–453 (2005) 27. Ruelle, D.: Work in progress (personal communication, 2006) 28. Rychlik, M., Sorets, E.: Regularity and other properties of absolutely continuous invariant measures for the quadratic family. Commun. Math. Phys. 150, 217–236 (1992) 29. Smania, D.: On the hyperbolicity of the period-doubling fixed point. Trans. Amer. Math. Soc. 358, 1827–1846 (2006) 30. Tsujii, M.: On continuity of Bowen–Ruelle–Sinai measures in families of one dimensional maps. Commun. Math. Phys. 177, 1–11 (1996) 31. Young, L.-S.: What are SRB measures, and which dynamical systems have them?. J. Stat. Phys. 108, 733–754 (2002) Communicated by G. Gallavotti

Commun. Math. Phys. 275, 861–872 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0319-y

Communications in

Mathematical Physics

The Beale-Kato-Majda Criterion for the 3D Magneto-Hydrodynamics Equations Qionglei Chen1 , Changxing Miao1 , Zhifei Zhang2 1 Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, P. R. China.

E-mail: [email protected]; [email protected]

2 School of Mathematical Science, Peking University, Beijing 100871, P. R. China.

E-mail: [email protected] Received: 18 January 2007 / Accepted: 15 March 2007 Published online: 15 August 2007 – © Springer-Verlag 2007

Abstract: We study the blow-up criterion of smooth solutions to the 3D MHD equations. By means of the Littlewood-Paley decomposition, we prove a Beale-Kato-Majda type blow-up criterion of smooth solutions via the vorticity of velocity only, namely
T sup 
j (∇ × u)∞ dt, j∈Z 0

where
j is the frequency localization operator in the Littlewood-Paley decomposition. 1. Introduction We consider the 3D incompressible magneto-hydrodynamics (MHD) equations ⎧ ∂u 1 2 ⎪ ⎪ ∂t − ν
u + u · ∇u = −∇ p − 2 ∇b + b · ∇b, ⎪ ⎪ ⎨ ∂b − η
b + u · ∇b = b · ∇u, ∂t (MHD) ⎪ ∇ · u = ∇ · b = 0, ⎪ ⎪ ⎪ ⎩ u(0, x) = u 0 (x), b(0, x) = b0 (x).

(1.1)

Here u, b denote the flow velocity vector and the magnetic field vector respectively, p is a scalar pressure, ν > 0 is the kinematic viscosity and η > 0 is the magnetic diffusivity, while u 0 and b0 are the given initial velocity and initial magnetic field respectively, with ∇ · u 0 = ∇ · b0 = 0. If ν = η = 0, (1.1) is called the ideal MHD equations. Using the standard energy method, it can be easily proved that for given initial data (u 0 , b0 ) ∈ H s (R3 ) with s > 21 , there exists a positive time T = T ((u 0 , b0 ) H s ) and a unique smooth solution (u(t, x), b(t, x)) on [0, T ) to the MHD equations satisfying (u, b) ∈ C([0, T ); H s ) ∩ C 1 ((0, T ); H s ) ∩ C((0, T ); H s+2 ).

862

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Whether smooth solutions of (1.1) on [0, T ) will lead to a singularity at t = T is an outstanding open problem, see Sermange and Temam [17]. Caflisch, Klapper and Steele [2] extended the well-known result of Beale-Kato-Majda [1] for the incompressible Euler equations to the 3D ideal MHD equations. Precisely, under the condition:
T (∇ × u(t)∞ + ∇ × b(t)∞ )dt < ∞, (1.2) 0

then smooth solutions (u, b) can be extended beyond t = T . Recently, there are some researches which have refined (1.2) such as
T (∇ × u(t) B˙ 0 + ∇ × b(t) B˙ 0 )dt < ∞, (see [24]), ∞,∞

0




T

∞,∞

(
j (∇ × u)(t)∞ + 
j (∇ × b)(t)∞ )dt = δ < M, (see [3])

lim sup

ε→0 j∈Z T −ε

for some positive constant M, and
j is a frequency localization on |ξ | ≈ 2 j . These results can be easily extended to (1.1) with ν, η > 0. Wu [22] also extended some Serrin type criteria for the Navier-Stokes equations to the MHD equations. Many relevant results can be found in [20, 21] and references therein. However, some numerical experiments [7, 16] seem to indicate that the velocity field plays a more important role than the magnetic field in the regularity theory of solutions to the MHD equations. Recently, inspired by the pioneering work of Constantin and Fefferman [6] where the regularity condition of the direction of vorticity was used to describe the regularity criterion to the Navier-Stokes equations, He and Xin [8] extended it to the MHD equations, but did not impose any condition on the magnetic field b which was consistent with the result of numerical experiments. Precisely, they showed that the solution remains smooth on [0, T ] if the vorticity of the velocity w = ∇ × u satisfies the following condition: 1

|w(x + y, t) − w(x, t)| ≤ K |w(x + y, t)||y| 2 if |y| ≤ ρ |w(x + y, t)| ≥ ,

(1.3)

for t ∈ [0, T ] and three positive constants K , ρ, . In addition, He and Xin [8] and Zhou [25] obtained some integrability condition on the magnitude of the velocity u alone, or the gradient of the velocity ∇u alone to characterize the regularity criterion to the MHD equations, i.e.
T 2 3 q + ≤ 1, 3 < p ≤ ∞ (1.4) u(t) p dt < ∞, q p 0 or


0

T

q

∇u(t) p dt < ∞,

2 3 + ≤ 2, q p

3 < p ≤ ∞. 2

(1.5)

We intend to obtain a criterion replacing ∇u by the vorticity w in (1.5). In the case p < ∞, using the Biot-Savart law [12] and the bounds of the Riesz transforms [18] on L p (1 < p < ∞), the condition (1.5) can be replaced by
T 2 3 3 q ∇ × u(t) p dt < ∞, + ≤ 2, < p < ∞. (1.6) q p 2 0

Beale-Kato-Majda Criterion for 3D Magneto-Hydrodynamics Equations

863

However, since the lack of continuity of Riesz transforms on L ∞ , their results missed the important marginal case p = ∞ which exactly corresponds to the Beale-Kato-Majda criterion. In the case of the Euler equations, Beale, Kato, and Majda got around this difficulty by using the logarithmic Sobolev inequality: ∇u∞ ≤ C(1 + ∇ × u∞ log(e + u H s )), s > 5/2.

(1.7)

For a refined form of this inequality, one can refer to [11, 14]. In order to make use of (1.7), one needs to estimate the higher order derivatives of the solution (at least in H s , s > 5/2). But, in the case of the MHD equations, it seems difficult to control them by the ∇ × u∞ alone. Therefore, as in [2], if the logarithmic Sobolev inequality (1.7) is used, one can only derive a criterion described by the vorticity of u and b. This difficulty is avoided by the following two observations. On one hand, the H 1 norm of the solution can be used to control any H s norm of the solution, while the H 1 norm of the solution (u, b) can be controlled by ∇u∞ . On the other hand, if we make use of the Littlewood-Paley decomposition to decompose the nonlinear terms into three parts: low frequency, middle frequency and high frequency, and deal with each part by virtue of different estimates, we can substitute ∇u∞ with ∇ × u∞ . It should be pointed out that we do not apply the Littlewood-Paley decomposition to the equation itself as some researches have done before, since if we localize the equation on a dyadic partition, we cannot control the H 1 norm of the solution (u, b) via ∇ × u∞ in the end when summing up every dyadic partition. Finally, we remark that the blow-up criterion we will establish in the framework of mixed time-space Besov spaces may be the most relaxed in some sense as for the incompressible Euler equations [15] and the Ideal MHD equations [3], where the losing estimate for the solution and the logarithmic Sobolev inequality are applied to set up the blow-up criterion, but in this paper, if we follow their method, as we mentioned above, we cannot characterize the blow-up of smooth solutions by ∇ × u only. Now we state our result as follows. Theorem 1.1. Let (u 0 , b0 ) ∈ H s , s > 21 with ∇ · u 0 = ∇ · b0 = 0. Suppose that (u, b) ∈ C([0, T ); H s ) ∩ C 1 ((0, T ); H s ) ∩ C((0, T ); H s+2 ) is the smooth solution to (1.1). If there exists an absolute constant M > 0 such that if
T lim sup 
j (∇ × u)∞ dt = δ < M, (1.8) ε→0 j∈Z T −ε

then δ = 0, and the solution (u, b) can be extended past time t = T . In other words, if
T lim sup 
j (∇ × u)∞ dt ≥ M, (1.9) ε→0 j∈Z T −ε

then the solution blows up at t = T . Here
j is a frequency localization on |ξ | ≈ 2 j , see Sect. 2 for its definition. Remark 1.1. For the Navier-Stokes equations (with b = 0 in (1.1)), Kozono, Taniuchi [10], and Kozono, Ogawa, Taniuchi [11] refined the Beale-Kato-Majda criterion to
T
T ∇ × u(t) B M O dt < ∞ and ∇ × u(t) B˙ 0 dt < ∞, 0

0

∞,∞

864

Q. Chen, C. Miao, Z. Zhang

0 respectively. Here B˙ ∞,∞ stands for the homogenous Besov spaces, see Sect. 2 for the definition. The condition (1.8) is weaker than all the above mentioned conditions. Hence, this also improves the results of [10, 11]. For the further explanation of (1.8), one can refer to the remarks after Theorem 1 in [15].

Remark 1.2. Very recently, Wu [23] use the energy estimate combined with the Bony paraproduct technique to derive many interesting regularity criteria for the generalized MHD equations. In the marginal case, the regularity criterion obtained there can be expressed as
T
T 1+δ u(t) B 1 dt < ∞ or u(t) B∞,∞ 1+ dt < ∞, 0

∞,∞

0

stands for the inhomogenous Besov spaces, see Sect. 2 for some δ, > 0. Here for the definition. But the velocity u cannot be replaced by its vorticity, since the Riesz s transformation is not bounded in B∞,∞ . B sp,q

Remark 1.3. For the Ideal MHD equations, whether a similar result holds is still open, since the viscous term plays an important role in our proof. Notation. Throughout the paper, C stands for a “harmless” constant, and changes from line to line;  ·  p denotes the norm of the Lebesgue space L p . 2. Preliminaries Let S(R3 ) be the Schwartz class of rapidly decreasing functions. Given f ∈ S(R3 ), its Fourier transform F f = fˆ is defined by
− 23 ˆ e−i x·ξ f (x)d x. f (ξ ) = (2π ) R3

Now let us recall the Littlewood-Paley decomposition (see [4, 19]). Choose two nonnegative radial functions χ , ϕ ∈ S(R3 ) supported respectively in B = {ξ ∈ R3 , |ξ | ≤ 43 } and C = {ξ ∈ R3 , 43 ≤ |ξ | ≤ 83 } such that  χ (ξ ) + ϕ(2− j ξ ) = 1, ξ ∈ R3 , 

j≥0

ϕ(2− j ξ ) = 1, ξ ∈ R3 \{0}.

j∈Z

Let h = F −1 ϕ and h˜ = F −1 χ ; the frequency localization operator is defined by
−j 3j h(2 j y) f (x − y)dy,
j f = ϕ(2 D) f = 2 R3
˜ j y) f (x − y)dy. S j f = χ (2− j D) f = 23 j h(2 R3

Formally,
j is a frequency projection to the annulus {|ξ | ≈ 2 j }, and S j is a frequency projection to the ball {|ξ |
2 j }. Observe that
j = S j − S j−1 . Also, if f is an L 2 function then S j f → 0 in L 2 as j → −∞ and S j f → f in L 2 as j → +∞ (this is

Beale-Kato-Majda Criterion for 3D Magneto-Hydrodynamics Equations

865

an easy consequence of Parseval’s theorem). By telescoping the series, we thus have the homogeneous Littlewood-Paley decomposition +∞ 

f =


j f,

(2.1)

j=−∞

for all f ∈ L 2 , where the summation is in the L 2 sense. Let s ∈ R, 1 ≤ p, q ≤ ∞, the homogenous Besov space B˙ sp,q is defined by B˙ sp,q = { f ∈ Z (R3 );  f  B˙ s

p,q

< ∞},

where ⎛  f  B˙ s

p,q

∞ 

=⎝

⎞1 q

q 2 jsq 
j f  p ⎠

,

j=−∞

(usual modification if q = ∞), and Z (R3 ) can be identified by the quotient space of S /P with the polynomials space P. The inhomogenous Besov space B sp,q is defined by B sp,q = { f ∈ S (R3 );  f  B sp,q = S0 f  p + {2 js 
j f  p } j≥0 q < ∞}. 1

We now denote the operator (I −
) 2 by  which is defined by f (ξ ) = (1 + |ξ |2 ) 2 fˆ(ξ ).  1

More generally, s f for s ∈ R can be identified with the Fourier Transform s f (ξ ) = (1 + |ξ |2 ) 2 fˆ(ξ ).   s

For s ∈ R, we define
 f  H s   f  L 2  s

(1 + |ξ | ) | fˆ(ξ )|2 dξ

1

2 s

R3

2

,

and the Sobolev space H s is denoted by H s  { f ∈ S (R3 );  f  H s < ∞}. The usual Sobolev space H s, p is endowed with the norm  f  H s, p  s f  L p . We can refer to [19] for more details. Lemma 2.1. Let k ∈ N. There exist constants C independent of f , j such that for all 1 ≤ p ≤ q ≤ ∞, sup ∂ α
j f q ≤ C2

jk+3 j ( 1p − q1 )

|α|=k

Rk
j f q ≤ C2

3 j ( 1p − q1 )


j f  p ,


j f  p .

Here Rk (k = 1, 2, 3) is the Riesz transform in R3 .

(2.2) (2.3)

866

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The proof of this lemma can be found in [4, 13]. Remark 2.1. Suppose that the vector function f is divergence-free, and set g = ∇ × f . Then there exist constants C independent of f such that ∇ f  p ≤ Cg p , ∀ 1 < p < ∞.

(2.4)

If the frequency of f is restricted to some annulus {|ξ | ≈ 2 j }, then there holds ∇ f  p ≤ Cg p , ∀ 1 ≤ p ≤ ∞.

(2.5)

Indeed, the inequality (2.4) can be derived from the Biot-Savart law [12] and the bounds of the Riesz transforms [18] on L p (1 < p < ∞), while the inequality (2.5) can be deduced from the Biot-Savart law and (2.3). Lemma 2.2 (Commutator estimate). Let 1 < p < ∞, s > 0. Assume that f, g ∈ H s, p , then there exists a constant C independent of f , g such that s ( f g) − f s g L p ≤ C(∇ f  L p1 g H s−1, p2 +  f  H s, p3 g L p4 )

(2.6)

with p2 , p3 ∈ (1, +∞) such that 1 1 1 1 1 + = + . = p p1 p2 p3 p4 This lemma is well-known and for a proof, see [9]. 3. Proof of Theorem 1.1 We will divide the proof of Theorem 1.1 into two steps. Step 1. H 1 estimates. In this step we will show there exists ε > 0,     sup u(t) H 1 + b(t) H 1 ≤ C u(T − ε) H 1 + b(T − ε) H 1 + e . t∈[T −ε,T )

(3.1)

Let w(t, x) = ∇ × u(t, x) and J (t, x) = ∇ × b(t, x). Taking the curl on both sides of (1.1), it can be written as  ∂w ∂t − ν
w + (u · ∇)w − (w · ∇)u − (b · ∇)J + (J · ∇)b = 0, (3.2) ∂J − η
J + (u · ∇)J − (J · ∇)u − (b · ∇)w + (w · ∇)b = 2T (b, u) ∂t with

⎞ ∂2 b · ∂3 u − ∂3 b · ∂2 u T (b, u) = ⎝∂3 b · ∂1 u − ∂1 b · ∂3 u ⎠ . ∂1 b · ∂2 u − ∂2 b · ∂1 u ⎛

Multiplying the first equation of (3.2) by w, the second one of (3.2) by J , then adding the resulting equations yields that
1 d (w(t)22 + J (t)22 ) + ν∇w(t)22 + η∇ J (t)22 = (w · ∇)u · wd x 2 dt R3


+ (J · ∇)u · J d x − T (b, u) · J d x ((J · ∇)b · w + (w · ∇)b · J ) d x + 2 R3

 I + I I + I I I + 2 I V,

R3

R3

(3.3)

Beale-Kato-Majda Criterion for 3D Magneto-Hydrodynamics Equations

867

where we have used the facts

(u · ∇)w · wd x = (u · ∇)J · J d x = 0 R3

and

R3


R3

((b · ∇)J · w + (b · ∇)w · J ) d x = 0,

which can be deduced from the divu = divb = 0 and integrating by parts. In what follows, we will deal with each term on the right-hand side of (3.3) separately below. Let us begin with estimating the term I . Using the Littlewood-Paley decomposition (2.1), we have    
jw =
jw +
jw +
j w, (3.4) w= j∈Z

−N ≤ j≤N

j<−N

j>N

where N is a positive integer to be determined later. Plugging (3.4) into I produces that I =


R3

j<−N

+

(w · ∇)u ·
j wd x +




R3

j>N

N


R3

j=−N

(w · ∇)u ·
j wd x

(w · ∇)u ·
j wd x  I1 + I2 + I3 .

Using the Hölder inequality, (2.4) and (2.2) to obtain that |I1 | ≤ −w2 ∇u2

 j<−N



|I2 | ≤ w2 ∇u2




j w∞ ≤ Cw22

3

3

2 2 j 
j w2 ≤ C2− 2 N w32 ,

j<−N


j w∞ ≤

Cw22

−N ≤ j≤N

(3.5)




j w∞ .

(3.6)

−N ≤ j≤N

From the Hölder inequality, (2.4), (2.2) and the Gagliardo-Nirenberg inequality, it follows that   j |I3 | ≤ w3 ∇u3 
j w3 ≤ Cw23 2 2 
j w2 j>N

⎛ ≤

Cw23 ⎝



j>N

⎞1 ⎛ 2

2

−j⎠

⎝

j>N

By summing up (3.5)–(3.7), we get ⎛ |I | ≤ C ⎝2

− 23 N

w32 + w22



⎞1 2

2

2j


j w22 ⎠

N

≤ C2− 2 w2 ∇w22 . (3.7)

j>N

 −N ≤ j≤N

⎞ 
j w∞ + 2

− N2

w2 ∇w22 ⎠ .

(3.8)

868

Q. Chen, C. Miao, Z. Zhang

Using the Littlewood-Paley decomposition (2.1) to ∇u, I I can be written as 


II =

j<−N

+

R3

(J · ∇)
j u · J d x +


j>N

R3

N
 j=−N

R3

(J · ∇)
j u · J d x

(J · ∇)
j u · J d x.

Then the Hölder inequality, (2.2), (2.5) and the Gagliardo-Nirenberg inequality allow us to show that ⎛ ⎞  3 N |I I | ≤ C ⎝2− 2 N w2 J 22 + J 22 
j w∞ + 2− 2 J 2 ∇w2 ∇ J 2 ⎠ . −N ≤ j≤N

(3.9) Similarly, using the Littlewood-Paley decomposition (2.1), I I I and I V can be written respectively as III =


j<−N

+

R3




R3

j>N

IV =


j<−N

+

R3


j>N

R3

N
   J · ∇b + (∇b)T ·
j wd x +

  J · ∇b + (∇b)T ·
j wd x, T (b,
j u) · J d x +

N
 j=−N

R3

j=−N

R3

  J · ∇b + (∇b)T ·
j wd x

T (b,
j u) · J d x

T (b,
j u) · J d x.

Then exactly as in the derivation of (3.8), (3.9), we can deduce that   3 |I I I | + 2|I V | ≤ C 2− 2 N w2 J 22 + J 22 
j w∞ −N ≤ j≤N

+2

− N2

 J 2 ∇w2 ∇ J 2 .

(3.10)

Combining (3.8)–(3.10) with (3.3), the Young inequality yields that for t ∈ [0, T ),  d  w(t)22 + J (t)22 + 2ν∇w(t)22 + 2η∇ J (t)22 dt ⎛      3 ≤ C ⎝2− 2 N w(t)32 + J (t)32 + 
j w(t)∞ w(t)22 + J (t)22 −N ≤ j≤N

+2

− N2

⎞   (w(t)2 + J (t)2 ) ∇w(t)22 + ∇ J (t)22 ⎠ .

(3.11)

Beale-Kato-Majda Criterion for 3D Magneto-Hydrodynamics Equations

869 N

Now let us choose a fixed positive integer N such that C2− 2 (w(t)2 + J (t)2 ) ≤ min(ν, η), i.e. 

 2 C N≥ (3.12) log+ (w(t)2 + J (t)2 ) + 1, log 2 min(ν, η) where log+ x = log(e + x). Thus (3.11) and (3.12) imply that for t ∈ [0, T ),  d  w(t)22 + J (t)22 + ν∇w(t)22 + η∇ J (t)22 dt N    ≤C 
j w(t)∞ w(t)22 + J (t)22 + C,

(3.13)

j=−N

which together with the Gronwall inequality gives that for t ∈ [0, T ), ⎛ ⎞ N
t  √ w(t)2 + J (t)2 ≤ exp ⎝C 
j w(t )∞ dt ⎠ ( Ct + w(0)2 + J (0)2 ). j=−N

0

Recalling the choice of N in (3.12), it follows from the above estimate that  
t +



w(t)2 + J (t)2 ≤ exp C log (w(t)2 + J (t)2 ) sup 
j w(t )∞ dt √ ×( Ct + w(0)2 + J (0)2 ).

j∈Z 0

For simplicity, let E(t)  w(t)2 + J (t)2 , the above inequality implies that    
T √ +



sup E(t) ≤ exp C log 
j w(t )∞ dt ( C T + E(0)). sup E(t) sup [0,T )

[0,T )

j∈Z 0

(3.14) We point out that the inequality (3.14) still holds if the time interval is replaced by [T − ε, T ). It follows from (3.14) that  
T +



sup E(t) ≤ exp log sup E(t) sup 
j w(t )∞ dt t∈[T −ε,T )

Defining Z (T )  log

t∈[T −ε,T )



j∈Z T −ε

√ ×( Cε + w(T − ε)2 + J (T − ε)2 ).  sup E(t) + e , the above estimate means that

t∈[T −ε,T )


√  Z (T ) ≤ log Cε + E(T − ε) + e + C Z (T ) sup

T

j∈Z T −ε


j w(t )∞ dt . (3.15)

1 If we choose M = 2C in Theorem 1.1, the condition (1.8) ensures that there exists a small positive ε0 such that
T 1 ∀ ε ∈ (0, ε0 ). 
j w(t )∞ dt ≤ , C sup 2 j∈Z T −ε

870

Q. Chen, C. Miao, Z. Zhang

Then the inequality (3.15) implies that Z (T ) ≤ 2 log (E(T − ε) + e) ,

∀ ε ∈ (0, ε0 ).

(3.16)

On the other hand, it is easy to prove that the solution (u, b) satisfies the energy inequality
t  2 2 ν∇u(t )22 + η∇b(t )22 dt ≤ u(s)22 + b(s)22 , u(t)2 + b(t)2 + 2 s

which together with (3.16), (2.4) implies (3.1). Step 2. H s (s > 1) estimates. For completeness, we will show how to deduce H s estimates from H 1 estimates. Taking the operation s on both sides of (1.1), multiplying (s u, s b) to the resulting equation, and integrating over R3 , we get 1 d (s u(t)22 + s b(t)22 ) + ν∇s u(t)22 + η∇s b(t)22 2 dt


=− s (u · ∇u)s ud x + s (b · ∇b)s ud x − s (u · ∇b)s bd x 3 3 3 R R
R s s +  (b · ∇u) bd x. (3.17) R3

Noting that divu = divb = 0 and integrating by parts, we rewrite (3.17) as 1 d (s u(t)22 + s b(t)22 ) + ν∇s u(t)22 + η∇s b(t)22 2 dt

=− (s (u · ∇u) − u · s ∇u)s ud x − (s (u · ∇b) − u · s ∇b)s bd x 3 3 R
R s s s s + ( (b · ∇b) − b ·  ∇b) u + ( (b · ∇u) − b · s ∇u)s bd x R3

 1 + 2 + 3 .

(3.18)

For the term 1 , it follows from (2.6), the Hölder inequality, the Gagliardo-Nirenberg inequality and the Young inequality that |1 | ≤ C(∇u2 ∇u H s−1,4 + u H s,4 ∇u2 )u H s,4 1 3 ν ≤ C∇u2 u H2 s ∇u H2 s ≤ C∇u42 u2H s + ∇u2H s . 2

(3.19)

The other terms can be treated in the same way: |2 | + |3 | ≤ C(∇u2 ∇b H s−1,4 + u H s,4 ∇b2 )b H s,4 +C(∇b2 ∇b H s−1,4 + b H s,4 ∇b2 )u H s,4 +C(∇b2 ∇u H s−1,4 + b H s,4 ∇u2 )b H s,4 ν η ≤ C(∇u42 + ∇b42 )(u2H s + b2H s ) + ∇u2H s + ∇b2H s . (3.20) 2 2 By summing up (3.19) and (3.20) with (3.18), we get d (u(t)2H s + b(t)2H s ) + ν∇u(t)2H s + η∇b(t)2H s dt ≤ C(∇u(t)42 + ∇b(t)42 )(u(t)2H s + b(t)2H s ).

Beale-Kato-Majda Criterion for 3D Magneto-Hydrodynamics Equations

Then the Gronwall inequality yields that
t u(t)2H s + b(t)2H s + (ν∇u(t )2H s + η∇b(t )2H s )dt

0

2 2



4 ≤ C(u(0) H s + b(0) H s ) exp t sup (u(t ), b(t )) H 1 . t ∈[0,t)

871

(3.21)

Hence we have the H s regularity for the solution at t = T and the solution can be continued after t = T . This completes the proof of Theorem 1.1. Acknowledgements. Q. Chen and C. Miao were partially supported by the NSF of China (No.10571016) and The Institute of Mathematical Sciences, The Chinese University of Hong Kong. Z. Zhang is supported by the the National Natural Science Foundation of China (No.10601002). The authors wish to thank both Prof. Zhouping Xin for stimulating discussion about this problem, and the referee for his valuable advice.

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